Nucleus Accumbens and Its Role in Reward and Emotional Circuitry: A Potential Hot Mess in Substance Use and Emotional Disorders

Mani Pavuluri; Kelley Volpe; Alexander Yuen; Mani Pavuluri; Kelley Volpe; Alexander Yuen

doi:10.3934/Neuroscience.2017.1.52

AIMS Neuroscience

2017, Volume 4, Issue 1: 52-70. doi: 10.3934/Neuroscience.2017.1.52

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Review

Nucleus Accumbens and Its Role in Reward and Emotional Circuitry: A Potential Hot Mess in Substance Use and Emotional Disorders

Department of Psychiatry, University of Illinois at Chicago, USA

Received: 02 January 2017 Accepted: 10 April 2017 Published: 18 April 2017

Nucleus accumbens (NAc) is a key region in the brain that is integral to both the reward and the emotional systems. The aim of the current paper is to synthesize the basic and the clinical neuroscience discoveries relevant to the NAc for the purpose of two-way translation. Selected literature on the structure and the functionality of the NAc is reviewed across animal and human studies. Dopamine, gamma-aminobutyric acid (GABA) and glutamate are the three key neurotransmitters that modulate the reward function and the motor activity. Dissociative roles of the core and the shell of the NAc include getting to the reward and staying on task with discretion, respectively. NAc shows decreased activation to reward in the individuals with major depressive disorder and the bipolar disorder, relative to that healthy controls (HC). The “difficult to please” or insatiability in response to reward in the emotional disorders may possibly be explained by such a neural pattern. Furthermore, it is likely that the increased amygdala activity reported in mood disorders could be accentuating the “wanting” of the reward by the virtue of its connections with the NAc, explaining the potential “hot mess”. In contrast, the NAc shows increased reward response in substance use disorders, relative to HC, in response to reward and emotional tasks. Accurate characterization of the NAc and its functionality in the human imaging studies of mood and substance use has important treatment implications.

Keywords:

Citation: Mani Pavuluri, Kelley Volpe, Alexander Yuen. Nucleus Accumbens and Its Role in Reward and Emotional Circuitry: A Potential Hot Mess in Substance Use and Emotional Disorders[J]. AIMS Neuroscience, 2017, 4(1): 52-70. doi: 10.3934/Neuroscience.2017.1.52

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Abstract

1. Introduction

It is well-established that fractional derivatives offer greater flexibility and accuracy compared to integer-order derivatives, particularly when modeling nonclassical engineering and scientific phenomena. Examples include fractional-order recurrent neural networks and various dynamical topics, as discussed in ^[1], a new extension of the fractality concept introduced in financial mathematics by Laskin et al. ^[2], frequency dependance of cell rheological behavior studied in ^[3], and the HIV/AIDS model and the present scenario of COVID-19 discussed in ^[4,5]. Fractional derivatives enable accurate modeling of systems requiring accurate modeling of damping. Numerous examples demonstrate the superiority of fractional calculus over integer-order calculus in such applications. Examples illustrating the application of fractional derivatives include: modeling non-linear earthquake fluctuations and developing fluid dynamic traffic models that mitigate problems caused by high traffic flow ^[6,7]. Numerous mathematical equations are increasingly being formulated as fractional partial differential equations, finding applications in diverse fields such as mathematical biology, transport, finance, visco-elasticity, particle chemistry, and population dynamics ^[8,9,10]. Wave breaking models have two most important equations, which are Kortewegde Varies (KdV) and the Banjamin-Bona-Mahony-Burger (BBM-Burger) equation ^[11]. In mathematics, KdV is a model of wave and shallow water surface. These equations are not applicable to certain long-wave physical systems, which led to the introduction of the BBM-Burger equations. The BBM-Burger equations serve as a refinement of the KdV equations. Specifically, the integer-order nonlinear BBM-Burger equation, which accounts for both dissipative and dispersive effects, is presented in ^[12].

The BBM-Burger equations have numerous applications across various fields, including cracked rock, acoustic-gravity waves in fluids, acoustic waves in a harmonic crystal, and thermodynamics ^[13]. Several methods have been proposed to solve fractional partial-integro differential equations and fractional differential equations, including finite element method with cubic B-spline (CBS) functions for numerical solutions of Burger and Fisher equations ^[14], CBS collocation method for numerical solutions of nonlinear inhomogeneous time-fractional (TF) Burger-Huxley equations ^[15], CBS functions for solving TF diffusion equations involving Caputo-Fabrizio fractional derivatives ^[16], and Atangana-Baleanu fractional derivative (ABFD) with CBS functions for solving TF Burger's equations ^[17]. Many fractional equations have been solved by using the quartic B-spline collocation method ^[18], an extended CBS technique ^[19], shifted Legendre polynomials involved in orthogonal basis function method ^[20,21], homotopy analysis method ^[22], and the fifth-kind Chebyshev polynomial collocation method ^[23]. For discussing dynamic physical structures, a nonhomogeneous fractional BBM-Burger model with a nonlocal viscous term is proposed ^[24],

$\begin{eqnarray} v_{s} +v_\mathfrak{u} -{\phi} v_{\mathfrak{u} \mathfrak{u} s} +\rho^{\psi} D_s ^{\psi} v + \tau v v_\mathfrak{u} -\chi v_{\mathfrak{u} \mathfrak{u}} = g(\mathfrak{u},s),\; \; s\in [0,S],\; \mathfrak{u} \in [\mathfrak{a},\mathfrak{b}],\; 0 < \psi \leq 1, \end{eqnarray}$

(1.1)

with initial condition (IC) and boundary conditions (BCs)

$\begin{equation} \begin{cases} v(\mathfrak{u},0) = v_0 (\mathfrak{u}),\\ v(\mathfrak{a},s) = f_1 (s),\; v(\mathfrak{b},s) = f_2 (s), \end{cases} \end{equation}$

(1.2)

where $\tau$ , ${\phi}$ , $\rho$ and $\chi$ are positive parameters. Due to the complexity of the nonhomogeneous BBM-Burger equation, finding an analytical solution is often challenging. Consequently, recent studies have focused on developing numerical methods to solve this equation, exploring alternative approaches to approximate its solution. Linearized difference schemes for the BBM-Burger equation incorporating a fractional nonlocal viscous term have been investigated in ^[25]. The homotopy analysis method was employed to investigate the BBM-Burger equation by Fakhari ^[26]. Song et al. ^[27] obtained an approximate solution for the fractional BBM-Burgers equation using the homotopy analysis method. Kumar et al. ^[28] solved non-integer order BBM-Burger equations using the novel homotopy analysis transform method. Shakeel et al. ^[29] obtained the exact solution of fractional BBM-Burger equation by using the $(G'/G)$ expansion method. The generalized Atangana-Baleanu BBM-Burgers equation involving dissipative term has been solved by using the modified sub-equation method and new $G'/(b G'+G+a)$ expansion schemes ^[30]. The residual power series technique for the series solution of the TF BBM-Burger equation has been used by Zhang et al. ^[31]. Salih et al. ^[32] applied cubic trigonometric B-spline functions to solve the BBM-Burger equation. A hybrid numerical technique has been used to solve the BBM-Burger equation in ^[33]. Karakoc et al. ^[34] obtained the exact traveling wave solution and computational solutions of the BBM-Burger equation by using the modified Kudryashov method and septic B-spline finite element method, respectively. Omrani ^[35] employed the Crank-Nicolson finite difference method to obtain a numerical solution for the BBM-Burger equation. Karakoc et al. ^[36] applied the CBS finite element method, in conjunction with a lumped Galerkin scheme, to spatially approximate the solitary-wave solution of the nonlinear BBM-Burger equation. Majeed et al. ^[37] employed the Caputo fractional derivative and CBS functions to solve the nonhomogeneous TF BBM-Burger equation. Lin et al. suggested an iterative RBF-based approach in order to solve the nonlinear TF BBM-Burger equation in any domain ^[38]. Mohsin Kamran et al. solved the BBM-Burger equation numerically using Caputo derivative and B-spline basis functions in ^[39]. Hamad Salih ^[40] solved the one-dimensional nonlinear BBM-Burger problem using the novel quartic trigonometric B-spline method based on finite difference. Lalit Mohan and Amit prakash provided a very effective method for analyzing the diffusion wave equation and the fractional BBM-Burger equation in ^[41]. Vineesh Kumar used the ansatz method and Adomian decomposition method on the BBM-Burger equation in ^[42]. Atallah El-shenawy et al. ^[43] uses the collocation approach, which is based on the cubic trigonometric B-Spline methodology, to numerically investigate Troesch's problem. Atallah El-shenawy et al. ^[44] developed a numerical method based on B-spline to solve the time-dependent Emden-Fow-ler-type equations.

The given study is inspired by recent developments in the investigation of computational solution of the nonhomogeneous time-fractional BBM-Burger equation. The objective of this study is to apply the $\theta$ -weighted scheme with CBS functions to the nonhomogeneous BBM-Burger equation for obtaining the numerical solution. In the past, many researchers have used B-spline techniques to solve non-homogeneous BBM-Burger equations. However, no one has ever used the CBS method together with the ABFD on the TF term involved in the BBM-Burger equation. The usage of non-singular kernel operator in B-spline methods is novel. Convergence and stability of the proposed problem are analyzed. By presenting few numerical examples, the efficiency and applicability of the proposed scheme is also analyzed. By contrasting analytical and numerical solutions, one can find that the present approach has provided the more effective results. The present scheme is novel for the approximate solution of the nonhomogeneous time-fractional BBM-Burger equation involving ABFD, and, as far as authors are aware, it has never been employed for this purpose before.

This paper has been presented in this way: Basic definitions utilized in this work are given in Section 2. This section contains an important definition of ABFD, which is the core of our work. Section 3 contains the description of the proposed method. Section 4 gives the answer of "how to get the values of the initial vector". Stability and convergence of this problem are given in Sections 5 and 6, respectively. Section 7 contains numerical examples and their discussions. Concluding remarks are presented in Section 8.

2. Basic definitions

The fundamental definitions including the ABFD with several properties ^[45] are a part of this section.

Definition 1. The ABFD of order $\psi$ and $v\in H'(0, 1)$ is defined as ^[46]:

$\qquad \mathcal{D}^{\psi}_{s} v(\mathfrak{u},s) = \frac{J(\psi)}{1-\psi}\int_{0}^{s}\frac{\partial^{n}}{\partial c^{n}}v^{n}(\mathfrak{u},c) E_{\psi}\bigg(\frac{-\psi}{n-\psi} (s-c)^{\psi}\bigg)dc, \qquad \mathfrak{n}-1 < \psi \leq \mathfrak{n},$

where $E_{\psi}(\nu)$ is the Mittag-Leffler function given as:

$\begin{align} \qquad E_{\psi}(\nu) = \sum\limits_{\kappa = 0}^{\infty} \frac{\nu^{\kappa}}{\Gamma (\psi \kappa+1)}, \end{align}$

and the standardization function $J(\psi)$ can be described as:

$\begin{align} \qquad J(\psi) = 1-\psi +\frac{\psi}{\Gamma(\psi)}. \end{align}$

For $0 < \psi \leq 1$ , it becomes

$\begin{align} \qquad \qquad \mathcal{D}^{\psi}_{s} v(\mathfrak{u},s) = \frac{J(\psi)}{1-\psi}\int_{0}^{s}\frac{\partial}{\partial c}v'(\mathfrak{u},c) E_{\psi}\bigg(\frac{-\psi}{1-\psi} (s-c)^{\psi}\bigg)dc,\; 0 < \psi \leq 1. \end{align}$

(2.1)

Definition 2. If $\hat{q}\in L^{2} [\mathfrak{a}, \mathfrak{b}]$ , Parseval's identity is given as ^[47]:

$\sum\limits_{\acute{n} = -\infty}^{\infty}|\tilde{q}(\acute{n})|^{2} = \int_{\mathfrak{a}}^{\mathfrak{b}} |\hat{q}(r)|^{2} dr,$

where $\tilde{q}(\acute{n}) = \int_{\mathfrak{a}}^{\mathfrak{b}} \hat{q}(r) e^{2 \pi i \acute{n} r}$ is the Fourier transform for every integer $\acute{n}$ .

2.1. Basis functions

The spatial domain $[\mathfrak{a}, \mathfrak{b}]$ is partitioned as $\mathfrak{a} = \mathfrak{u}_{0} < \mathfrak{u}_{1} < \cdots < \mathfrak{u}_{\hat{N}} = \mathfrak{b}$ , where $\mathfrak{u}_{j} = \mathfrak{u}_{0}+j h, \; j = 0(1)\hat{N}$ . Domain is divided into $\hat{N}$ equal subintervals of length $h = \frac{\mathfrak{b}-\mathfrak{a}}{\hat{N}}$ . Now, assume that $\Psi(\mathfrak{u}, s)$ is the CBS approximation for $v(\mathfrak{u}, s)$ s.t.

$\begin{align} \qquad \Psi(\mathfrak{u},s) = \sum\limits_{j = -1}^{\hat{N}+1} y_{j}^{\hat{m}}(s)F_{j}(\mathfrak{u}), \end{align}$

(2.2)

where $F_{j}(\mathfrak{u})$ are CBS functions and $y_{j}^{\hat{m}}$ are control points that will be computed at each time interval. The CBS functions are defined as ^[48]:

$\begin{equation} {F_{j}(\mathfrak{u})} = {\frac{1}{6h^3}} \begin{cases} (\mathfrak{u}-\mathfrak{u}_{j-2})^{3}, & \mathfrak{u}\in [\mathfrak{u}_{j-2},\mathfrak{u}_{j-1}), \\h^3+3h^2(\mathfrak{u}-\mathfrak{u}_{j-1})+3h(\mathfrak{u}-\mathfrak{u}_{j-1})^{2}-3(\mathfrak{u}-\mathfrak{u}_{j-1})^{3}, & \mathfrak{u}\in[\mathfrak{u}_{j-1},\mathfrak{u}_j), \\h^3+3h^2(\mathfrak{u}_{j+1}-\mathfrak{u})+3h(\mathfrak{u}_{j+1}-\mathfrak{u})^{2}-3(\mathfrak{u}_{j+1}-\mathfrak{u})^{3}, & \mathfrak{u}\in[\mathfrak{u}_{j},\mathfrak{u}_{j+1}), \\(\mathfrak{u}_{j+2}-\mathfrak{u})^{3}, & \mathfrak{u}\in[\mathfrak{u}_{j+1},\mathfrak{u}_{j+2}),\\ 0, & \text{otherwise}. \end{cases} \end{equation}$

(2.3)

Domain for time $[0, S]$ can break using knots $0 = s_{0} < s_{1} < \cdots < s_{\hat{M}} = S$ in $\hat{M}$ uniform subintervals $[s_{\hat{m}}, s_{\hat{m}+1}]: s_{\hat{m}} = \hat{m}\Delta s, \; \hat{m} = 0, 1, \cdots, \hat{M}$ , where $\Delta s = S/\hat{M}$ implements numerical scheme. The values of $\Psi$ , $\Psi_{\mathfrak{u}}$ , and $\Psi_{\mathfrak{u} \mathfrak{u}}$ at nodal points can be expressed in terms of the parameter $y_{j}$ by combining Eq (2.2) with expression $F_{j}(\mathfrak{u})$ . These values are summarized in the following Table 1:

Table 1. Values of

$(\Psi)_{j}^{\hat{m}}$ ,

$(\Psi_{\mathfrak{u}})_{j}^{\hat{m}}$ , and

$(\Psi_{\mathfrak{u} \mathfrak{u}})_{j}^{\hat{m}}$ at the knots.

	$y_{j-1}$	$y_j$	$y_{j+1}$	Otherwise
$(\Psi)_{j}^{\hat{m}}$	$\frac{1}{6}$	$\frac{4}{6}$	$\frac{1}{6}$	0
$(\Psi_{\mathfrak{u}})_{j}^{\hat{m}}$	$-\frac{1}{2h}$	0	$\frac{1}{2h}$	0
$(\Psi_{\mathfrak{u} \mathfrak{u}})_{j}^{\hat{m}}$	$\frac{1}{h^{2}}$	$-\frac{2}{h^{2}}$	$\frac{1}{h^{2}}$	0

| Show Table

DownLoad: CSV

3. Description of the method

The ABFD used in (1.1) is discretized at $s = s_{\hat{m}+1}$ as ^[49]:

$\begin{align} \frac{\partial^{\psi}}{\partial s^{\psi}}v(\mathfrak{u},s_{\hat{m}+1}) = \frac{J(\psi)}{1-\psi}\sum\limits^{\hat{m}}_{\zeta = 0} l_{\zeta}[v(\mathfrak{u},s_{\hat{m}-\zeta+1})-v(u,s_{\hat{m}-\zeta})]+ \lambda^{\hat{m}+1}_{\Delta s}, \end{align}$

(3.1)

where $l_{\zeta} = (\zeta+1)E_{\zeta+1}-\zeta E_{\zeta}$ and $E_{\zeta} = E_{\psi, 2}[-\frac{\psi}{1-\psi}(\zeta \Delta s)^{\psi}]$ . It is easy to see that

● $l_{\zeta} > 0$ and $l_{0} = E_{1}$ , $\zeta = 1:1: \hat{m}$ ,

● $l_{0} > l_{1} > l_{2} > \cdots > l_{\zeta}$ , $l_{\zeta}\rightarrow 0$ as $\zeta \rightarrow \infty$ ,

● $\sum\limits^{\hat{m}}_{\zeta = 0}(l_{\zeta}-l_{\zeta+1})+l_{\hat{m}+1} = (E_{1}-l_{1})+\sum\limits^{\hat{m}-1}_{\zeta = 1}(l_{\zeta}-l_{\zeta+1})+ l_{\hat{m}} = E_{1}$ .

Moreover, truncation error $\lambda^{\hat{m}+1}_{\Delta s}$ is given by ^[49]:

$\begin{equation} \lambda^{\hat{m}+1}_{\Delta s}\leq \frac{J(\psi)}{1-\psi} \frac{(\Delta s)^{2}}{2}\bigg[\max\limits_{0\leq {s} \leq {s_{\hat{m}}}} \frac{\partial^{2} v (\mathfrak{u},s)}{\partial s^{2}}\bigg]c_{1}, \end{equation}$

$c_{1}$ is constant, and

$\begin{equation} \mid \lambda^{\hat{m}+1}_{\Delta s} \mid \leq \hat{\vartheta} (\Delta s)^{2}, \end{equation}$

(3.2)

where $\hat{\vartheta}$ is a constant. The nonlinear term in Eq (1.1) can be linearized by the generalized formula used in ^[50].

$\begin{equation} (v^{\varrho}v_{\mathfrak{u}})^{\hat{m}+1}_{j} = \varrho v^{\hat{m}+1}_{j}((v)^{\varrho -1}v_{\mathfrak{u}})^{\hat{m}}_{j}+(v_{\mathfrak{u}})^{\hat{m}+1}_{j}(v^{\varrho})^{\hat{m}}_{j}-\varrho(v^{\varrho}v_{\mathfrak{u}})^{\hat{m}}_{j}, \end{equation}$

(3.3)

where $\varrho$ is a positive integer. Applying the $\theta$ -weighted scheme for spatial discretization, forward difference for temporal discretization, and Eq (3.1) for the ABFD to Eq (1.1) yields:

$\begin{eqnarray} \frac{v^{\hat{m}+1}-v^{\hat{m}}}{\Delta s}-{\phi} \frac{v^{\hat{m}+1}_{\mathfrak{u} \mathfrak{u}}-v^{\hat{m}}_{\mathfrak{u} \mathfrak{u}}}{\Delta s}+ \rho^{\psi} \gamma \sum\limits^{\hat{m}}_{\zeta = 0} l_{\zeta}[v^{\hat{m}-\zeta+1}-v^{\hat{m}-\zeta}] + \theta (v_{\mathfrak{u}}^{\hat{m}+1}+ \tau(v v_{\mathfrak{u}})^{\hat{m}+1}-\chi v_{\mathfrak{u} \mathfrak{u}}^{\hat{m}+1})\\ +(1-\theta) (v_{\mathfrak{u}}^{\hat{m}}+\tau(v v_{\mathfrak{u}})^{\hat{m}}-\chi v_{\mathfrak{u} \mathfrak{u}}^{\hat{m}}) = g(\mathfrak{u},s_{\hat{m}+1}), \end{eqnarray}$

(3.4)

where $\gamma = \frac{J(\psi)}{1-\psi}$ and $v^{\hat{m}} = v(\mathfrak{u}, s_{\hat{m}})$ . For $\theta = 0$ , this is an explicit scheme. When $\theta = 0.5$ , it becomes the Crank-Nicolson scheme, and for $\theta = 1$ , it is fully implicit. This problem is solved here for $\theta = 1$ , which yields better results. Substituting Eq (3.3) into (3.4) and simplifying yields:

$\begin{eqnarray} v^{\hat{m}+1}(1+\tau \Delta s {\phi} v^{\hat{m}}_{\mathfrak{u}}+\rho ^{\psi}\Delta s \gamma E_{1})+v^{\hat{m}+1}_{\mathfrak{u}}(\Delta s \theta+\tau \Delta s \theta v^{\hat{m}})+v^{\hat{m}+1}_{\mathfrak{u} \mathfrak{u}}(-{\phi} -\chi \theta \Delta s) \\ = v^{\hat{m}}(1+ \rho^{\psi}\Delta s\gamma E_{1})+v^{\hat{m}+1}_{\mathfrak{u}}(-(1-\theta)\Delta s+(2\theta-1)\tau \Delta s v^{\hat{m}}) +v^{\hat{m}}_{\mathfrak{u} \mathfrak{u}}(-{\phi}+\chi (1-\theta)\Delta s)\\ -\rho^{\psi}\gamma\Delta s\bigg(\sum\limits^{\hat{m}}_{\zeta = 1}l_{\zeta}(v^{\hat{m}-\zeta+1}-v^{\hat{m}-\zeta})\bigg)+\Delta s g(\mathfrak{u},s_{\hat{m}+1}). \end{eqnarray}$

(3.5)

Combining Eqs (2.2) and (3.5) with the data in Table 1 results in the following system of equations:

$\begin{eqnarray} y^{\hat{m}+1}_{j-1}(\kappa_{0})+y^{\hat{m}+1}_{j}(\kappa_{1})+y^{\hat{m}+1}_{j+1}(\kappa_{2}) = y^{\hat{m}}_{j-1}(\varpi_{1})+ y^{\hat{m}}_{j}(\varpi_{2})+y^{\hat{m}}_{j+1}(\varpi_{3}) \\ +\rho^{\psi}\gamma E_{1} \bigg(y^{\hat{m}}_{j-1}(\frac{1}{6})+y^{\hat{m}}_{j}(\frac{4}{6})+y^{\hat{m}}_{j+1}(\frac{1}{6})\bigg) -\rho^{\psi} \gamma \bigg[\sum\limits^{\hat{m}}_{\zeta = 1}l_{\zeta}\bigg((y^{\hat{m}-\zeta+1}_{j-1}-y^{\hat{m}-\zeta}_{j-1})+4(y^{\hat{m}-\zeta+1}_{j} -y^{\hat{m}-\zeta}_{j}) \\ +(y^{\hat{m}-\zeta+1}_{j+1}-y^{\hat{m}-\zeta}_{j+1})\big)/6\bigg]+g(\mathfrak{u},s_{\hat{m}+1}), \end{eqnarray}$

(3.6)

where,

$\begin{align} \kappa_{0}& = \frac{1}{6 \Delta s}+\frac{\tau \theta R_{1}}{12 h}+\frac{\rho^{\psi}\gamma E_{1}}{6}-\frac{\theta}{2h}-\frac{\tau \theta R_{2}}{12h}-\frac{{\phi}}{h^{2}\Delta s}-\frac{\chi \theta}{h^{2}}, \\ \kappa_{1}& = \frac{4}{6 \Delta s}+\frac{\tau \theta R_{1}}{3 h}+\frac{4 \rho^{\psi}\gamma E_{1}}{6}+\frac{2 {\phi}}{h^{2}\Delta s}+\frac{2 \chi \theta}{h^{2}}, \\ \kappa_{2}& = \frac{1}{6 \Delta s}+\frac{\tau \theta R_{1}}{12 h}+\frac{\rho^{\psi}\gamma E_{1}}{6}+\frac{\theta}{2h}+\frac{\tau \theta R_{2}}{12h}-\frac{{\phi}}{h^{2}\Delta s}-\frac{\chi \theta}{h^{2}}, \\ \varpi_{1}& = \frac{1}{6 \Delta s}+\frac{(1-\theta)}{2h}-\frac{\tau (2\theta-1) R_{2}}{12h}-\frac{{\phi}}{h^{2}\Delta s}+\frac{\chi (1-\theta)}{h^{2}}, \\ \varpi_{2}& = \frac{4}{6 \Delta s}+\frac{2 {\phi}}{h^{2}\Delta s}-\frac{2 \chi (1-\theta)}{h^{2}}, \\ \varpi_{3}& = \frac{1}{6 \Delta s}-\frac{(1-\theta)}{2h}+\frac{\tau (2\theta-1) R_{2}}{12h}-\frac{{\phi}}{h^{2}\Delta s}+\frac{\chi (1-\theta)}{h^{2}}, \\ R_{1}& = y^{\hat{m}}_{j+1}-y^{\hat{m}}_{j-1}, \\ R_{2}& = y^{\hat{m}}_{j-1}+4 y^{\hat{m}}_{j}+y^{\hat{m}}_{j+1}. \end{align}$

The summation term from the right side of Eq (3.6) is dropped for $\zeta = 0$ , and this equation is considered for the iteration process for $\zeta \geq 1$ . Here, a system of $\hat{N}+1$ linear equations can be obtained for $\hat{N}+3$ unknown parameters $(y_{-1}, y_{0}, y_{1}, y_{2}, y_{3}, \cdots, y_{\hat{N}+1})^{T}$ . To get $\hat{N}+3$ equations, boundary conditions will be used. Boundary conditions are

$\begin{equation} \begin{cases} (y^{\hat{m}+1}_{-1}+4 y^{\hat{m}+1}_{0}+y^{\hat{m}+1}_{1})/6 = f^{\hat{m}+1}_{1},\\ (y^{\hat{m}+1}_{\hat{N}-1}+4 y^{\hat{m}+1}_{\hat{N}}+y^{\hat{m}+1}_{\hat{N}+1})/6 = f^{\hat{m}+1}_{2}. \end{cases} \end{equation}$

(3.7)

Equation (3.6) in matrix form can be written as:

$\begin{equation} C Y^{\hat{m}+1} = Z Y^{\hat{m}}+ R \bigg(\sum\limits^{\hat{m}-1}_{\zeta = 0}(l_{\zeta}-l_{\zeta+1})Y^{\hat{m}-\zeta} +l_{\hat{m}}Y^{0}\bigg)+B^{\hat{m}+1}, \end{equation}$

(3.8)

where

$\begin{equation*} Y^{\hat{m}+1} = (y^{\hat{m}+1}_{-1},y^{\hat{m}+1}_{0},y^{\hat{m}+1}_{1},\cdots,y^{\hat{m}+1}_{\hat{N}-1},y^{\hat{m}+1}_{\hat{N}} ,y^{\hat{m}+1}_{\hat{N}+1})^{T}, \end{equation*}$

$\begin{equation*} C = \begin{pmatrix} \frac{1}{6} & \frac{4}{6} & \frac{1}{6} & 0 & \cdots & 0 & 0 & 0 \\ \kappa^{\hat{m}+1}_{0} & \kappa^{\hat{m}+1}_{1} & \kappa^{\hat{m}+1}_{2} & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \kappa^{\hat{m}+1}_{0} & \kappa^{\hat{m}+1}_{1} & \kappa^{\hat{m}+1}_{2} \\ 0 & 0 & 0 & 0 & \cdots & \frac{1}{6} & \frac{4}{6} & \frac{1}{6} \end{pmatrix}, \end{equation*}$

$\begin{equation*} Z = \begin{pmatrix} 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ \varpi^{\hat{m}}_{1} & \varpi^{\hat{m}}_{2} & \varpi^{\hat{m}}_{3} & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \varpi^{\hat{m}}_{1} & \varpi^{\hat{m}}_{2} & \varpi^{\hat{m}}_{3} \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \end{pmatrix}, R = \begin{pmatrix} 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ \frac{1}{6} & \frac{4}{6}& \frac{1}{6} & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots &\frac{1}{6} & \frac{4}{6} & \frac{1}{6} \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \end{pmatrix}\\ B = \begin{pmatrix} f_{1}^{\hat{m}+1} \\ g_{0}^{\hat{m}+1} \\ \vdots \\ g_{\hat{N}}^{\hat{m}+1} \\ f_{2}^{\hat{m}+1} \end{pmatrix}. \end{equation*}$

Insert the vector $Y^{\hat{m}+1}$ in (2.2), and the numerical solution at $(\hat{m}+1)^{th}$ time stage can be obtained for $\hat{m} = 0, 1, \cdots, \hat{M}$ .

4. Initial vector

By using the IC, the initial vector $Y^{0} = (y^{0}_{-1}, y^{0}_{0}, y^{0}_{1}, ..., y^{0}_{\hat{N}-1}, y^{0}_{\hat{N}}, y^{0}_{\hat{N}+1})^{T}$ of the given problem can be calculated. Derivatives are approximated at boundary points and the IC is used to know the value of $Y^{0}$ as:

$\begin{align} & \qquad (v_{\mathfrak{u}})^{0}_{j} = v'_{0}(\mathfrak{u}_{j})\; \text{for}\; j = 0 \; \text{and}\; \hat{N}, \\ & \qquad (v_{\mathfrak{u}})^{0}_{j} = v_{0}(\mathfrak{u}_{j})\; \text{for}\; j = 0,1,2,\cdots,\hat{N}. \end{align}$

This system gives a matrix of dimension $(\hat{N}+3) \times (\hat{N}+3)$ ,

$\begin{align} P Y^{0} = Z_{0}, \end{align}$

where,

$\begin{equation*} P = \begin{pmatrix} \frac{-1}{2h} & 0 & \frac{1}{2h} & 0 & \cdots & 0 & 0 & 0 \\ \frac{1}{6} & \frac{4}{6} & \frac{1}{6} & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \frac{1}{6} & \frac{4}{6} & \frac{1}{6} \\ 0 & 0 & 0 & 0 & \cdots & \frac{-1}{2h} & 0 & \frac{1}{2h} \end{pmatrix},\; Y^{0} = \begin{pmatrix} y^{0}_{-1} \\ y^{0}_{0} \\ \vdots \\ y^{0}_{\hat{N}} \\ y^{0}_{\hat{N}+1} \end{pmatrix}, \; \text{and}\; Z_{0} = \begin{pmatrix} v'_{0}(\mathfrak{u}_{0}) \\ v_{0}(\mathfrak{u}_{0}) \\ \vdots \\ v_{0}(\mathfrak{u}_{\hat{N}}) \\ v'_{0}(\mathfrak{u}_{\hat{N}}) \end{pmatrix}. \end{equation*}$

5. The stability analysis

The von Neumann stability analysis is based on the decomposition of numerical errors of numerical approximations into Fourier series ^[51]. This is employed in this section to examine the stability of the proposed numerical scheme. For this, suppose $y^{\hat{m}}_{j}$ symbolizes the growth factor in Fourier mode as:

$\begin{align} y^{\hat{m}}_{j} = \acute{A}\xi^{\hat{m}} exp(ij\mu), \end{align}$

(5.1)

where $\mu = \mathfrak{p} h$ , $\acute{A}$ is the harmonic amplitude, and $\mathfrak{p}$ is the mode number. Now using Eq (5.1) in Eq (3.6), the result is

$\begin{eqnarray} (\kappa_{0})(\acute{A}\xi^{\hat{m}+1}e^{i(j-1)\mu})+(\kappa_{1})(\acute{A}\xi^{\hat{m}+1}e^{i(j)\mu}) +(\kappa_{2})(\acute{A}\xi^{\hat{m}+1}e^{i(j+1)\mu}) = (\varpi_{1})(\acute{A}\xi^{\hat{m}}e^{i(j-1)\mu}) \\ +(\varpi_{2}) (\acute{A}\xi^{\hat{m}}e^{i(j)\mu}) +(\varpi_{3})(\acute{A}\xi^{\hat{m}}e^{i(j+1)\mu})-\rho^{\psi} \gamma\bigg[\sum\limits^{\hat{m}}_{\zeta = 1} l_{\zeta}\big((\acute{A}\xi^{\hat{m}-\zeta+1}e^{i(j-1)\mu}-\acute{A}\xi^{\hat{m}-\zeta}e^{i(j-1)\mu}) \\ +4(\acute{A} \xi^{\hat{m}-\zeta+1}e^{i(j)\mu}-\acute{A}\xi^{\hat{m} -\zeta}e^{i(j)\mu})+(\acute{A}\xi^{\hat{m}-\zeta+1}e^{i(j+1)\mu}-\acute{A}\xi^{\hat{m}-\zeta}e^{i(j+1)\mu})/6\big)\bigg] +g(\mathfrak{u},s_{\hat{m}+1}). \end{eqnarray}$

(5.2)

After further simplification, Eq (5.2) becomes

$\begin{eqnarray} \xi^{\hat{m}+1}[e^{-i \mu}\kappa_{0}+\kappa_{1}+e^{i \mu} \kappa_{2}] = \xi^{\hat{m}}[e^{-i \mu}\varpi_{1}+\varpi_{2}+e^{i \mu} \varpi_{3}]+ (\acute{A})^{-1}e^{-i j \mu}f(\mathfrak{u},s_{\hat{m}+1})-\rho^{\psi}\gamma \\ \bigg[\sum\limits^{\hat{m}}_{\zeta = 0} l_{\zeta}\big((\xi^{\hat{m}-\zeta+1}e^{-i\mu}-\xi^{\hat{m}-\zeta}e^{-i\mu}) +4(\xi^{\hat{m}-\zeta+1}-\xi^{\hat{m} -\zeta})+(\xi^{\hat{m}-\zeta+1}e^{i \mu}-\xi^{\hat{m}-\zeta}e^{i \mu})/6\big)\bigg]. \end{eqnarray}$

(5.3)

Using the values of $\kappa_{0}, \; \kappa_{1}, \; \kappa_{2}, \; \varpi_{1}, \; \varpi_{2}$ , and $\varpi_{3}$ in Eq (5.3), we have

$\begin{eqnarray} \xi^{\hat{m}+1} = \xi^{\hat{m}} \frac{\varpi_{4}2cos \mu+\varpi_{5}2 i sin\mu+\varpi_{2}}{[2 \kappa_{4}cos \mu+\kappa_{5}2 i sin\mu+\kappa_{1}]}\\ +\frac{A^{-1}e^{-ij \mu}f(\mathfrak{u},s_{\hat{m}+1}) -\rho^{\psi}\gamma\sum\limits^{\hat{m}}_{\zeta = 0} l_{\zeta}[((\xi^{\hat{m}-\zeta+1}-\xi^{\hat{m}-\zeta})(2+ cos\mu))/3]}{[2 \kappa_{4}cos\mu+\kappa_{5}2 i sin\mu+\kappa_{1}]}, \end{eqnarray}$

(5.4)

where

$\begin{align} \kappa_{4} = &\frac{1}{6 \Delta s}+\frac{\tau \theta R_{1}}{12 h}+\frac{\rho^{\psi}\gamma E_{1}}{6}-\frac{{\phi}}{h^{2} \Delta s}-\frac{\chi \theta}{h^{2}}, \\ \kappa_{5} = &\frac{\theta}{2 h}+\frac{\tau \theta R_{2}}{12 h}, \\ \varpi_{4} = &\frac{1}{6 \Delta s}+\frac{\rho^{\psi}\gamma E_{1}}{6}-\frac{{\phi}}{h^{2} \Delta s}+\frac{\chi (1-\theta)}{h^{2}}, \\ \varpi_{5} = &\frac{2\theta-1}{12 h}-\frac{1-\theta}{12 h}. \end{align}$

Denominator value is greater than the numerator value. Thus, $| \xi|^2 \leq 1$ .

Since the modulus of the eigenvalues must be less than one, the suggested approach for the time-fractional BBM-Burger equation is, therefore, unconditionally stable from (5.4). This indicates that the grid size $h$ and step size $\Delta s$ in the time level are not limited; rather, we should favor the values of $h$ and $\Delta s$ that yield the highest scheme accuracy.

6. The convergence analysis

The convergence of the proposed technique is examined using the approach presented in ^[52]. We first introduce Theorem 1 and Lemma 6.1, which are based on the work of Hall ^[53] and Boor ^[54].

Theorem 1. Assume that $g(\mathfrak{u}, s)$ and $v(\mathfrak{u}, s)$ belong to $C^{2}(\mathfrak{a}, \mathfrak{b})$ and $C^{4}(\mathfrak{a}, \mathfrak{b})$ respectively. The equidistance partition of $[\mathfrak{a}, \mathfrak{b}]$ is $\Upsilon = [\mathfrak{a} = \mathfrak{u}_{0}, \mathfrak{u}_{1}, \mathfrak{u}_{2}, \cdots, \mathfrak{u}_{\hat{N}} = \mathfrak{b}]$ with stepsize $h$ . If $\hat{\Psi}$ is the unique spline interpolation of the given problem at knots $\mathfrak{u}_{0}, \mathfrak{u}_{1}, \mathfrak{u}_{2}, \cdots, \mathfrak{u}_{\hat{N}}\in \Upsilon$ , then there exists a constant $\Im_{j}$ independent of $h$ in which $\mathfrak{u}_{j} = \mathfrak{a}+j h$ , $j = 0, 1, 2, \cdots, \hat{N}$ , then for every $s\geq 0$ , it can be obtained that

$\begin{align} \parallel D^{j}(v(\mathfrak{u},s)-\hat{\Psi}(\mathfrak{u},s)) \parallel \leq \Im_{j} h^{4-j},\; \; \; j = 0,1,2. \end{align}$

(6.1)

Lemma 6.1. The cubic B-spline set ${F_{-1}, F_{0}, F_{1}, \cdots, F_{\hat{N}+1}}$ in Eq (2.3) satisfies the inequality

$\begin{align} \sum\limits_{j = -1}^{\hat{N}+1} |F_{j}(\mathfrak{u})| \leq \frac{5}{3}, \; \; 0 \leq \mathfrak{u} \leq 1. \end{align}$

Theorem 2. For the BBM-Burger Eq (1.1) and BCs (1.2), there is a computational approximation $\Psi(\mathfrak{u}, s)$ to the analytical solution $v(\mathfrak{u}, s)$ . Moreover, if $g\in C^{2}[\mathfrak{a}, \mathfrak{b}]$ , then

$\begin{equation*} \parallel v(\mathfrak{u},s)-\Psi(\mathfrak{u},s) \parallel _{\infty} \leq \tilde{\Im} h^{2}, \; \; \forall \; \; s\geq 0, \end{equation*}$

where $h$ is relatively small and $\tilde{\Im} > 0$ is free of $h$ .

Proof. Assume $\Psi(\mathfrak{u}, s)$ is approximated as $\hat{\Psi}(\mathfrak{u}, s) = \sum\limits^{\hat{N}+1}_{j = -1} w_{j}^{\hat{m}}(t)F_{j}(\mathfrak{u})$ . From triangular inequality:

$\begin{equation} \parallel v(\mathfrak{u},s)- \Psi(\mathfrak{u},s)\parallel _{\infty} \leq \parallel v(\mathfrak{u},s)- \hat{\Psi}(\mathfrak{u},s)\parallel _{\infty}+\parallel \hat{\Psi}(\mathfrak{u},s)-\Psi(\mathfrak{u},s)\parallel _{\infty}. \end{equation}$

(6.2)

By using Theorem 1, Eq (6.2) becomes:

$\begin{align} \parallel v(\mathfrak{u},s)- \Psi(\mathfrak{u},s)\parallel _{\infty} \leq \Im_{0}h^{4}+\parallel \hat{\Psi}(\mathfrak{u},s)-\Psi(\mathfrak{u},s)\parallel _{\infty}. \end{align}$

(6.3)

The proposed scheme has collocation conditions as: $L v(\mathfrak{u}_{j}, s) = L \Psi(\mathfrak{u}_{j}, s) = g(\mathfrak{u}_{j}, s), \; \; j = 0(1) \hat{N}$ . Suppose that $L \hat{\Psi}(\mathfrak{u_{j}}, s) = \hat{g}(\mathfrak{u}_{j}, s)$ , $j = 0(1)\hat{N}.$ Thus, for any temporal level, the difference $\hat{\Psi}(\mathfrak{u}_{j}, s)-\Psi(\mathfrak{u}_{j}, s)$ with $\theta = 1$ in linear form can be given as:

(6.4)

where

$\begin{align} \kappa_{6} & = \frac{1}{6}-\frac{{\phi}}{h^{2}}-\frac{\Delta s \tau \psi}{2h}-\frac{\Delta s \chi}{h^{2}}-\frac{\Delta s}{2h}+\frac{\rho^{\psi} \Delta s E_{1} \gamma}{6}, \\ \kappa_{7} & = \frac{4}{6}+\frac{ 2 {\phi}}{h^{2}}+\frac{2 \Delta s \chi}{h^{2}}+\frac{4 \rho^{\psi} \Delta s E_{1} \gamma}{6}, \\ \kappa_{8} & = \frac{1}{6}-\frac{{\phi}}{h^{2}}+\frac{\Delta s \tau \psi}{2h}-\frac{\Delta s \chi}{h^{2}}+\frac{\Delta s}{2h}+\frac{\rho^{\psi} \Delta s E_{1} \gamma}{6}. \end{align}$

BCs can be written as:

$\begin{align} \frac{1}{6}\Omega^{\hat{m}+1}_{j-1}+\frac{4}{6}\Omega^{\hat{m}+1}_{j}+\frac{1}{6}\Omega^{\hat{m}+1}_{j+1} = 0, \; \; j = 0,\hat{N}, \end{align}$

where

$\begin{equation*} \Omega^{\hat{m}}_{j} = y^{\hat{m}}_{j}-w_{j}^{\hat{m}}, \; \; \; j = -1: 0: \hat{N}+1, \end{equation*}$

and

$\begin{equation*} \beta^{\hat{m}}_{j} = h^{2}[g^{\hat{m}}_{j}-\tilde{g}^{\hat{m}}_{j}],\; \; j = 0:1:\hat{N}. \end{equation*}$

From inequality (6.1), it is clear that

$\begin{equation*} \mid \beta^{\hat{m}}_{j}\mid = h^{2}\mid g^{\hat{m}}_{j}-\tilde{g}^{\hat{m}}_{j}\mid \leq \Im h^{4}. \end{equation*}$

Define $\beta^{\hat{m}} = max \{ { \mid \beta^{\hat{m}}_{j} \mid; 0\leq j \leq \hat{N} } \}, \; e^{\hat{m}}_{j} = \mid \Omega^{\hat{m}}_{j}\mid$ and $e^{\hat{m}} = max \{\mid e^{\hat{m}}_{j} \mid; 0\leq j \leq \hat{N}\}$ . When $\hat{m} = 0$ , Eq (6.4) becomes

$\begin{eqnarray} \Omega^{1}_{j-1}(\kappa_{6}) +\Omega^{1}_{j}(\kappa_{7})+\Omega^{1}_{j+1}(\kappa_{8}) = \bigg(\frac{1}{6}\Omega^{0}_{j-1}+ \frac{4}{6}\Omega^{0}_{j}+\frac{1}{6} \Omega^{0}_{j+1} \bigg) -{\phi} \bigg(\frac{1}{h^{2}}\Omega^{0}_{j-1}-\frac{2}{h^{2}}\Omega^{0}_{j}+\frac{1}{h^{2}}\Omega^{0}_{j+1}\bigg)\\ + \rho^{\psi} \Delta s \gamma l_{0} \bigg(\frac{1}{6}\Omega^{0}_{j-1}+ \frac{4}{6}\Omega^{0}_{j}+\frac{1}{6} \Omega^{0}_{j+1} \bigg) +\Delta s \frac{\beta_{j}^{1}}{h^{2}}. \end{eqnarray}$

(6.5)

From IC, $e^{0} = 0$ . For sufficiently small mesh spacing $h$ and norms of $\beta^{1}_{i}, \; \Omega^{1}_{i}$ , Eq (6.5) yields:

$\begin{equation*} e^{1}_{j}\leq \frac{3 \Delta s \Im h^{4}}{ h^{2}+12 {\phi}+12 \Delta s \chi+ h^{2} \Delta s \rho^{\psi}\gamma E_{1}}. \end{equation*}$

From BCs $e^{1}_{-1}$ and $e^{1}_{\hat{N}+1}$ , it can be written as:

$\begin{equation*} e^{1}_{-1}\leq \frac{15 \Delta s \Im h^{4}}{ h^{2}+12 {\phi}+12 \Delta s \chi+ h^{2} \Delta s \rho^{\psi}\gamma E_{1}}, \end{equation*}$

$\begin{equation*} e^{1}_{\hat{N}+1}\leq \frac{15 \Delta s \Im h^{4}}{ h^{2}+12 {\phi}+12 \Delta s \chi+ h^{2} \Delta s \rho^{\psi}\gamma E_{1}}. \end{equation*}$

From above conditions, it is carried out that

$\begin{equation*} e^{1}\leq \Im_{1} h^{2}, \end{equation*}$

where $\Im_{1}$ is not depending on $h$ . To prove this theorem, mathematical induction is applied to $\hat{m}$ . It is considered that the term $e^{z}_{j}\leq \Im_{z}h^{2}$ is true, when $z = 1, 2, 3, \cdots, \hat{m}$ and $\Im = max \{\Im_{z}: z = 0, 1, 2, \cdots, \hat{m} \}$ , then Eq (6.4) becomes:

$\begin{eqnarray} \Omega^{\hat{m}+1}_{j-1}(\kappa_{6})+\Omega^{\hat{m}+1}_{j}(\kappa_{7})+\Omega^{\hat{m}+1}_{j+1}(\kappa_{8}) = \bigg(\frac{1}{6}\Omega^{\hat{m}}_{j-1}+ \frac{4}{6}\Omega^{\hat{m}}_{j}+\frac{1}{6}\Omega^{\hat{m}}_{j+1}\bigg)-{\phi}\bigg(\frac{1}{h^{2}}\Omega^{\hat{m}}_{j-1} -\frac{2}{h^{2}}\Omega^{\hat{m}}_{j}+\frac{1}{h^{2}}\Omega^{\hat{m}}_{j+1}\bigg)\\ + \rho^{\psi} \Delta s \gamma (l_{0}-l_{1})\bigg(\frac{1}{6}\Omega^{\hat{m}}_{j-1}+\frac{4}{6}\Omega^{\hat{m}}_{j}+\frac{1}{6}\Omega^{\hat{m}}_{j+1} \bigg)+ \rho^{\psi} \Delta s \gamma (l_{1}-l_{2}) \bigg(\frac{1}{6}\Omega^{\hat{m}-1}_{j-1}+\frac{4}{6}\Omega^{\hat{m}-1}_{j}+\frac{1}{6}\Omega^{\hat{m}-1}_{j+1} \bigg)\\ + \rho^{\psi} \Delta s \gamma (l_{2}-l_{3})\bigg(\frac{1}{6}\Omega^{\hat{m}-2}_{j-1}+\frac{4}{6}\Omega^{\hat{m}-2}_{j}+\frac{1}{6}\Omega^{\hat{m}-2}_{j+1} \bigg) +\cdots+ \rho^{\psi} \Delta s \gamma (l_{\hat{m}-2}-l_{\hat{m}-1})\bigg(\frac{1}{6}\Omega^{2}_{j-1}+\frac{4}{6}\Omega^{2}_{j}+\frac{1}{6}\Omega^{2}_{j+1} \bigg)\\ +\rho^{\psi} \Delta s \gamma (l_{\hat{m}-1}-l_{\hat{m}})\bigg(\frac{1}{6}\Omega^{1}_{j-1} +\frac{4}{6}\Omega^{1}_{j}+\frac{1}{6}\Omega^{1}_{j+1} \bigg) + \rho^{\psi} \Delta s \gamma l_{\hat{m}}\bigg(\frac{1}{6}\Omega^{0}_{j-1}+\frac{4}{6}\Omega^{0}_{j}+\frac{1}{6}\Omega^{0}_{j+1} \bigg) +\Delta s \frac{\beta_{j}^{\hat{m}+1}}{h^{2}}. \end{eqnarray}$

(6.6)

Apply the norm again on $\Omega^{\hat{m}+1}_{j}$ and $\beta^{\hat{m}+1}_{j}$ , and Eq (6.6) gives

$\begin{equation*} e^{\hat{m}+1}_{j}\leq \frac{3 \Im h^{4}}{( h^{2}+12 {\phi}+12 \Delta s \chi+ h^{2} \Delta s \rho^{\psi}\gamma E_{1})} \bigg(1+\Delta s+ \Delta s \rho^{\psi} \gamma\sum\limits^{\hat{m}-1}_{\zeta = 0}(l_{\zeta}-l_{\zeta+1}) \bigg). \end{equation*}$

Similarly, from BCs $e^{\hat{m +1}}_{-1}$ and $e^{\hat{m}+1}_{\hat{N}+1}$ , it can be taken as:

$\begin{equation*} e^{\hat{m}+1}_{-1} \leq \frac{15 \Im h^{4}}{( h^{2}+12 {\phi}+12 \Delta s \chi+ h^{2} \Delta s \rho^{\psi}\gamma E_{1})}\bigg(1+\Delta s+ \Delta s \rho^{\psi} \gamma\sum\limits^{\hat{m}-1}_{\zeta = 0}(l_{\zeta}-l_{\zeta+1}) \bigg), \end{equation*}$

$\begin{equation*} e^{\hat{m}+1}_{\hat{N}+1} \leq \frac{15 \Im h^{4}}{( h^{2}+12 {\phi}+12 \Delta s \chi+ h^{2} \Delta s \rho^{\psi}\gamma E_{1})}\bigg(1+\Delta s+ \Delta s \rho^{\psi} \gamma\sum\limits^{\hat{m}-1}_{\zeta = 0}(l_{\zeta}-l_{\zeta+1}) \bigg). \end{equation*}$

For all $\hat{m}$ , we get

$\begin{align} e^{\hat{m}+1} \leq \Im h^{2}. \end{align}$

(6.7)

Particularly,

$\begin{equation*} \hat{\Psi}(\mathfrak{u},s)-\Psi(\mathfrak{u},s) = \sum\limits^{\hat{N}+1}_{j = -1}(w_{j}(s)-y_{j}(s))F_{j}(\mathfrak{u}). \end{equation*}$

Therefore, the inequality (6.7) and Lemma 6.1 gives

$\begin{align} \parallel \hat{\Psi}(\mathfrak{u},s)-\Psi(\mathfrak{u},s)\parallel _{\infty} \leq \frac{5}{3}\Im h^{2}. \end{align}$

(6.8)

From inequalities (6.3) and (6.8), we obtain:

$\begin{align} \parallel v(\mathfrak{u},s)-\Psi(\mathfrak{u},s)\parallel _{\infty} \leq \Im_{0}h^{4}+\frac{5}{3}\Im h^{2} = \check{\Im}h^{2}, \end{align}$

where $\check{\Im} = \Im_{0}h^{2}+\frac{5}{3}\Im$ . □

Theorem 3. The BBM-Burger equation is convergent with the initial and BCs.

Proof. Consider the BBM-Burger equation has analytical solution $v(\mathfrak{u}, s)$ and numerical solution $\Psi(\mathfrak{u}, s)$ . Therefore, the previous theorem and inequality (3.2) validate that

$\begin{equation} \parallel v(\mathfrak{u},s)-\Psi(\mathfrak{u},s)\parallel_{\infty} \leq \check{\Im} h^{2}+ \hat{\vartheta} (\Delta s)^{2}, \end{equation}$

(6.9)

where $\check{\Im}$ and $\hat{\vartheta}$ are arbitrary constants. Consequently, the present scheme is second order convergent in both the spatial and temporal directions. □

7. Examples and discussion

This section includes numerical results and their consistency to show how accurate our results are. Error norms $L_{2}(\hat{N})$ and $L_{\infty}(\hat{N})$ are defined as:

$\begin{align} L_{2}(\hat{N}) = & \|v(\mathfrak{u}_{j},s)-\Psi(\mathfrak{u}_{j},s) \|_{2} = \sqrt{h \sum\limits^{\hat{N}}_{j = 0}\mid v(\mathfrak{u}_{j},s)-\Psi(\mathfrak{u}_{j},s) \mid ^{2}}, \\ L_{\infty}(\hat{N}) = & \|v(\mathfrak{u}_{j},s)-\Psi(\mathfrak{u}_{j},s) \|_{\infty} = \max\limits_{0\leq j\leq \hat{N}}\mid v(\mathfrak{u}_{j},s)-\Psi(\mathfrak{u}_{j},s) \mid. \end{align}$

The convergence order of the present scheme can be calculated as ^[55]:

$\begin{equation*} log \bigg(\frac{L_{\infty}(\hat{N})}{L_{\infty}(\hat{N}+1)} \bigg) \diagup log \bigg( \frac{ (\hat{N}+1)}{(\hat{N})} \bigg). \end{equation*}$

Normalization function in all examples is considered as $R(\psi) = 1$ and $\theta = 1$ . Mathematica $9$ is used for numerical calculations on an Intel(R)Core(TM) i5-3437U CPU@2.60GHz, 2712Mhz with 16GB RAM, SSD and 64-bit operating system (Windows 11 pro). Processing to compute numerical results takes less than a minute for all computed results given in tables.

Example 7.1. Consider the nonhomogeneous time-fractional BBM-Burger Eq (1.1) with specific parameter values, namely, ${\phi} = \rho = \tau = \chi = 1.$

$\begin{equation*} v_{s} +v_\mathfrak{u} - v_{\mathfrak{u} \mathfrak{u} s} +\rho^{\psi} D_s ^{\psi} v + v v_\mathfrak{u} - v_{\mathfrak{u} \mathfrak{u}} = g(\mathfrak{u},s),\; \; \; \; \; \; s\in [0,S],\; \mathfrak{u} \in [0,2],\; 0 < \psi \leq 1, \end{equation*}$

with IC and BCs

$\begin{equation*} \begin{cases} v(\mathfrak{u},0) = 0,\\ v(0,s) = s^{2},\; v(2,s) = s^{2} e^{2}, \end{cases} \end{equation*}$

and the source term is $g(\mathfrak{u}, s) = 2 \bigg(\frac{R(\psi)}{1-\psi} \bigg)s^{2} e^{\mathfrak{u}} E_{\psi, 3}\bigg[\frac{-\psi}{1-\psi} s^{\psi} \bigg]+s^{4} e^{2\mathfrak{u}}$ .

The exact solution is $v(\mathfrak{u}, s) = s^{2} e^{\mathfrak{u}}$ . and compare exact and computational solutions, along with absolute errors, for different choices of $\psi$ and $s = 0.25, 0.75$ , $\psi = 0.5$ with $\hat{N} = 80$ and $\hat{N} = 60$ , $\Delta s = 0.001$ , respectively, demonstrating good agreement between the exact and approximate solutions. Error norms for various $\psi$ values are displayed in at different time stages. The comparison of error norms and the convergence order for different choices of $\Delta s$ is represented in . illustrates convergence order and error norm for different values of $h$ . A very good agreement between exact and spline solutions at various time levels for fixed $\hat{N} = 40$ with $\psi = 0.3$ and $\Delta s = 0.01$ is shown in . This figure also demonstrates good results for $\hat{N} = 80$ , $\psi = 0.7$ , and $\Delta s = 0.002$ . The correctness of the suggested approach is demonstrated in using 3D graphs comparing analytical and numerical answers at $\hat{N} = 60$ , $\psi = 0.5$ , $\Delta s = 0.001$ , and $s = 1$ . At $s = 1$ , 2D and 3D error profiles are demonstrated in Figure 3. All tables and figures guarantee the exactness of the scheme.

Table 2. Absolute error for different choices of

$\psi$ with

$\hat{N} = 80$ ,

$s = 1$ , and

$\Delta s = 0.001$ of Example 7.1.

	Approximate Solution			Absolute Error
$\mathfrak{u}$	Exact Solution	$\psi=0.2$	$\psi=0.7$	$\psi=0.2$	$\psi=0.7$
0.0	1	0.999999999999	0.999999999999	$3.78141\times10^{-13}$	$5.09814\times10^{-13}$
0.2	1.221402758160	1.221392348771	1.221392808075	$1.04093\times10^{-5}$	$9.95008\times10^{-6}$
0.4	1.491824697641	1.491804802626	1.491805721466	$1.98950\times10^{-5}$	$1.89761\times10^{-5}$
0.6	1.822118800390	1.822090386954	1.822091728899	$2.84134\times10^{-5}$	$2.70714\times10^{-5}$
0.8	2.225540928492	2.225505139956	2.225506833758	$3.57885\times10^{-5}$	$3.40947\times10^{-5}$
1.0	2.718281828459	2.718240173922	2.718242113097	$4.16545\times10^{-5}$	$3.97153\times10^{-5}$
1.2	3.320116922736	3.320071566803	3.320073606814	$4.53559\times10^{-5}$	$4.33159\times10^{-5}$
1.4	4.055199966844	4.055154197542	4.055156150339	$4.57693\times10^{-5}$	$4.38165\times10^{-5}$
1.6	4.953032424395	4.952991444977	4.952993070875	$4.09794\times10^{-5}$	$3.93535\times10^{-5}$
1.8	6.049647464412	6.049619788339	6.049620786205	$2.76760\times10^{-5}$	$2.66782\times10^{-5}$
2.0	7.389056098930	7.389056098930	7.389056098930	0	0

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Table 3. Absolute error norms for different choices of

$s$ with

$\hat{N} = 60$ ,

$\psi = 0.5$ , and

$\Delta s = 0.001$ of Example 7.1.

	$s=0.25$	$s=0.75$		$s=0.25$	$s=0.75$
Exact	Approximate	Exact	Approximate	Absolute Error	Absolute Error
0.0625000000	0.0625000000	0.5625000000	0.5625000000	$9.603\times10^{-15}$	$1.479\times10^{-13}$
0.0763376723	0.0763360796	0.6870390514	0.6870267888	$1.592\times10^{-6}$	$1.226\times10^{-5}$
0.0932390436	0.0932360506	0.8391513924	0.8391280897	$2.992\times10^{-6}$	$2.330\times10^{-5}$
0.1138824250	0.1138782314	1.0249418252	1.0249087651	$4.193\times10^{-6}$	$3.306\times10^{-5}$
0.1390963080	0.1390911427	1.2518667722	1.2518254715	$5.165\times10^{-6}$	$4.130\times10^{-5}$
0.1698926142	0.1698867632	1.5290335285	1.5289859675	$5.851\times10^{-6}$	$4.756\times10^{-5}$
0.2075073076	0.2075011476	1.8675657690	1.8675147100	$6.160\times10^{-6}$	$5.105\times10^{-5}$
0.2534499979	0.2534440400	2.2810499813	2.2809994257	$5.957\times10^{-6}$	$5.055\times10^{-5}$
0.3095645265	0.3095594719	2.7860807387	2.7860366082	$5.054\times10^{-6}$	$4.413\times10^{-5}$
0.3781029665	0.3780997789	3.4029266987	3.4028978806	$3.187\times10^{-6}$	$2.881\times10^{-5}$
0.4618160061	0.4618160061	4.1563440556	4.1563440556	$1.110 \times10^{-16}$	$8.881 \times10^{-16}$

| Show Table

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Table 4. Error norms for different values of

$\psi$ , whereas

$\Delta s = 0.002$ , and

$\hat{N} = 120$ of Example 7.1.

	$L_{\infty}(\hat{N})$			$L_{2}(\hat{N})$
$s$	$\psi=0.1$	$\psi=0.5$	$\psi=0.9$	$\psi=0.1$	$\psi=0.5$	$\psi=0.9$
0.2	$9.94079\times10^{-7}$	$9.81215\times10^{-7}$	$8.84006\times10^{-7}$	$1.01832\times10^{-6}$	$1.00509\times10^{-6}$	$9.05239\times10^{-7}$
0.4	$3.77369\times10^{-6}$	$3.68997\times10^{-6}$	$3.13395\times10^{-6}$	$3.86106\times10^{-6}$	$3.77527\times10^{-6}$	$3.20615\times10^{-6}$
0.6	$7.36909\times10^{-6}$	$7.14659\times10^{-6}$	$5.75676\times10^{-6}$	$7.54395\times10^{-6}$	$7.31676\times10^{-6}$	$5.90012\times10^{-6}$
0.8	$9.65892\times10^{-6}$	$9.27306\times10^{-6}$	$6.84334\times10^{-6}$	$9.91894\times10^{-6}$	$9.52635\times10^{-6}$	$7.04179\times10^{-6}$
1.0	$7.88589\times10^{-6}$	$7.43121\times10^{-6}$	$4.52746\times10^{-6}$	$7.95573\times10^{-6}$	$7.47227\times10^{-6}$	$4.29801\times10^{-6}$

| Show Table

DownLoad: CSV

Table 5. Comparison of error norm with distinct values of

$\Delta s = \frac{1}{\hat{M}}$ , when

$s = 1$ ,

$\hat{N} = 60$ , and

$h = \frac{2}{N}$ for Example 7.1.

$\psi$	$\hat{M}$	$L_{\infty}(\hat{N})$	$L_{2}(\hat{N})$	Order
0.2	$10$	$3.67022\times10^{-2}$	$3.46856\times10^{-2}$	...
	$20$	$9.42441\times10^{-3}$	$8.9247\times10^{-3}$	$1.96139$
	$40$	$2.33138\times10^{-3}$	$2.20614\times10^{-3}$	$2.01522$
	$80$	$5.23722\times10^{-4}$	$4.90554\times10^{-4}$	$2.15431$
	$160$	$6.85997\times10^{-5}$	$5.80062\times10^{-5}$	$2.93253$
0.3	$10$	$3.67142\times10^{-2}$	$3.46928\times10^{-2}$	...
	$20$	$9.43083\times10^{-3}$	$8.93013\times10^{-3}$	$1.96088$
	$40$	$2.33379\times10^{-3}$	$2.2083\times10^{-3}$	$2.01471$
	$80$	$5.24667\times10^{-4}$	$4.91436\times10^{-4}$	$2.15327$
	$160$	$6.90639\times10^{-5}$	$5.84713\times10^{-5}$	$2.92546$
0.4	$10$	$3.67304\times10^{-2}$	$3.47023\times10^{-2}$	...
	$20$	$9.43919\times10^{-3}$	$8.93661\times10^{-3}$	$1.96024$
	$40$	$2.33666\times10^{-3}$	$2.2108\times10^{-3}$	$2.01421$
	$80$	$5.25812\times10^{-4}$	$4.92489\times10^{-4}$	$2.15183$
	$160$	$6.96661\times10^{-5}$	$5.9075\times10^{-5}$	$2.91602$
0.5	$10$	$3.6767\times10^{-2}$	$3.473\times10^{-2}$	...
	$20$	$9.45364\times10^{-3}$	$8.94847\times10^{-3}$	$1.95947$
	$40$	$2.34118\times10^{-3}$	$2.21475\times10^{-4}$	$2.01363$
	$80$	$5.27459\times10^{-4}$	$4.94012\times10^{-4}$	$2.15011$
	$160$	$7.04993\times10^{-5}$	$5.99106\times10^{-5}$	$2.90338$

| Show Table

DownLoad: CSV

Table 6. Comparison of error norms with distinct values of h, when

$s = 1$ ,

$\Delta s = \frac{1}{\hat{M}}$ , and

$\hat{N} = 100$ for Example 7.1.

$\psi$	$h$	$L_{\infty}(\hat{N})$	$L_{2}(\hat{N})$	Order
0.2	$\frac{1}{2}$	$2.07143\times10^{-2}$	$2.10695\times10^{-2}$	...
	$\frac{1}{4}$	$4.64235\times10^{-3}$	$4.73467\times10^{-3}$	$2.15776$
	$\frac{1}{8}$	$8.71729\times10^{-4}$	$8.93474\times10^{-4}$	$2.41294$
	$\frac{1}{16}$	$1.00684\times10^{-4}$	$7.38965\times10^{-5}$	$3.11404$
0.3	$\frac{1}{2}$	$2.06423\times10^{-2}$	$2.09873\times10^{-2}$	...
	$\frac{1}{4}$	$4.62263\times10^{-3}$	$4.71485\times10^{-3}$	$2.15882$
	$\frac{1}{8}$	$8.66622\times10^{-4}$	$8.88302\times10^{-4}$	$2.41524$
	$\frac{1}{16}$	$1.01944\times10^{-4}$	$7.50392\times10^{-5}$	$3.08763$
0.4	$\frac{1}{2}$	$2.05389\times10^{-2}$	$2.08701\times10^{-2}$	...
	$\frac{1}{4}$	$4.59445\times10^{-3}$	$4.68654\times10^{-3}$	$2.16043$
	$\frac{1}{8}$	$8.59434\times10^{-4}$	$8.81023\times10^{-4}$	$2.41843$
	$\frac{1}{16}$	$1.03629\times10^{-4}$	$7.65899\times10^{-5}$	$3.05196$
0.5	$\frac{1}{2}$	$2.03979\times10^{-2}$	$2.07107\times10^{-2}$	...
	$\frac{1}{4}$	$4.55611\times10^{-3}$	$4.64804\times10^{-3}$	$2.16255$
	$\frac{1}{8}$	$8.49624\times10^{-4}$	$8.71091\times10^{-4}$	$2.42291$
	$\frac{1}{16}$	$1.06076\times10^{-4}$	$7.87782\times10^{-5}$	$3.00173$

| Show Table

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Figure 1. Exact and numerical solutions for Example 7.1 at different temporal stages.

DownLoad: Full-Size Img PowerPoint

Figure 2. 3D exact and approximate solution for Example 7.1, when

$\hat{N} = 60$ ,

$\psi = 0.5$ ,

$\Delta s = 0.001$ , and

$s = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. 2D and 3D error profiles for Example 7.1, when

$\hat{N} = 32$ ,

$\psi = 0.6$ ,

$\Delta s = 0.001$ , and

$s = 1$ .

DownLoad: Full-Size Img PowerPoint

Example 7.2. Consider the nonhomogeneous time-fractional BBM-Burger Eq (1.1) with specific parameter values, namely, ${\phi} = \rho = \tau = \chi = 1.$

$\begin{equation*} v_{s} +v_\mathfrak{u} - v_{\mathfrak{u} \mathfrak{u} s} +\rho^{\psi} D_s ^{\psi} v + v v_\mathfrak{u} - v_{\mathfrak{u} \mathfrak{u}} = g(\mathfrak{u},s),\; \; \; s\in [0,S],\; \mathfrak{u} \in [0,2],\; 0 < \psi \leq 1, \end{equation*}$

with IC and BCs

$\begin{equation*} \begin{cases} v(\mathfrak{u},0) = 0,\\ v(0,s) = 0,\; v(2,s) = s^{3} \sin (2), \end{cases} \end{equation*}$

and the source term is $g(\mathfrak{u}, s) = 6 \bigg(\frac{R(\psi)}{1-\psi} \bigg)s^{3} \sin(\mathfrak{u}) E_{\psi, 4}\bigg[\frac{-\psi}{1-\psi} s^{\psi} \bigg]+3 s^{2} \sin (\mathfrak{u})+s^{3} cos (\mathfrak{u})+3 s^{2} \sin(\mathfrak{u})+(s^{3} \sin(\mathfrak{u}))(s^{3} \cos(\mathfrak{u}))+ s^{3} \sin(\mathfrak{u})$ .

The exact solution is $\psi(\mathfrak{u}, s) = s^{3} sin(\mathfrak{u})$ . The absolute errors for various values of $\mathfrak{u}$ setting $\Delta s \; = \; 0.001$ , $s = 1$ , $\psi = 0.5$ , and $\hat{N} = 30$ are reported in of Example 7.2. presents the exact solutions, approximate results, and absolute errors for the proposed problem when $\Delta s \; = \; 0.001$ , $s = 0.25$ , $\psi = 0.2$ , $\hat{N} = 50$ . Comparison of error norms is tabulated in Table 9. Tables 10 and 11 exhibit error norms and their convergence orders for various choices of parameters in temporal and spatial directions. Figure 4 shows the significant agreement between the numerical findings and exact solutions of the suggested scheme at different time levels. A 3D representation of exact and computational solutions at distinct values of parameters is given in Figure 5. The 2D and 3D error profile is exhibited in Figure 6, which demonstrates procedure of best accuracy.

Table 7. Absolute error of Example 7.2 with

$\Delta s = 0.001$ ,

$s = 1$ ,

$\psi = 0.5$ , and

$\hat{N} = 30$ .

$\mathfrak{u}$	Exact solution	Approximate solution	Absolute Error
0.0	0	$1.41512\times10^{-14}$	$1.41512\times10^{-14}$
0.2	0.198669330	0.198782763	0.000113432
0.4	0.389418342	0.389644040	0.000225698
0.6	0.564642473	0.564970890	0.000328417
0.8	0.717356090	0.717768999	0.000412908
1.0	0.841470984	0.841941093	0.000470108
1.2	0.932039085	0.932529613	0.000490527
1.4	0.985449729	0.985913966	0.000464236
1.6	0.999573603	0.999954505	0.000380902
1.8	0.973847630	0.974077462	0.000229831
2.0	0.909297426	0.909297426	0

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Table 8. Absolute error of Example 7.2 for

$\Delta s = 0.001$ ,

$s = 0.25$ ,

$\psi = 0.2$ , and

$\hat{N} = 50$ .

$\mathfrak{u}$	Exact solution	Approximate solution	Absolute Error
0.0	0	$-4.99873\times10^{-16}$	$4.99873\times10^{-16}$
0.2	0.003104208	0.003107163	$2.95558\times10^{-6}$
0.4	0.006084661	0.006090412	$5.75135\times10^{-6}$
0.6	0.008822538	0.008830720	$8.18207\times10^{-6}$
0.8	0.011208688	0.011218739	$1.00503\times10^{-5}$
1.0	0.013147984	0.013159153	$1.11693\times10^{-5}$
1.2	0.014563110	0.014574477	$1.13666\times10^{-5}$
1.4	0.015397652	0.015408136	$1.04846\times10^{-5}$
1.6	0.015618337	0.015626718	$8.38092\times10^{-6}$
1.8	0.015216369	0.015221295	$4.92614\times10^{-6}$
2.0	0.014207772	0.014207772	$1.73472\times10^{-18}$

| Show Table

DownLoad: CSV

Table 9. Error norms for distinct values of

$\psi$ , where

$\hat{N} = 40$ , and

$\Delta s = 0.001$ of Example 7.2.

	$L_{\infty}(\hat{N})$			$L_{2}(\hat{N})$
$s$	$\psi=0.2$	$\psi=0.5$	$\psi=0.8$	$\psi=0.2$	$\psi=0.5$	$\psi=0.8$
0.2	$5.77880\times10^{-6}$	$5.72266\times10^{-6}$	$5.51977\times10^{-6}$	$5.85582\times10^{-6}$	$5.79904\times10^{-6}$	$5.59391\times10^{-6}$
0.4	$4.30703\times10^{-5}$	$4.24272\times10^{-5}$	$4.03835\times10^{-5}$	$4.35810\times10^{-5}$	$4.29325\times10^{-5}$	$4.08728\times10^{-5}$
0.6	$1.35549\times10^{-4}$	$1.33144\times10^{-4}$	$1.26329\times10^{-4}$	$1.36972\times10^{-4}$	$1.34554\times10^{-4}$	$1.27708\times10^{-4}$
0.8	$2.99387\times10^{-4}$	$2.93729\times10^{-4}$	$2.79364\times10^{-4}$	$3.02142\times10^{-4}$	$2.96474\times10^{-4}$	$2.82029\times10^{-4}$
1.0	$5.44153\times10^{-4}$	$5.34025\times10^{-4}$	$5.10643\times10^{-4}$	$5.47508\times10^{-4}$	$5.37333\times10^{-4}$	$5.13911\times10^{-4}$

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Table 10. Comparison of error norm with distinct values of

$\Delta s = \frac{1}{\hat{M}}$ , when

$s = 1$ ,

$\hat{N} = 10$ , and

$h = \frac{2}{\hat{N}}$ for Example 7.2.

$\psi$	$\hat{M}$	$L_{\infty}(\hat{N})$	$L_{2}(\hat{N})$	Order
0.2	100	$5.10677\times10^{-3}$	$5.13719\times10^{-3}$	...
	200	$2.09648\times10^{-3}$	$2.10751\times10^{-3}$	$1.28444$
	400	$5.92019\times10^{-4}$	$5.93432\times10^{-4}$	$1.82425$
	800	$1.59913\times10^{-4}$	$1.63783\times10^{-4}$	$1.88836$
0.3	100	$5.08562\times10^{-3}$	$5.11596\times10^{-3}$	...
	200	$2.08753\times10^{-3}$	$2.09853\times10^{-3}$	$1.28463$
	400	$5.89267\times10^{-4}$	$5.90684\times10^{-4}$	$1.82480$
	800	$1.59537\times10^{-4}$	$1.63384\times10^{-4}$	$1.88503$
0.4	100	$5.05523\times10^{-3}$	$5.8544\times10^{-3}$	...
	200	$2.07469\times10^{-3}$	$2.08565\times10^{-3}$	$1.28488$
	400	$5.85347\times10^{-4}$	$5.86771\times10^{-4}$	$1.82553$
	800	$1.58962\times10^{-4}$	$1.62774\times10^{-4}$	$1.88061$
0.5	100	$5.01418\times10^{-3}$	$5.04424\times10^{-3}$	...
	200	$2.05742\times10^{-3}$	$2.06834\times10^{-3}$	$1.28518$
	400	$5.80171\times10^{-4}$	$5.8161\times10^{-4}$	$1.82629$
	800	$1.58041\times10^{-4}$	$1.61803\times10^{-4}$	$1.87618$

| Show Table

DownLoad: CSV

Table 11. Comparison of error norm with distinct values of

$h$ , when

$s = 1$ ,

$h = \frac{2}{\hat{N}}$ ,

$\hat{M} = 1200$ for Example 7.2.

$\psi$	$\hat{N}$	$L_{\infty}(\hat{N})$	$L_{2}(\hat{N})$	Order
0.2	2	$2.19161\times10^{-2}$	$2.19161\times10^{-2}$	...
	4	$5.04859\times10^{-3}$	$5.25733\times10^{-3}$	$2.11804$
	8	$9.17818\times10^{-4}$	$9.33769\times10^{-4}$	$2.45965$
	16	$1.45032\times10^{-4}$	$1.45084\times10^{-4}$	$2.66184$
0.3	2	$2.18407\times10^{-2}$	$2.18407\times10^{-2}$	...
	4	$5.02905\times10^{-3}$	$5.23767\times10^{-3}$	$2.11867$
	8	$9.14482\times10^{-4}$	$9.30322\times10^{-4}$	$2.45926$
	16	$1.44311\times10^{-4}$	$1.44359\times10^{-4}$	$2.66378$
0.4	2	$2.17315\times10^{-2}$	$2.17315\times10^{-2}$	...
	4	$5.00074\times10^{-3}$	$5.20919\times10^{-3}$	$2.11957$
	8	$9.09634\times10^{-4}$	$9.25313\times10^{-4}$	$2.45878$
	16	$1.43289\times10^{-4}$	$1.43332\times10^{-4}$	$2.66636$
0.5	2	$2.15801\times10^{-2}$	$2.15801\times10^{-2}$	...
	4	$4.96149\times10^{-3}$	$5.16970\times10^{-3}$	$2.12085$
	8	$9.02844\times10^{-4}$	$9.18299\times10^{-4}$	$2.45823$
	16	$1.41956\times10^{-4}$	$1.41997\times10^{-4}$	$2.66903$

| Show Table

DownLoad: CSV

Figure 4. Exact and computational solutions for Example 7.2 at distinct time stages.

DownLoad: Full-Size Img PowerPoint

Figure 5. 3D exact and approximate solution for Example 7.2, when

$\hat{N} = 20$ ,

$\psi = 0.5$ ,

$\Delta s = 0.001$ , and

$s = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. 2D and 3D error profiles for Example 7.2, when

$\hat{N} = 25$ ,

$\psi = 0.5$ ,

$\Delta s = 0.001$ , and

$s = 1$ .

DownLoad: Full-Size Img PowerPoint

8. Conclusions

An accurate numerical solution of the nonhomogeneous time-fractional BBM-Burger equation based on CBS functions has been determined in this study. For this purpose, CBS functions have been used to build up a collocation technique for the nonhomogeneous time-fractional BBM-Burger equation. The TF derivative has been approximated by the typical finite difference scheme and Atangana-Baleanu fractional operator, whereas the spatial derivative was discretized by using the $\theta$ -weighted scheme with CBS functions. Two test problems have been solved, and their graphical and numerical comparison uncovers that the proposed method is computationally very effective. The scheme has possessed the second order temporal and spatial convergence, as well as unconditionally stable. This method can be used to get efficient approximations for a large number of fractional differential equations. It has provided the results in the form of an improved solution for the problems for which there is no precise solution. This method offers effective solutions to numerous problems. In the future, we may consider the solution of higher dimensional and higher order BBM-Burger equations by using spline functions of higher degrees.

Author contributions

Muserat Shaheen: Methodology, writing-original draft; Muhammad Abbas: Supervision, methodology, writing-original draft; Miguel Vivas-Cortez: Software, formal analysis, writing-review & editing; M. R. Alharthi: Visualization, writing-review & editing; Y. S. Hamed: Software, formal analysis, writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The authors would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University for funding this work.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

[1]	Floresco SB (2015) The nucleus accumbens: an interface between cognition, emotion, and action. Annu Rev Psychol 66: 25–52. doi: 10.1146/annurev-psych-010213-115159
[2]	Diekhof EK, Falkai P, Gruber O (2008) Functional neuroimaging of reward processing and decision-making: a review of aberrant motivational and affective processing in addiction and mood disorders. Brain Res Rev 59: 164–184. doi: 10.1016/j.brainresrev.2008.07.004
[3]	Salgado S, Kaplitt MG (2015) The Nucleus Accumbens: A Comprehensive Review. Stereotact Funct Neurosurg 93: 75–93. doi: 10.1159/000368279
[4]	Mogenson GJ, Jones DL, Yim CY (1980) From motivation to action: functional interface between the limbic system and the motor system. Prog Neurobiol 14: 69–97. doi: 10.1016/0301-0082(80)90018-0
[5]	Zahm DS, Brog JS (1992) On the significance of subterritories in the "accumbens" part of the rat ventral striatum. Neuroscience 50: 751–767. doi: 10.1016/0306-4522(92)90202-D
[6]	Baliki MN, Mansour A, Baria AT, et al. (2013) Parceling human accumbens into putative core and shell dissociates encoding of values for reward and pain. J Neurosci Off J Soc Neurosci 33: 16383–16393. doi: 10.1523/JNEUROSCI.1731-13.2013
[7]	Voorn P, Brady LS, Schotte A, et al. (1994) Evidence for two neurochemical divisions in the human nucleus accumbens. Eur J Neurosci 6: 1913–1916. doi: 10.1111/j.1460-9568.1994.tb00582.x
[8]	Meredith GE (1999) The synaptic framework for chemical signaling in nucleus accumbens. Ann N Y Acad Sci 877: 140–156. doi: 10.1111/j.1749-6632.1999.tb09266.x
[9]	Francis TC, Lobo MK (2016) Emerging Role for Nucleus Accumbens Medium Spiny Neuron Subtypes in Depression. Biol Psychiatry.
[10]	Lu XY, Ghasemzadeh MB, Kalivas PW (1998) Expression of D1 receptor, D2 receptor, substance P and enkephalin messenger RNAs in the neurons projecting from the nucleus accumbens. Neuroscience 82: 767–780.
[11]	Shirayama Y, Chaki S (2006) Neurochemistry of the nucleus accumbens and its relevance to depression and antidepressant action in rodents. Curr Neuropharmacol 4: 277–291. doi: 10.2174/157015906778520773
[12]	Ding Z-M, Ingraham CM, Rodd ZA, et al. (2015) The reinforcing effects of ethanol within the nucleus accumbens shell involve activation of local GABA and serotonin receptors. J Psychopharmacol Oxf Engl 29: 725–733. doi: 10.1177/0269881115581982
[13]	Voorn P, Brady LS, Berendse HW, et al. (1996) Densitometrical analysis of opioid receptor ligand binding in the human striatum-I. Distribution of mu-opioid receptor defines shell and core of the ventral striatum. Neuroscience 75: 777–792.
[14]	Schoffelmeer ANM, Hogenboom F, Wardeh G, et al. (2006) Interactions between CB1 cannabinoid and mu opioid receptors mediating inhibition of neurotransmitter release in rat nucleus accumbens core. Neuropharmacology 51: 773–781. doi: 10.1016/j.neuropharm.2006.05.019
[15]	O'Neill RD, Fillenz M (1985) Simultaneous monitoring of dopamine release in rat frontal cortex, nucleus accumbens and striatum: effect of drugs, circadian changes and correlations with motor activity. Neuroscience 16: 49–55. doi: 10.1016/0306-4522(85)90046-6
[16]	Haralambous T, Westbrook RF (1999) An infusion of bupivacaine into the nucleus accumbens disrupts the acquisition but not the expression of contextual fear conditioning. Behav Neurosci 113: 925–940. doi: 10.1037/0735-7044.113.5.925
[17]	Levita L, Hoskin R, Champi S (2012) Avoidance of harm and anxiety: a role for the nucleus accumbens. NeuroImage 62: 189–198. doi: 10.1016/j.neuroimage.2012.04.059
[18]	Parkinson JA, Olmstead MC, Burns LH, et al. (1999) Dissociation in effects of lesions of the nucleus accumbens core and shell on appetitive pavlovian approach behavior and the potentiation of conditioned reinforcement and locomotor activity by D-amphetamine. J Neurosci Off J Soc Neurosc i 19: 2401–2411.
[19]	Feja M, Hayn L, Koch M (2014) Nucleus accumbens core and shell inactivation differentially affects impulsive behaviours in rats. Prog Neuropsychopharmacol Biol Psychiatry 54: 31–42. doi: 10.1016/j.pnpbp.2014.04.012
[20]	Fernando ABP, Murray JE, Milton AL (2013) The amygdala: securing pleasure and avoiding pain. Front Behav Neurosci 7: 190.
[21]	Di Ciano P, Cardinal RN, Cowell RA, et al. (2001) Differential involvement of NMDA, AMPA/kainate, and dopamine receptors in the nucleus accumbens core in the acquisition and performance of pavlovian approach behavior. J Neurosci Off J Soc Neurosci 21: 9471–9477.
[22]	Parkinson JA, Willoughby PJ, Robbins TW, et al. (2000) Disconnection of the anterior cingulate cortex and nucleus accumbens core impairs Pavlovian approach behavior: further evidence for limbic cortical-ventral striatopallidal systems. Behav Neurosci 114: 42–63. doi: 10.1037/0735-7044.114.1.42
[23]	Saunders BT, Robinson TE (2012) The role of dopamine in the accumbens core in the expression of Pavlovian-conditioned responses. Eur J Neurosci 36: 2521–2532. doi: 10.1111/j.1460-9568.2012.08217.x
[24]	Stopper CM, Floresco SB (2011) Contributions of the nucleus accumbens and its subregions to different aspects of risk-based decision making. Cogn Affect Behav Neurosci 11: 97–112. doi: 10.3758/s13415-010-0015-9
[25]	Deutch AY, Lee MC, Iadarola MJ (1992) Regionally specific effects of atypical antipsychotic drugs on striatal Fos expression: The nucleus accumbens shell as a locus of antipsychotic action. Mol Cell Neurosci 3: 332–341. doi: 10.1016/1044-7431(92)90030-6
[26]	Ma J, Ye N, Cohen BM (2006) Typical and atypical antipsychotic drugs target dopamine and cyclic AMP-regulated phosphoprotein, 32 kDa and neurotensin-containing neurons, but not GABAergic interneurons in the shell of nucleus accumbens of ventral striatum. Neuroscience 141: 1469–1480. doi: 10.1016/j.neuroscience.2006.05.013
[27]	Pierce RC, Kalivas PW (1995) Amphetamine produces sensitized increases in locomotion and extracellular dopamine preferentially in the nucleus accumbens shell of rats administered repeated cocaine. J Pharmacol Exp Ther 275: 1019–1029.
[28]	Park SY, Kang UG (2013) Hypothetical dopamine dynamics in mania and psychosis--its pharmacokinetic implications. Prog Neuropsychopharmacol Biol Psychiatry 43: 89–95. doi: 10.1016/j.pnpbp.2012.12.014
[29]	Mosholder AD, Gelperin K, Hammad TA, et al. (2009) Hallucinations and other psychotic symptoms associated with the use of attention-deficit/hyperactivity disorder drugs in children. Pediatrics 123: 611–616. doi: 10.1542/peds.2008-0185
[30]	Bassareo V, De Luca MA, Di Chiara G (2002) Differential Expression of Motivational Stimulus Properties by Dopamine in Nucleus Accumbens Shell versus Core and Prefrontal Cortex. J Neurosci Off J Soc Neurosci 22: 4709–4719.
[31]	Di Chiara G, Bassareo V, Fenu S, et al. (2004) Dopamine and drug addiction: the nucleus accumbens shell connection. Neuropharmacology 47: 227–241. doi: 10.1016/j.neuropharm.2004.06.032
[32]	Di Chiara G, Bassareo V (2007) Reward system and addiction: what dopamine does and doesn't do. Curr Opin Pharmacol 7: 69–76. doi: 10.1016/j.coph.2006.11.003
[33]	Basar K, Sesia T, Groenewegen H, et al. (2010) Nucleus accumbens and impulsivity. Prog Neurobiol 92: 533–557. doi: 10.1016/j.pneurobio.2010.08.007
[34]	Ahima RS, Harlan RE (1990) Charting of type II glucocorticoid receptor-like immunoreactivity in the rat central nervous system. Neuroscience 39: 579–604. doi: 10.1016/0306-4522(90)90244-X
[35]	Barrot M, Marinelli M, Abrous DN, et al. (2000) The dopaminergic hyper-responsiveness of the shell of the nucleus accumbens is hormone-dependent. Eur J Neurosci 12: 973–979. doi: 10.1046/j.1460-9568.2000.00996.x
[36]	Piazza PV, Rougé-Pont F, Deroche V, et al. (1996) Glucocorticoids have state-dependent stimulant effects on the mesencephalic dopaminergic transmission. Proc Natl Acad Sci U S A 93: 8716–8720. doi: 10.1073/pnas.93.16.8716
[37]	van der Knaap LJ, Oldehinkel AJ, Verhulst FC, et al. (2015) Glucocorticoid receptor gene methylation and HPA-axis regulation in adolescents. The TRAILS study. Psychoneuroendocrinology 58: 46–50. doi: 10.1016/j.psyneuen.2015.04.012
[38]	Bustamante AC, Aiello AE, Galea S, et al. (2016) Glucocorticoid receptor DNA methylation, childhood maltreatment and major depression. J Affect Disord 206: 181–188. doi: 10.1016/j.jad.2016.07.038
[39]	Roozendaal B, de Quervain DJ, Ferry B, et al. (2001) Basolateral amygdala-nucleus accumbens interactions in mediating glucocorticoid enhancement of memory consolidation. J Neurosci Off J Soc Neurosci 21: 2518–2525.
[40]	Schwarzer C, Berresheim U, Pirker S, et al. (2001) Distribution of the major gamma-aminobutyric acid(A) receptor subunits in the basal ganglia and associated limbic brain areas of the adult rat. J Comp Neurol 433: 526–549. doi: 10.1002/cne.1158
[41]	Van Bockstaele EJ, Pickel VM (1995) GABA-containing neurons in the ventral tegmental area project to the nucleus accumbens in rat brain. Brain Res 682: 215–221. doi: 10.1016/0006-8993(95)00334-M
[42]	Root DH, Melendez RI, Zaborszky L, et al. (2015) The ventral pallidum: Subregion-specific functional anatomy and roles in motivated behaviors. Prog Neurobiol 130: 29–70. doi: 10.1016/j.pneurobio.2015.03.005
[43]	Cho YT, Fromm S, Guyer AE, et al. (2013) Nucleus accumbens, thalamus and insula connectivity during incentive anticipation in typical adults and adolescents. NeuroImage 66: 508–521. doi: 10.1016/j.neuroimage.2012.10.013
[44]	Kelley AE, Baldo BA, Pratt WE, et al. (2005) Corticostriatal-hypothalamic circuitry and food motivation: integration of energy, action and reward. Physiol Behav 86: 773–795. doi: 10.1016/j.physbeh.2005.08.066
[45]	Rada PV, Mark GP, Hoebel BG (1993) In vivo modulation of acetylcholine in the nucleus accumbens of freely moving rats: II. Inhibition by gamma-aminobutyric acid. Brain Res 619: 105–110.
[46]	Wong LS, Eshel G, Dreher J, et al. (1991) Role of dopamine and GABA in the control of motor activity elicited from the rat nucleus accumbens. Pharmacol Biochem Behav 38: 829–835. doi: 10.1016/0091-3057(91)90250-6
[47]	Pitman KA, Puil E, Borgland SL (2014) GABA(B) modulation of dopamine release in the nucleus accumbens core. Eur J Neurosci 40: 3472–3480. doi: 10.1111/ejn.12733
[48]	Kim JH, Vezina P (1997) Activation of metabotropic glutamate receptors in the rat nucleus accumbens increases locomotor activity in a dopamine-dependent manner. J Pharmacol Exp Ther 283: 962–968.
[49]	Angulo JA, McEwen BS (1994) Molecular aspects of neuropeptide regulation and function in the corpus striatum and nucleus accumbens. Brain Res Brain Res Rev 19: 1–28. doi: 10.1016/0165-0173(94)90002-7
[50]	Vezina P, Kim JH (1999) Metabotropic glutamate receptors and the generation of locomotor activity: interactions with midbrain dopamine. Neurosci Biobehav Rev 23: 577–589. doi: 10.1016/S0149-7634(98)00055-4
[51]	Khamassi M, Humphries MD (2012) Integrating cortico-limbic-basal ganglia architectures for learning model-based and model-free navigation strategies. Front Behav Neurosci 6: 79.
[52]	Williams MJ, Adinoff B (2008) The role of acetylcholine in cocaine addiction. Neuropsychopharmacol Off Publ Am Coll Neuropsychopharmacol 33: 1779–1797. doi: 10.1038/sj.npp.1301585
[53]	Avena NM, Bocarsly ME (2012) Dysregulation of brain reward systems in eating disorders: neurochemical information from animal models of binge eating, bulimia nervosa, and anorexia nervosa. Neuropharmacology 63: 87–96. doi: 10.1016/j.neuropharm.2011.11.010
[54]	Balleine BW, Delgado MR, Hikosaka O (2007) The role of the dorsal striatum in reward and decision-making. J Neurosci Off J Soc Neurosci 27: 8161–8165. doi: 10.1523/JNEUROSCI.1554-07.2007
[55]	Liljeholm M, O'Doherty JP (2012) Contributions of the striatum to learning, motivation, and performance: an associative account. Trends Cogn Sci 16: 467–475. doi: 10.1016/j.tics.2012.07.007
[56]	Asaad WF, Eskandar EN (2011) Encoding of both positive and negative reward prediction errors by neurons of the primate lateral prefrontal cortex and caudate nucleus. J Neurosci Off J Soc Neurosci 31: 17772–17787. doi: 10.1523/JNEUROSCI.3793-11.2011
[57]	Burton AC, Nakamura K, Roesch MR (2015) From ventral-medial to dorsal-lateral striatum: neural correlates of reward-guided decision-making. Neurobiol Learn Mem 117: 51–59. doi: 10.1016/j.nlm.2014.05.003
[58]	Mattfeld AT, Gluck MA, Stark CEL (2011) Functional specialization within the striatum along both the dorsal/ventral and anterior/posterior axes during associative learning via reward and punishment. Learn Mem Cold Spring Harb N 18: 703–711. doi: 10.1101/lm.022889.111
[59]	Ikemoto S (2007) Dopamine reward circuitry: two projection systems from the ventral midbrain to the nucleus accumbens-olfactory tubercle complex. Brain Res Rev 56: 27–78. doi: 10.1016/j.brainresrev.2007.05.004
[60]	Matsumoto M, Hikosaka O (2009) Two types of dopamine neuron distinctly convey positive and negative motivational signals. Nature 459: 837–841. doi: 10.1038/nature08028
[61]	Gottfried JA, O'Doherty J, Dolan RJ (2003) Encoding predictive reward value in human amygdala and orbitofrontal cortex. Science 301: 1104–1107. doi: 10.1126/science.1087919
[62]	Stefani MR, Moghaddam B (2016) Rule learning and reward contingency are associated with dissociable patterns of dopamine activation in the rat prefrontal cortex, nucleus accumbens, and dorsal striatum. J Neurosci Off J Soc Neurosci 26: 8810–8818.
[63]	Castro DC, Cole SL, Berridge KC (2015) Lateral hypothalamus, nucleus accumbens, and ventral pallidum roles in eating and hunger: interactions between homeostatic and reward circuitry. Front Syst Neurosci 9: 90.
[64]	Peciña S, Smith KS, Berridge KC (2006) Hedonic hot spots in the brain. Neurosci Rev J Bringing Neurobiol Neurol Psychiatry 12: 500–511.
[65]	Smith KS, Berridge KC, Aldridge JW (2011) Disentangling pleasure from incentive salience and learning signals in brain reward circuitry. Proc Natl Acad Sci U S A 108: E255-264. doi: 10.1073/pnas.1101920108
[66]	Berridge KC, Robinson TE (1998) What is the role of dopamine in reward: hedonic impact, reward learning, or incentive salience? Brain Res Brain Res Rev 28: 309–369. doi: 10.1016/S0165-0173(98)00019-8
[67]	Smith KS, Berridge KC (2007) Opioid limbic circuit for reward: interaction between hedonic hotspots of nucleus accumbens and ventral pallidum. J Neurosci Off J Soc Neurosci 27: 1594–1605. doi: 10.1523/JNEUROSCI.4205-06.2007
[68]	Belujon P, Grace AA (2016) Hippocampus, amygdala, and stress: interacting systems that affect susceptibility to addiction. Ann N Y Acad Sci 1216: 114–121.
[69]	Weinshenker D, Schroeder JP (2007) There and back again: a tale of norepinephrine and drug addiction. Neuropsychopharmacol Off Publ Am Coll Neuropsychopharmacol 32: 1433–1451. doi: 10.1038/sj.npp.1301263
[70]	Everitt BJ, Hutcheson DM, Ersche KD, et al. (2007) The orbital prefrontal cortex and drug addiction in laboratory animals and humans. Ann N Y Acad Sci 1121: 576–597. doi: 10.1196/annals.1401.022
[71]	Britt JP, Benaliouad F, McDevitt RA, et al. (2012) Synaptic and behavioral profile of multiple glutamatergic inputs to the nucleus accumbens. Neuron 76: 790–803. doi: 10.1016/j.neuron.2012.09.040
[72]	Asher A, Lodge DJ (2012) Distinct prefrontal cortical regions negatively regulate evoked activity in nucleus accumbens subregions. Int J Neuropsychopharmacol 15: 1287–1294. doi: 10.1017/S146114571100143X
[73]	Ishikawa A, Ambroggi F, Nicola SM, et al. (2008) Dorsomedial prefrontal cortex contribution to behavioral and nucleus accumbens neuronal responses to incentive cues. J Neurosci Off J Soc Neurosci 28: 5088–5098. doi: 10.1523/JNEUROSCI.0253-08.2008
[74]	Connolly L, Coveleskie K, Kilpatrick LA, et al. (2013) Differences in brain responses between lean and obese women to a sweetened drink. Neurogastroenterol Motil Off J Eur Gastrointest Motil Soc 25: 579–e460. doi: 10.1111/nmo.12125
[75]	Robbins TW, Ersche KD, Everitt BJ (2008) Drug addiction and the memory systems of the brain. Ann N Y Acad Sci 1141: 1–21. doi: 10.1196/annals.1441.020
[76]	Müller CP (2013) Episodic memories and their relevance for psychoactive drug use and addiction. Front Behav Neurosci 7: 34.
[77]	Naqvi NH, Bechara A (2010) The insula and drug addiction: an interoceptive view of pleasure, urges, and decision-making. Brain Struct Funct 214: 435–450. doi: 10.1007/s00429-010-0268-7
[78]	Satterthwaite TD, Kable JW, Vandekar L, et al. (2015) Common and Dissociable Dysfunction of the Reward System in Bipolar and Unipolar Depression. Neuropsychopharmacol Off Publ Am Coll Neuropsychopharmacol 40: 2258–2268. doi: 10.1038/npp.2015.75
[79]	Surguladze S, Brammer MJ, Keedwell P, et al. (2005) A differential pattern of neural response toward sad versus happy facial expressions in major depressive disorder. Biol Psychiatry 57: 201–209. doi: 10.1016/j.biopsych.2004.10.028
[80]	Elliott R, Rubinsztein JS, Sahakian BJ, et al. (2002) The neural basis of mood- congruent processing biases in depression. Arch Gen Psychiatry 59: 597–604. doi: 10.1001/archpsyc.59.7.597
[81]	Keedwell PA, Andrew C, Williams SCR, et al. (2005) A double dissociation of ventromedial prefrontal cortical responses to sad and happy stimuli in depressed and healthy individuals. Biol Psychiatry 58: 495–503. doi: 10.1016/j.biopsych.2005.04.035
[82]	Yurgelun-Todd DA, Gruber SA, Kanayama G, et al. (2000) fMRI during affect discrimination in bipolar affective disorder. Bipolar Disord 2: 237–248. doi: 10.1034/j.1399-5618.2000.20304.x
[83]	Caseras X, Murphy K, Lawrence NS, et al. (2015) Emotion regulation deficits in euthymic bipolar I versus bipolar II disorder: a functional and diffusion-tensor imaging study. Bipolar Disord 17: 461–470. doi: 10.1111/bdi.12292
[84]	Redlich R, Dohm K, Grotegerd D, et al. (2015) Reward Processing in Unipolar and Bipolar Depression: A Functional MRI Study. Neuropsychopharmacol Off Publ Am Coll Neuropsychopharmacol 40: 2623–2631. doi: 10.1038/npp.2015.110
[85]	Namburi P, Beyeler A, Yorozu S, et al. (2015) A circuit mechanism for differentiating positive and negative associations. Nature 520: 675–678. doi: 10.1038/nature14366
[86]	Mahon K, Burdick KE, Szeszko PR (2010) A Role for White Matter Abnormalities in the Pathophysiology of Bipolar Disorder. Neurosci Biobehav Rev 34: 533–554. doi: 10.1016/j.neubiorev.2009.10.012
[87]	Franklin TR, Wang Z, Wang J, et al. (2007) Limbic activation to cigarette smoking cues independent of nicotine withdrawal: a perfusion fMRI study. Neuropsychopharmacol Off Publ Am Coll Neuropsychopharmacol 32: 2301–2309. doi: 10.1038/sj.npp.1301371
[88]	Garavan H, Pankiewicz J, Bloom A, et al. (2000) Cue-induced cocaine craving: neuroanatomical specificity for drug users and drug stimuli. Am J Psychiatry 157(11): 1789–1798.
[89]	Diekhof EK, Falkai P, Gruber O (2008) Functional neuroimaging of reward processing and decision-making: a review of aberrant motivational and affective processing in addiction and mood disorders. Brain Res Rev 59: 164–184. doi: 10.1016/j.brainresrev.2008.07.004
[90]	White NM, Packard MG, McDonald RJ (2013) Dissociation of memory systems: The story unfolds. Behav Neurosci 127: 813–834. doi: 10.1037/a0034859
[91]	Wrase J, Schlagenhauf F, Kienast T, et al. (2007) Dysfunction of reward processing correlates with alcohol craving in detoxified alcoholics. NeuroImage 35: 787–794. doi: 10.1016/j.neuroimage.2006.11.043
[92]	Drevets WC, Gautier C, Price JC, et al. (2001) Amphetamine-induced dopamine release in human ventral striatum correlates with euphoria. Biol Psychiatry 49: 81–96. doi: 10.1016/S0006-3223(00)01038-6
[93]	Ding YS, Logan J, Bermel R, et al. (2000) Dopamine receptor-mediated regulation of striatal cholinergic activity: positron emission tomography studies with norchloro[18F]fluoroepibatidine. J Neurochem 74: 1514–1521.
[94]	Greenberg BD, Gabriels LA, Malone DA, et al. (2010) Deep brain stimulation of the ventral internal capsule/ventral striatum for obsessive-compulsive disorder: worldwide experience. Mol Psychiatry 15: 64–79. doi: 10.1038/mp.2008.55
[95]	Denys D, Mantione M, Figee M, van den Munckhof P, et al. (2010) Deep brain stimulation of the nucleus accumbens for treatment-refractory obsessive-compulsive disorder. Arch Gen Psychiatry 67: 1061-1068. doi: 10.1001/archgenpsychiatry.2010.122
[96]	Scott DJ, Stohler CS, Egnatuk CM, et al. (2008) Placebo and nocebo effects are defined by opposite opioid and dopaminergic responses. Arch Gen Psychiatry 65: 220–231. doi: 10.1001/archgenpsychiatry.2007.34

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Nucleus Accumbens and Its Role in Reward and Emotional Circuitry: A Potential Hot Mess in Substance Use and Emotional Disorders

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Nucleus Accumbens and Its Role in Reward and Emotional Circuitry: A Potential Hot Mess in Substance Use and Emotional Disorders

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