On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

Khalid K. Ali; K. R. Raslan; Amira Abd-Elall Ibrahim; Mohamed S. Mohamed; Khalid K. Ali; K. R. Raslan; Amira Abd-Elall Ibrahim; Mohamed S. Mohamed

doi:10.3934/math.2023925

AIMS Mathematics

2023, Volume 8, Issue 8: 18206-18222. doi: 10.3934/math.2023925

Previous Article Next Article

Research article

On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

1.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt
2.
October High Institute for Engineering and Technology
3.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 12 March 2023 Revised: 08 May 2023 Accepted: 14 May 2023 Published: 26 May 2023
MSC : 26Axx, 34Axx, 45Jxx, 45Lxx

The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.

Keywords:

Citation: Khalid K. Ali, K. R. Raslan, Amira Abd-Elall Ibrahim, Mohamed S. Mohamed. On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type[J]. AIMS Mathematics, 2023, 8(8): 18206-18222. doi: 10.3934/math.2023925

Related Papers:

[1]	Rubayyi T. Alqahtani, Jean C. Ntonga, Eric Ngondiep . Stability analysis and convergence rate of a two-step predictor-corrector approach for shallow water equations with source terms. AIMS Mathematics, 2023, 8(4): 9265-9289. doi: 10.3934/math.2023465
[2]	Abdulhamed Alsisi . The powerful closed form technique for the modified equal width equation with numerical simulation. AIMS Mathematics, 2025, 10(5): 11071-11085. doi: 10.3934/math.2025502
[3]	Kamel Mohamed, H. S. Alayachi, Mahmoud A. E. Abdelrahman . The mR scheme to the shallow water equation with horizontal density gradients in one and two dimensions. AIMS Mathematics, 2023, 8(11): 25754-25771. doi: 10.3934/math.20231314
[4]	Mingming Li, Shaoyong Lai . The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model. AIMS Mathematics, 2024, 9(1): 1772-1782. doi: 10.3934/math.2024086
[5]	Mohammad Alqudah, Safyan Mukhtar, Haifa A. Alyousef, Sherif M. E. Ismaeel, S. A. El-Tantawy, Fazal Ghani . Probing the diversity of soliton phenomena within conformable Estevez-Mansfield-Clarkson equation in shallow water. AIMS Mathematics, 2024, 9(8): 21212-21238. doi: 10.3934/math.20241030
[6]	Christelle Dleuna Nyoumbi, Antoine Tambue . A fitted finite volume method for stochastic optimal control problems in finance. AIMS Mathematics, 2021, 6(4): 3053-3079. doi: 10.3934/math.2021186
[7]	Mostafa M. A. Khater, S. H. Alfalqi, J. F. Alzaidi, Samir A. Salama, Fuzhang Wang . Plenty of wave solutions to the ill-posed Boussinesq dynamic wave equation under shallow water beneath gravity. AIMS Mathematics, 2022, 7(1): 54-81. doi: 10.3934/math.2022004
[8]	M. Mossa Al-Sawalha, Rasool Shah, Adnan Khan, Osama Y. Ababneh, Thongchai Botmart . Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives. AIMS Mathematics, 2022, 7(10): 18334-18359. doi: 10.3934/math.20221010
[9]	Youngjin Hwang, Jyoti, Soobin Kwak, Hyundong Kim, Junseok Kim . An explicit numerical method for the conservative Allen–Cahn equation on a cubic surface. AIMS Mathematics, 2024, 9(12): 34447-34465. doi: 10.3934/math.20241641
[10]	Jin Hong, Shaoyong Lai . Blow-up to a shallow water wave model including the Degasperis-Procesi equation. AIMS Mathematics, 2023, 8(11): 25409-25421. doi: 10.3934/math.20231296

Abstract

1. Introduction

Hydraulic jumps are frequently observed in lab studies, river flows, and coastal environments. The shallow water equations, commonly known as Saint-Venant equations, have important applications in oceanography and hydraulics. These equations are used to represent fluid flow when the depth of the fluid is minor in comparison to the horizontal scale of the flow field fluctuations ^[1]. They have been used to simulate flow at the atmospheric and ocean scales, and they have been used to forecast tsunamis, storm surges, and flow around constructions, among other things ^[2,3]. Shallow water is a thin layer of constant density fluid in hydrostatic equilibrium that is bordered on the bottom by a rigid surface and on the top by a free surface ^[4].

Nonlinear waves have recently acquired importance due to their capacity to highlight several complicated phenomena in nonlinear research with spirited applications ^[5,6,7,8]. Teshukov in ^[9] developed the equation system that describes multi-dimensional shear shallow water (SSW) flows. Some novel solitary wave solutions for the ill-posed Boussinesq dynamic wave model under shallow water beneath gravity were constructed ^[10]. The effects of vertical shear, which are disregarded in the traditional shallow water model, are included in this system to approximate shallow water flows. It is a non-linear hyperbolic partial differential equations (PDE) system with non-conservative products. Shocks, rarefactions, shear, and contact waves are all allowed in this model. The SSW model can capture the oscillatory character of turbulent hydraulic leaps, which corrects the conventional non-linear shallow water equations' inability to represent such occurrences ^[11]. Analytical and numerical techniques for solving such problems have recently advanced; for details, see ^{[12,13,14,15,16,17]} and the references therein.

The SSW model consists of six non-conservative hyperbolic equations, which makes its numerical solution challenging since the concept of a weak solution necessitates selecting a path that is often unknown. The physical regularization mechanism determines the suitable path, and even if one knows the correct path, it is challenging to construct a numerical scheme that converges to the weak solution since the solution is susceptible to numerical viscosity ^[18].

In the present work, we constructed the generalized Rusanov (G. Rusanov) scheme to solve the SSW model in 1D of space. This technique is divided into steps for predictors and correctors ^{[19,20,21,22,23]}. The first one includes a numerical diffusion control parameter. In the second stage, the balance conservation equation is retrieved. Riemann solutions were employed to determine the numerical flow in the majority of the typical schemes. Unlike previous schemes, the interesting feature of the G. Rusanov scheme is that it can evaluate the numerical flow even when the Riemann solution is not present, which is a very fascinating advantage. In actuality, this approach may be applied as a box solver for a variety of non-conservative law models.

The remainder of the article is structured as follows. Section 2 offers the non-conservative shear shallow water model. Section 3 presents the structure of the G. Rusanov scheme to solve the 1D SSW model. Section 4 shows that the G. Rusanov satisfies the C-property. Section 5 provides several test cases to show the validation of the G. Rusanov scheme versus Rusanov, Lax-Friedrichs, and reference solutions. Section 6 summarizes the work and offers some conclusions.

2. Mathematical formulation for the shear shallow water model

The one-dimensional SSW model without a source term is

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + \frac{\partial \mathit{F}(\mathrm{W})}{\partial x}+\mathrm{K( \mathrm{W})}\frac{\partial \mathrm{h}}{\partial x} = \mathrm{S(\mathrm{W})}, \end{equation}$

(2.1)

$\begin{equation*} \label{equa3} \mathrm{W} = \left(\begin{array}{c} \mathrm{h} \\ \mathrm{h v_1}\\ \mathrm{h v_2}\\ \mathrm{E_{11}}\\ \mathrm{E_{12}}\\ \mathrm{E_{22}}\\ \end{array}\right),\qquad {\mathit{F}(\mathrm{W})} = \left(\begin{array}{c} \mathrm{h v_1} \\ \mathrm{R_{11}}+\mathrm{h v^2_{1}}+g\frac{\mathrm{h^2}}{2} \\ \mathrm{R_{12}}+\mathrm{h v_{1}v_2} \\ (\mathrm{E_{11}}+\mathrm{R_{11}})\mathrm{v_1} \\ \mathrm{E_{12} v_1}+\frac{1}{2}(\mathrm{R_{11}v_2}+\mathrm{R_{12}}v_1) \\ \mathrm{E_{22} v_1}+\mathrm{R_{12}v_2} \\ \end{array}\right), \end{equation*}$

$\begin{equation*} \label{equa3k} \mathrm{K(\mathrm{W})} = \left(\begin{array}{c} 0 \\ 0\\ 0\\ g\mathrm{h v_1}\\ \frac{1}{2}g\mathrm{hv_2}\\ 0\\ \end{array}\right), \mathrm{S(\mathrm{W})} = \left(\begin{array}{c} 0 \\ -gh \frac{\partial \mathrm{B}}{\partial x}- C_f|v|v_1\\ -C_f|v|v_2\\ -ghv_1 \frac{\partial \mathrm{B}}{\partial x}+\frac{1}{2} D_{11} -C_f|v|v_{1}^2\\\\ -\frac{1}{2} ghv_2 \frac{\partial \mathrm{B}}{\partial x}+\frac{1}{2} D_{12}-C_f|v|v_1v_2\\\\ \frac{1}{2} D_{22}-C_f|v|v_{2}^2\\ \end{array}\right), \end{equation*}$

while $\mathrm{R_{11}}$ and $\mathrm{R_{12}}$ are defined by the following tensor $\mathrm{R_{ij}} = \mathrm{h p_{i j}}$ , and also, $\mathrm{E_{11}}$ , $\mathrm{E_{12}}$ and $\mathrm{E_{22}}$ are defined by the following tensor $\mathrm{E_{i j}} = \frac{1}{2}\mathrm{R_{ij}}+\frac{1}{2}\mathrm{\mathrm{h v_i v_j}}$ with $i\ge 1, j\le 2$ . We write the previous system 2.1 in nonconservative form with non-dissipative $(C_f = 0, D_{11} = D_{12} = D{22} = 0)$ , see ^[24], which can be written as follows

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + (\nabla{\mathit{F}(\mathrm{W})} +\mathit{C(\mathrm{W}})) \frac{\partial \mathrm{W}}{\partial x} = \mathrm{S_1(\mathrm{W})}. \end{equation}$

(2.2)

Also, we can write the previous system as the following

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + \mathit{A(\mathrm{W})} \frac{\partial \mathrm{W}}{\partial x} = \mathrm{S_1(\mathrm{W})} \end{equation}$

(2.3)

with

$\begin{equation*} \mathit{A(\mathrm{W})} = \begin{bmatrix} 0 & 1 & 0 & 0 &0 & 0 \\ \mathrm{gh} & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ -3\frac{\mathrm{E_{11} v_1}}{\mathrm{h}}+2v^{3}_1 + \mathrm{g h v_2} &3\frac{\mathrm{E_{11}}}{\mathrm{h}}-3v^{2}_1 & 0 & 3 \mathrm{v_1} & 0 & 0 \\ -2\frac{\mathrm{E_{12} v_1}}{\mathrm{h}}-\frac{\mathrm{E_{11}v_2}}{\mathrm{h}}+ \mathrm{2v^{2}_1 v_2}+\frac{\mathrm{g h v_2}}{2} & 2\frac{\mathrm{E_{12}}}{h}-2\mathrm{v_1 v_2} & \frac{\mathrm{E_{11}}}{h}- \mathrm{v^{2}_1} & \mathrm{v_2} & 2\mathrm{v_1} & 0 \\ -\frac{\mathrm{E_{22} v_1}}{\mathrm{h}}-2\frac{\mathrm{E_{12}v_2}}{\mathrm{h}}+ \mathrm{v^{2}_2 v_1}& \frac{\mathrm{E_{22}}}{\mathrm{h}}+ \mathrm{v^{2}_2}& 2\frac{\mathrm{E_{12}}}{\mathrm{h}}-2\mathrm{v_2 v_1}& 0 &2\mathrm{v_2}& \mathrm{v_1} \\ \end{bmatrix} \end{equation*}$

and

$\begin{equation*} \label{equa3km} \mathrm{S_1(\mathrm{W})} = \left(\begin{array}{c} 0 \\ -gh \frac{\partial \mathrm{B}}{\partial x}\\ 0\\ -ghv_1 \frac{\partial \mathrm{B}}{\partial x}\\\\ -\frac{1}{2} ghv_2 \frac{\partial \mathrm{B}}{\partial x}\\\\ 0\\ \end{array}\right). \end{equation*}$

System (2.3) is a hyperbolic system and has the following eigenvalues

$\lambda_1 = \lambda_2 = v_1, \quad \lambda_3 = v_1-\sqrt{2\frac{\mathrm{E_{11}}}{h}-v^2_1}, \quad \lambda_4 = v_1+\sqrt{2\frac{\mathrm{E_{11}}}{h}-v^2_1},$

$\lambda_5 = v_1-\sqrt{\mathrm{g h}+3(v^2_1-2\frac{\mathrm{E_{11}}}{h})}, \quad \lambda_6 = v_1+\sqrt{\mathrm{g h}+3(v^2_1-2\frac{\mathrm{E_{11}}}{h})}.$

3. The G. Rusanov scheme for SSWE

In order deduce the G. Rusanov scheme, we rewrite the system (2.1) as follows

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + \frac{\partial \mathit{F}(\mathrm{W})}{\partial x} = -\mathrm{K( \mathrm{W})}\frac{\partial \mathrm{h}}{\partial x}+\mathrm{S_1(\mathrm{W})} = \mathrm{Q(\mathrm{W})}. \end{equation}$

(3.1)

Integrating Eq (3.1) over the domain $[t_n, t_{n+1}]\times[x_{i-\frac{1}{2}}, x_{i+\frac{1}{2}}]$ gives the finite-volume scheme

$\begin{equation} \mathrm{W_i^{n+1}} = \mathrm{W_i^n} - \frac{\Delta{t}}{\Delta{x}}\left(\mathit{F}\left(\mathrm{W^n_{i+1/2}}\right) - \mathit{F}\left(\mathrm{W^n_{i-1/2}}\right)\right)+\Delta t {\mathrm{Q_{i}}}^{n} \end{equation}$

(3.2)

in the interval $[x_{i-1/2}, x_{i+1/2}]$ at time $t_n$ . $\mathrm{W_i^n}$ represents the average value of the solution $\mathrm{W}$ as follows

$\mathrm{W_i^n} = \frac{1}{\Delta{x}}\int^{x_{i+1/2}}_{x_{i-1/2}}\mathrm{W}(t_{n},x)\;dx,$

and $\mathit{F}(\mathrm{W_{i\pm1/2}^n)}$ represent the numerical flux at time $t_n$ and space $x = x_{i\pm1/2}$ . It is necessary to solve the Riemann problem at $x_{i+1/2}$ interfaces due to the organization of the numerical fluxes. Assume that for the first scenario given below, there is a self-similar Riemann problem solution associated with Eq (3.1)

$\begin{equation} \mathrm{W(x,0)} = \begin{cases} \mathrm{W_L}, & \text{if}\quad x < 0,\\[2ex] \mathrm{W_R}, & \text{if}\quad x > 0 \end{cases} \end{equation}$

(3.3)

is supplied by

$\begin{equation} \mathrm{W(t,x)} = \mathrm{R_s}\left(\frac xt,\mathrm{W_L},\mathrm{W_R}\right). \end{equation}$

(3.4)

The difficulties with discretization of the source term in (3.2) could arise from singular values of the Riemann solution at the interfaces. In order to overcome these challenges and reconstruct a $\mathrm{W_{i+1/2}^n}$ approximation, we created a finite-volume scheme in ^{[19,20,21,22,23,25,26,27]} for numerical solutions of conservation laws including source terms and without source terms. The principal objective here is building the intermediate states $\mathrm{W_{i\pm1/2}^n}$ to be utilized in the corrector stage (3.2). The following is obtained by integrating Eq (3.1) through a control volume $[t_n, t_{n}+\theta^n_{i+1/2}]\times[x^-, x^+]$ , that includes the point $(t_n, x_{i+1/2})$ , with the objective to accomplish this:

$\begin{eqnarray} \int_{x^-}^{x^+}\mathrm{W(t_{n}+\theta^n_{i+1/2},x)}\;dx & = & \Delta x^-\mathrm{W_i^n} + \Delta x^+\mathrm{W_{i+1}^n} -\\ &&\theta^n_{i+1/2}\Bigl(\mathit{F}(\mathrm{W^n_{i+1}}) \!-\! \mathit{F}(\mathrm{W^n_{i}})\Bigr)\!+\!\theta^n_{i+1/2}(\Delta x^{-}\!-\!\Delta x^+)\mathrm{Q^n_{i\!+\!\frac{1}{2}}} \end{eqnarray}$

(3.5)

with $\mathrm{W_{i\pm1/2}^n}$ denoting the approximation of Riemann solution $\mathrm{R_s}$ across the control volume $[x^-, x^+]$ at time $t_{n}+\theta^n_{i+1/2}$ . While calculating distances $\Delta x^-$ and $\Delta x^+$ as

$\Delta x^- = \left|x^- - x_{i+1/2}\right|,\qquad \Delta x^+ = \left|x^+ - x_{i+1/2}\right|,$

$\mathrm{Q^n_{i+\frac{1}{2}}}$ closely resembles the average source term $\mathrm{Q}$ and is given by the following

$\begin{equation} \mathrm{Q^n_{i+\frac{1}{2}}} = \frac{1}{\theta^n_{i+\frac{1}{2}}(\Delta x^- + \Delta x^+)} \int_{t_n}^{t_n+\theta^n_{i+\frac{1}{2}} }\int_{x^-}^{x^+}{\mathrm{Q}}(\mathrm{W})dt\;dx . \end{equation}$

(3.6)

Selecting $x^- = x_i$ , $x^+ = x_{i+1}$ causes the Eq (3.5) to be reduced to the intermediate state, which given as follows

$\begin{equation} \mathrm {W_{i+1/2}^{n}} = \frac12\left(\mathrm{W_i^n} + \mathrm{W_{i+1}^n}\right) - \frac{\theta^n_{i+1/2}}{\Delta{x}}\Bigl(\mathit{F}(\mathrm{W^n_{i+1}}) - \mathit{F}(\mathrm{W^n_{i}})\Bigr)+\theta^n_{i+\frac{1}{2}}\mathrm{Q^n_{i+\frac{1}{2}}}, \end{equation}$

(3.7)

where the approximate average value of the solution $\mathrm{W}$ in the control domain $[t_n, t_{n}+\theta^n_{i+1/2}]\times[x_{i}, x_{i+1}]$ is expressed by $\mathrm{W_{i+1/2}^n}$ as the following

$\begin{equation} \mathrm{{W}_{i+1/2}^n} = \frac{1}{\Delta x}\int_{x_i}^{x_{i+1}}{W}(x,t_n+\theta^n_{i+1/2})dx \end{equation}$

(3.8)

by selecting $\theta^n_{i+1/2}$ as follows

$\begin{equation} \theta^n_{i+1/2} = {\alpha^n_{i+1/2}}\bar{\theta}_{i+1/2}\quad ,\qquad \bar{\theta}_{i+1/2} = \frac{\Delta x}{2S_{i+1/2}^n}. \end{equation}$

(3.9)

This choice is contingent upon the results of the stability analysis^[19]. The following represents the local Rusanov velocity

$\begin{equation} S_{i+1/2}^n = \underset{k = 1,...,K}{max}(max(\mid \lambda^n_{k,i} \mid,\mid \lambda^n_{k,i+1} \mid )), \end{equation}$

(3.10)

where $\alpha^n_{i+1/2}$ is a positive parameter, and $\lambda^n_{k, i}$ represents the $k^{th}$ eigenvalues in (2.3) evaluated at the solution state $\mathrm{W^n_{i}}$ . Here, in our case, $k = 6$ for the shear shallow water model. One can use the Lax-Wendroff technique again for $\alpha^n_{i+1/2} = \frac{\Delta t}{\Delta x S^n_{i+1/2}}$ . Choosing the slope $\alpha^n_{i+1/2} = \tilde\alpha_{i+1/2}^n$ , the proposed scheme became a first-order scheme, where

$\begin{equation} \tilde\alpha_{i+1/2}^n = \frac{S_{i+1/2}^n}{s_{i+1/2}^n}, \end{equation}$

(3.11)

and

$\begin{equation} s_{i+1/2}^n = \underset{k = 1,...,K}{\min}(\max(\mid \lambda^n_{k,i} \mid,\mid \lambda^n_{k,i+1} \mid )). \end{equation}$

(3.12)

In this case, the control parameter can be written as follows:

$\begin{equation} \mathrm{\alpha_{i+1/2}^n} = \mathrm{\tilde\alpha_{i+1/2}^n} + \sigma_{i+1/2}^n\Phi\left(\mathrm{r^m_{i+1/2}}\right), \end{equation}$

(3.13)

where $\mathrm{\tilde \alpha_{i+1/2}^n}$ is given by (3.11), and $\Phi_{i+1/2} = \Phi\left(r_{i+1/2}\right)$ is a function that limits the slope. While for

$\begin{equation*} \mathrm{r_{i+1/2}} = \frac{\mathrm{W_{i+1-q}}-\mathrm{W_{i-q}}}{\mathrm{W_{i+1}}-\mathrm{W_{i}}},\qquad q = \mathrm{sgn}\left[\mathit{F'}(X^{n+1},\mathrm{W^n_{i+1/2}})\right] \end{equation*}$

and

$\begin{equation*} \sigma_{i+1/2}^n = \frac{\Delta t}{\Delta x}S_{i+1/2}^n - \frac{S_{i+1/2}^n}{s_{i+1/2}^n}. \end{equation*}$

One may use any slope limiter function, including minmod, superbee, and Van leer ^[28] and ^[29]. At the end, the G. Rusanov scheme for Eq (2.3) can be written as follows

$\begin{equation} \left \{ \begin{array}{ll} \mathrm{W}^n_{i+\frac{1}{2}} = \frac{1}{2}( \mathrm{W}^n_i+ \mathrm{W}^n_{i+1})-\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{j+\frac{1}{2}}} \left [\mathit{F}( \mathrm{W}^n_{i+1})-\mathit{F}( \mathrm{W}^n_i) \right ]+\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{j+\frac{1}{2}}} \mathrm{Q^n_{i+\frac{1}{2}}},\\ \mathrm{W}_{i}^{n+1} = \mathrm{W}_{i}^{n}-r^{n}\left[ \mathit{F}\left( \mathrm{W}_{i+{\frac{1}{2}}}^{n}\right) -\mathit{F}\left( \mathrm{W}_{i-{\frac{1}{2}}}^{n}\right) \right]+\Delta t^n {\mathrm{Q_{i}}}^{n}. \end{array} \right. \end{equation}$

(3.14)

4. How the source term is treated

With a definition of the C-property ^[30], the source term in (2.1) is discretized in the G. Rusanov method in a way that is well-balanced with the discretization of the flux gradients. According to ^[30,31], if the following formulas hold, a numerical scheme is said to achieve the C-property for the system (3.1).

$\begin{equation} (hp_{11})^n = \mbox{ constant},\quad \quad h^n+B = \mbox{ constant}\quad \mbox{and}\quad v^n_1 = v^n_2 = 0. \end{equation}$

(4.1)

When we set $v_1 = v_2 = 0$ in the stationary flow at rest, we get system (3.1), which can be expressed as follows.

$\begin{equation} \begin{array}{ll} \frac{\partial}{\partial t} \left(\begin{array}{c} h \\ 0 \\ 0\\ \frac{1}{2}h p_{11}\\ \frac{1}{2}h p_{12}\\ \frac{1}{2}h p_{22}\\ \end{array}\right)+ \frac{\partial}{\partial x} \left(\begin{array}{c} 0 \\ h p_{11}+\frac{1}{2}gh^2\\ h p_{12}\\ 0\\ 0\\ 0\\ \end{array}\right) = \left(\begin{array}{c} 0 \\ -gh\frac{\partial B}{\partial x} \\ 0\\ 0\\ 0\\ 0\\ \end{array}\right) \end{array} = Q(x,t). \end{equation}$

(4.2)

We can express the predictor stage (3.14) as follows after applying the G. Rusanov scheme to the previous system

$\begin{equation} \mathbb{W}^n_{i+\frac{1}{2}} = \left(\begin{array}{c} \frac{1}{2}({h}^n_i+{h}^n_{i+1}) \\ -\frac{\alpha^n_{i+\frac{1}{2}}}{4 S^n_{i+\frac{1}{2}}}({h}^n_i+{h}^n_{i+1})[({h}^n_{i+1}+ B_{i+1}) - ({h}^n_{i}+ B_{i})]- \frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{i+\frac{1}{2}}}( (hp_{11})^n_{i+1}- (hp_{11})^n_{i}\\ -\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{i+\frac{1}{2}}} ((hp_{12})^n_{i+1}- (hp_{12})^n_{i} )\\ \frac{1}{4}((hp_{11})^n_{i+1}+ (hp_{11})^n_{i})\\ \frac{1}{4}((hp_{12})^n_{i+1}+(hp_{12})^n_{i})\\ \frac{1}{4}((hp_{22})^n_{i+1}+(hp_{22})^n_{i})\\ \end{array}\right).\\ \end{equation}$

(4.3)

Also, we can write the previous equations as follows

$\begin{equation} \mathrm{W}^n_{i+\frac{1}{2}} = \left(\begin{array}{c} {h}^n_{i+\frac{1}{2}} \\ 0\\ -\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{i+\frac{1}{2}}} ((hp_{12})^n_{i+1}- (hp_{12})^n_{i} )\\ \frac{1}{2}(hp_{11})^n_{i+\frac{1}{2}}\\ \frac{1}{2}(hp_{12})^n_{i+\frac{1}{2}}\\ \frac{1}{2}(hp_{22})^n_{i+\frac{1}{2}}\\ \end{array}\right),\\ \end{equation}$

(4.4)

and the stage of the corrector updates the solution to take on the desired form.

$\begin{equation} \left(\begin{array}{c} (h)^{n+1}_i \\ (h v_1)^{n+1}_i\\ (h v_2)^{n+1}_i \\ (E_{11})^{n+1}_i\\ (E_{12})^{n+1}_i\\ (E_{22})^{n+1}_i \end{array}\right) = \left(\begin{array}{c} (h)^{n}_i \\ (h v_1)^{n}_i\\ (h v_2)^{n}_i \\ (E_{11})^{n}_i\\ (E_{12})^{n}_i\\ (E_{22})^{n}_i\\ \end{array}\right)-\frac{ \Delta t_n}{ \Delta x} \left(\begin{array}{c} 0 \\ \frac{g}{2} ( ({h}^n_{i+\frac{1}{2}})^2-({h}^n_{i-\frac{1}{2}})^2 )\\ (h p_{12})^n_{i+\frac{1}{2}}-(h p_{12})^n_{i-\frac{1}{2}}\\ 0\\ 0\\ 0 \end{array}\right) \end{equation}$

(4.5)

$\begin{equation*} + \left(\begin{array}{c} 0 \\ \Delta t_n Q^n_i\\ 0 \\ 0\\ 0\\ 0 \end{array}\right). \end{equation*}$

The solution is stationary when $\mathrm{W}^{n+1}_i = \mathrm{W}^n_i$ and this leads us to rewrite the previous system (4.5) as follows

$\frac{\Delta t_n}{2 \Delta x }g((h^n_{i+\frac{1}{2}})^2-(h^n_{i-\frac{1}{2}})^2 ) -\Delta t_n Q^n_i.$

This lead to the following

$\begin{equation} Q^n_i = -\frac{g}{8 \Delta x}(h^n_{i+1}+2 h^n_i+h^n_{i-1})(B_{i+1}-B_{i-1}). \end{equation}$

(4.6)

Afterward, when the source term in the corrector stage is discretized in the earlier equations, this leads to the G. Rusanov satisfying the C-property.

5. Numerical simulation for the SSW model

In order to simulate the SSW system numerically, we provide five test cases without a source term using the G. Rusanov, Rusanov, and Lax-Friedrichs schemes to illustrate the effectiveness and precision of the suggested G. Rusanov scheme. The computational domain for all cases is $\mathbb{L} = [0, 1]$ divided into $300$ gridpoints, and the final time is $t = 0.5s$ , except for test case $2$ , where the final time is $t = 10s$ . We evaluate the three schemes with the reference solution on the extremely fine mesh of $30000$ cells that was produced by the traditional Rusanov scheme. Also, we provide one test case with a source term using the G. Rusanov scheme in the domain $[0, 1]$ divided into $200$ gridpoints and final time $t = 0.5$ . We select the stability condition ^[19] in the following sense

$\begin{equation} \mathrm{\Delta t} = \mathrm{CFL}\frac{\mathrm{\Delta x}}{\max\limits_{i}\bigl(\left|\mathrm{\alpha_{i+\frac{1}{2}}^n}\mathrm{S_{i+\frac{1}{2}}^n}\right|\bigr)}, \end{equation}$

(5.1)

where a constant $\mathrm{CFL} = 0.5$ .

5.1. Test case 1

This test case was studied in ^[24] and ^[32], and the initial condition is given by

$\begin{equation} (h, v_1, v_2, p_{11}, p_{22}, p_{12}) = \begin{cases} (0.01,\quad 0.1,\quad 0.2,\quad 0.04,\quad 0.04,\quad 1\times 10^{-8}) \quad \mbox{if} \quad x \le \frac{\mathbb{L}}{2},\\[2ex] (0.02,\quad 0.1,\quad -0.2,\quad 0.04,\quad 0.04,\quad 1\times 10^{-8} ) \quad \mbox{if} \quad x > \frac{\mathbb{L}}{2}.\\ \end{cases} \end{equation}$

(5.2)

The solution include five waves, which are represented by $1-$ shock and $6-$ rarefaction. We compare the numerical results with the reference solution, which is calculated with the Rusanov scheme with very fine mesh of $30000$ cells. Figures 1–3 show the numerical results and variation of parameter of control. We note that all waves are captured by this scheme and the numerical solution agrees with the reference solution. Table 1 shows the CPU time of computation for the three schemes by using a Dell i5 laptop, CPU 2.5 GHZ.

Figure 1. High water

$h$ and velocity

$v_1$ .

Scheme	G. Rusanov	Rusanov	Lax-Friedrichs
CPU time	0.068005	0.056805	0.0496803

[1]	P. N. Duc, E. Nane, O. Nikan, N. A. Tuan, Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements, AIMS Mathematics, 7 (2022), 12620–12634. https://doi.org/10.3934/math.2022698 doi: 10.3934/math.2022698
[2]	T. N. Thach, N. H. Tuan, Stochastic pseudo-parabolic equations with fractional derivative and fractional Brownian motion, Stoch. Anal. Appl., 40 (2022), 328–351. https://doi.org/10.1080/07362994.2021.1906274 doi: 10.1080/07362994.2021.1906274
[3]	A. T. Nguyen, N. H. Tuan, C.Yang, On Cauchy problem for fractional parabolic-elliptic Keller-Segel model, Adv. Nonlinear Anal., 12 (2023), 97–116. https://doi.org/10.1515/anona-2022-0256 doi: 10.1515/anona-2022-0256
[4]	N. H. Tuan, N. D. Phuong, T. N. Thach, New well-posedness results for stochastic delay Rayleigh-Stokes equations, AIMS Mathematics, 28 (2023), 347–358. https://doi.org/10.3934/dcdsb.2022079 doi: 10.3934/dcdsb.2022079
[5]	F. M. Gaafar. The Existence of solution for a nonlocal problem of an implicit fractional-order differential equation, J. Egy. Math. Soc., 26 (2018), 184–195. https://doi.org/10.21608/JOEMS.2018.9473
[6]	S. Abbas, M. Benchohra, J. J. Nieto, Caputo-Fabrizio fractional differential equations with instantaneous impulses, AIMS Mathematics, 6 (2021), 2932–2946. https://doi.org/10.3934/math.2021177 doi: 10.3934/math.2021177
[7]	S. K. Ntouyas, M. E. Samei, Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus, Adv. Differ. Equ., 2019 (2019), 475. https://doi.org/10.1186/s13662-019-2414-8 doi: 10.1186/s13662-019-2414-8
[8]	M. E. Samei, L. Karimi, M. K. A. Kaabar, To investigate a class of multi-singular pointwise defined fractional q–integro-differential equation with applications, AIMS Mathematics, 7 (2022), 7781–7816. https://doi.org/10.3934/math.2022437 doi: 10.3934/math.2022437
[9]	E. Cuesta, C. Palencia, A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces, Appl. Numer. Math., 45 (2003), 139–159. https://doi.org/10.1016/S0168-9274(02)00186-1 doi: 10.1016/S0168-9274(02)00186-1
[10]	S. S. Ahmed, Solving a system of fractional-order Volterra integro-differential equations based on the explicit finite difference approximation via the trapezoid method with error analysis, Symmetry, 14 (2022), 575. https://doi.org/10.3390/sym14030575 doi: 10.3390/sym14030575
[11]	F. Mirzaee, S. Alipour, Cubic B-spline approximation for linear stochastic integro-differential equation of fractional order, J. Comput. Appl. Math., 366 (2020), 112440. https://doi.org/10.1016/j.cam.2019.112440 doi: 10.1016/j.cam.2019.112440
[12]	A. A. Ibrahim, A. A. S. Zaghrout, K. R. Raslan, K. K. Ali, On the analytical and numerical study for nonlinear Fredholm integro-differential equations, Appl. Math. Inform. Sci., 14 (2020), 921–929. https://doi.org/10.18576/amis/140520 doi: 10.18576/amis/140520
[13]	A. A. Ibrahim, A. A. S. Zaghrout, K. R. Raslan, K. K. Ali, On study nonlocal integro differetial equation involving the Caputo-Fabrizio Fractional derivative and q-integral of the Riemann Liouville Type, Appl. Math. Inform Sci., 16 (2022), 983–993. https://doi.org/10.18576/amis/160615 doi: 10.18576/amis/160615
[14]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[15]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92. https://doi.org/10.12785/pfda/010202
[16]	B. Ahmad, J. J. Nieto, A. Alsaedi, H. Al-Hutami, Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions, J. Franklin I., 351 (2014), 2890–2909. https://doi.org/10.1016/j.jfranklin.2014.01.020 doi: 10.1016/j.jfranklin.2014.01.020
[17]	S. Suantai, S. K. Ntouyas, S. Asawasamrit, J. Tariboon, A coupled system of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions, Adv. Differ. Equ., 2015 (2015), 124. https://doi.org/10.1186/s13662-015-0462-2 doi: 10.1186/s13662-015-0462-2
[18]	V. Kac, P. Cheung, Quantum calculus, New York: Springer, 2002. https://doi.org/10.1007/978-1-4613-0071-7
[19]	M. H. Annaby, Z. S. Mansour, q-fractional calculus and equations, Springer, 2012. https://doi.org/10.1007/978-3-642-30898-7
[20]	A. N. Kolomogorov, S. V. Fomin, Inroductory real analysis, Dover Publications, Inc, 1975.
[21]	R. Saadati, B. Raftari, H. Adibi, S. M. Vaezpour, S. Shakeri, A comparison between the variational iteration method and trapezoidal rule for solving linear integro-differential equations, Word Appl. Sci. J., 4 (2008), 321–325.
[22]	R. C. Mittal, R. K. Jain, Numerical solutions of nonlinear Burgers' equation with modified cubic B-splines collocation method, Appl. Math. Comput., 218 (2012), 7839–7855. https://doi.org/10.1016/j.amc.2012.01.059 doi: 10.1016/j.amc.2012.01.059
[23]	K. E. Atkinson, An introduction to numerical analysis, Cambridge University Press, 1988. https://doi.org/10.1017/CBO9780511801181

AIMS Mathematics

On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

Related Papers:

Abstract

1. Introduction

2. Mathematical formulation for the shear shallow water model

3. The G. Rusanov scheme for SSWE

4. How the source term is treated

5. Numerical simulation for the SSW model

5.1. Test case 1

5.2. Test case 2

5.3. Test case 3

5.4. Test case 4

5.5. Test case 5

5.6. Test case 6 with a source term

6. Conclusions

Author contributions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog