Stabilization for a degenerate wave equation with drift and potential term with boundary fractional derivative control

Ibtissam Issa; Zayd Hajjej; Ibtissam Issa; Zayd Hajjej

doi:10.3934/era.2024227

Electronic Research Archive

2024, Volume 32, Issue 8: 4926-4953. doi: 10.3934/era.2024227

Previous Article Next Article

Research article

Stabilization for a degenerate wave equation with drift and potential term with boundary fractional derivative control

Ibtissam Issa ^{1
,
,},
Zayd Hajjej ²

1.
Dipartimento di Matematica, Università degli studi di Bari Aldo Moro, Via E. Orabona 4, Bari 70125, Italy
2.
Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 08 June 2024 Revised: 23 July 2024 Accepted: 08 August 2024 Published: 16 August 2024

This paper explores the boundary stabilization of a degenerate wave equation in the non-divergence form, which includes a drift term and a singular potential term. Additionally, we introduce boundary fractional derivative damping at the endpoint where divergence is absent. Using semi-group theory and the multiplier method, we establish polynomial stability, with a decay rate depending upon the order of the fractional derivative.

Keywords:

Citation: Ibtissam Issa, Zayd Hajjej. Stabilization for a degenerate wave equation with drift and potential term with boundary fractional derivative control[J]. Electronic Research Archive, 2024, 32(8): 4926-4953. doi: 10.3934/era.2024227

Related Papers:

[1]	Muhammad Nawaz Khan, Imtiaz Ahmad, Mehnaz Shakeel, Rashid Jan . Fractional calculus analysis: investigating Drinfeld-Sokolov-Wilson system and Harry Dym equations via meshless procedures. Mathematical Modelling and Control, 2024, 4(1): 86-100. doi: 10.3934/mmc.2024008
[2]	Tugba Obut, Erkan Cimen, Musa Cakir . A novel numerical approach for solving delay differential equations arising in population dynamics. Mathematical Modelling and Control, 2023, 3(3): 233-243. doi: 10.3934/mmc.2023020
[3]	Xianyi Li, Luyao Lv . Global attractivity of a rational difference equation with higher order and its applications. Mathematical Modelling and Control, 2024, 4(3): 260-272. doi: 10.3934/mmc.2024021
[4]	Shipeng Li . Impulsive control for stationary oscillation of nonlinear delay systems and applications. Mathematical Modelling and Control, 2023, 3(4): 267-277. doi: 10.3934/mmc.2023023
[5]	Ruxin Zhang, Zhe Yin, Ailing Zhu . Numerical simulations of a mixed finite element method for damped plate vibration problems. Mathematical Modelling and Control, 2023, 3(1): 7-22. doi: 10.3934/mmc.2023002
[6]	Bharatkumar K. Manvi, Shravankumar B. Kerur, Jagadish V Tawade, Juan J. Nieto, Sagar Ningonda Sankeshwari, Hijaz Ahmad, Vediyappan Govindan . MHD Casson nanofluid boundary layer flow in presence of radiation and non-uniform heat source/sink. Mathematical Modelling and Control, 2023, 3(3): 152-167. doi: 10.3934/mmc.2023014
[7]	Ravindra Rao, Jagan Mohan Jonnalagadda . Existence of a unique solution to a fourth-order boundary value problem and elastic beam analysis. Mathematical Modelling and Control, 2024, 4(3): 297-306. doi: 10.3934/mmc.2024024
[8]	Daizhan Cheng, Ying Li, Jun-e Feng, Jianli Zhao . On numerical/non-numerical algebra: Semi-tensor product method. Mathematical Modelling and Control, 2021, 1(1): 1-11. doi: 10.3934/mmc.2021001
[9]	Ihtisham Ul Haq, Nigar Ali, Shabir Ahmad . A fractional mathematical model for COVID-19 outbreak transmission dynamics with the impact of isolation and social distancing. Mathematical Modelling and Control, 2022, 2(4): 228-242. doi: 10.3934/mmc.2022022
[10]	Mrutyunjaya Sahoo, Dhabaleswar Mohapatra, S. Chakraverty . Wave solution for time fractional geophysical KdV equation in uncertain environment. Mathematical Modelling and Control, 2025, 5(1): 61-72. doi: 10.3934/mmc.2025005

Abstract

1. Introduction

Nonlinear evolution equations (NEEs) are mathematical equations that describe the behavior of complex physical systems. These equations are often used in various fields of science, including material and biological sciences, physics, mathematics, and mechanics ^[1,2,3,4,5]. One of the most commonly used NEE is the nonlinear Schrödinger equation, which is used in quantum mechanics to describe the behavior of wave packets ^[6,7]. Another example is the Klein-Gordon equation, which is used in particle physics to describe the behavior of spinless particles. The Cahn-Hilliard equation is another important equation used in materials science to describe the phase separation behaviour of binary mixtures. This equation has applications in the fields of metallurgy, polymer science, and biomaterials. The Navier-Stokes equation is a well-known NEE that describes the behavior of fluids, and is used in fluid mechanics to study the flow of fluids in various applications, including aerospace engineering, civil engineering, and oceanography ^[8,9,10,11].

NEEs are important in many fields of science because they can provide a mathematical model to describe the behavior of complex physical systems. By studying these equations and developing new numerical techniques to solve them, scientists can gain a better understanding of the physical phenomena they describe, and develop new technologies and materials that are based on this knowledge. To study the behavior of the physical problem (see ^{[12,13,14,15]}), either analytical or numerical solutions are needed. However, finding an analytical solution for NNEs is not always simple. As a result, developing numerical methods for such problems is useful. In this article, we investigate the nonlinear Klein-Gordon and sinh-Gordon equations, which are given as follows:

$\begin{equation} u_{tt}+\alpha \Delta u+ \psi(u) = g(x, t), \end{equation}$

(1.1)

where $u = u({\bf x}, {\bf{t}}),$ $\alpha$ is constant, and $\psi$ is some nonlinear function. For

$\psi(u) = \sinh(u),$

the Eq (1.1) becomes the sinh-Gordon equation ^[16]. When

$\psi(u) = \beta u+\gamma u^2,$

the Eq (1.1) becomes the Klein-Gordon equation, which is given as follows:

$\begin{equation} u_{tt}+\alpha \Delta u+ \beta u+\gamma u^2 = g(x, t), \end{equation}$

(1.2)

with the following initial and boundary conditions:

$\begin{align} u(x, 0)& = g_1(x), \; \; u_t(x, 0) = g_2(x), \; \; a\leq x \leq b, \end{align}$

(1.3)

$\begin{align} u(a, t)& = f_{1}(t), \; \; u(b, t) = f_2(t), \; \; t > 0. \end{align}$

(1.4)

The Klein-Gordon equation has numerous applications in different areas of physics and engineering and plays a crucial role in our understanding of the behavior of particles and fields in the universe. The Klein-Gordon equation is a relativistic wave equation that describes the behavior of particles with a zero spin, such as the Higgs boson particle. It has several applications in different fields of physics; for example, the Klein-Gordon equation is fundamental in the quantum field theory. It describes the behavior of scalar fields, which are fields that have a single value at each point in space-time ^[17,18]. Additionally, it can be used to describe the behavior of phonons, which are the quantized vibrations of a crystal lattice, and is used to describe the behavior of the inflaton field, which is thought to be responsible for the rapid expansion of the universe during the Big Bang ^[19]. Due to the wide range of applications, numerous researchers have investigated the solution of this equation utilizing various methodologies, including collocation points with the thin-plate splines radial basis function ^[20], the sine-cosine and tanh methods for travelling wave solutions ^[19], and the generalized auxiliary equation methods in ^[21]. Pseduospectral formulation with a perfectly matched layer has been suggested for the non-relativistic Klein-Gordon equation in ^[22]. Three different numerical methods were introduced to solve the rotating Klein-Gordon equation from relativistic regimes to non-relativistic regimes in polar and Cartesian coordinates ^[23]. Various numerical methods, including finite difference time domain methods, the time-splitting method, an exponential wave integrator, a limit integrator, a multiscale time integrator, the two-scale formulation method, and an iterative exponential integrator, were compared in terms of their accuracy while solving the Klein-Gordon equation (see ^[24]). The authors used a finite element technique based on cubic B-splines, the decomposition method ^[25], the fourth-order implicit Runge-Kutta-Nystrom, the fourth-order compact finite difference method ^[26], the boundary integral equation method ^[27], and the cubic B-spline collocation approach ^[28] to solve the generalized form of the nonlinear Klein-Gordon equation. For the solution of this model, a variety of numerical techniques have been reported in ^[29,30]. The two-dimensional sinh-Gordon equation is an essential partial differential equation (PDE) model with applications in a variety of fields.

The following is a concise review of the sinh-Gordon equation. In ^[31], the (G'/G)-expansion scheme was proposed for the generalized form of the double sinh-Gordon equations. For the analytical solutions to the sinh-Gordon equation, the tanh method was used in ^[32]. The element-free Galerkin technique was suggested in ^[33] to solve the generalized sinh-Gordon equation. The radial basis function was utilized by the authors in ^[34] to solve the sinh-Gordon equation. In ^[35], the author employed an implicit Lie-group iterative technique for the solution of the sinh-Gordon equation, whereas the Pseudo spectral and Kansa's methods based on radial basis functions were used in ^[36] to solve the two-dimensional sinh-Gordon equation. In ^[37], two effective approaches, namely the generalized leap-frog and the method of lines, were utilized for the sinh-Gordon equation, whereas the polynomial differential quadrature method was examined in ^[38].

In this work, we utilize the Fibonacci polynomials combined with Störmer's method to compute the numerical solutions of the one-dimensional Klein-Gordon and the two-dimensional sinh-Gordon equations. One of the key advantages of the suggested method is the ease implementation in higher-order derivatives using the relation of the Lucas and Fibonacci polynomials. Furthermore, the proposed approach improves the accuracy even for a small number of collocation points, thus lowering the computing costs. These polynomials have a wide range of applications in differential equations; for instance, the relationship between Chebyshev and Lucas polynomials was discussed in ^[39], and accurate solutions for the boundary value problems were obtained. Higher-order differential equations were solved using the Lucas polynomial technique in ^[40], while the Volterra-Fredholm integral differential equations were solved using a Fibonacci polynomial approach in ^[41]. Delay difference equations were solved using a hybrid Taylor-Lucas polynomial approach ^[42]. The Fibonacci wavelet approach was implemented for non-linear reaction diffusion equations of a Ficher-type, non-linear Hunter-Saxton and time fractional telegraph equation in ^[43,44]. The author and his co-worker used the Gegenbauer wavelet method for time fractional PDEs in ^[45,46]. For the first time, the authors proposed a hybrid Lucas and Fibonacci polynomial scheme to solve the time-dependent PDEs in ^[16,47]. More publications using Lucas polynomials and finite-differences to find efficient numerical solutions for different types of PDE models can be found in ^{[48,49,50,51]}.

The rest of the paper is organized as follows: in Section 2, fundamental concepts and definitions are discussed; in Section 3, the proposed methodology of the scheme is described; the method is verified with the help of numerical experiment in Section 4, and finally, the concluding remarks are drawn at the end.

2. Fundamental concepts

In this section, we will go over several key definitions and properties related to the Fibonacci and Lucas polynomials.

2.1. Fibonacci polynomials

The Fibonacci polynomial is an extension of Fibonacci numbers described by the linear recurrence relation, which is shown as follows ^[52]:

$\begin{equation} F_k(x) = kF_{k-1}(x)+F_{k-2}(x), \; \; k\; \geq 2, \end{equation}$

(2.1)

with the initial values

$F_0(x) = 0 \; \; \text{and}\; \; F_1(x) = 1.$

Their explicit form is as follows:

$\begin{equation*} F_{k}(x) = \sum\limits_{n = 0}^{\lfloor \frac{k}{2}\rfloor}\binom{k-n}{n} x^{k-2n}, \end{equation*}$

where

$\binom{m}{n} = \frac{m!}{n!(m-n)!},$

and $\lfloor{k}\rfloor$ denotes the integer floor function. Equation (2.1) generates a sequence for $x = 1$ of the Fibonacci numbers.

2.2. Lucas polynomials

The Lucas polynomials can be defined as follows ^[52]:

$\begin{equation} L_k(x) = kL_{k-1}(x)+L_{k-2}(x), \; \; k\; \geq 2, \end{equation}$

(2.2)

with

$L_0(x) = 2\; \; \text{and}\; \; L_1(x) = x.$

Their explicit form is

$\begin{equation*} \label{S3} L_{k}(x) = \sum\limits_{n = 0}^{\lfloor \frac{k}{2}\rfloor}\frac{k}{k-n}\binom{k-n}{n} x^{k-2n}, \end{equation*}$

when $x = 1$ . Equation (2.2) generates a sequence of the Lucas numbers.

2.3. Function approximation

Let $u(x)$ be a continuous function that may be approximated in terms of the Lucas series in the following manner:

$\begin{equation} u(x) = \sum\limits_{k = 0}^{M}\lambda_{k}^{n+1} L_k(x), \end{equation}$

(2.3)

where $L(x)$ are the Lucas polynomials and $\lambda_k$ are the unknown coefficients. Utilizing the finite terms of the Lucas series, the $mth$ -order derivative of a function $u(x)$ can be approximated as follows:

$\begin{equation} u^{(m)}(x) = \sum\limits_{k = 0}^{M}\lambda_{k}^{n+1} L^{(m)}_k(x). \end{equation}$

(2.4)

Utilizing the differentiation matrix ${\bf D}$ and Fibonacci polynomials, $L^{(m)}_k(x)$ can be approximated ^[52] as follows:

$\begin{equation} L^{(m)}_k(x) = kF_{k}(x)D^{m-1}, \end{equation}$

(2.5)

where

$\begin{equation*} D = \begin{bmatrix} 0 & 0 & \dots & 0 \\ 0 \\ \vdots & \ & d\\ 0 \end{bmatrix}, \end{equation*}$

and $d$ is a $N\times N$ matrix

$\begin{equation} d_{m, n} = \begin{cases} m\sin\frac{(n-m)\pi}{2}, & \text{if}\ n > m , \\ 0, & \text{otherwise}. \end{cases} \end{equation}$

(2.6)

3. Proposed methodology

To describe the proposed hybrid scheme, we consider Eq (1.1)

$\begin{equation} u_{tt}+\alpha \Delta u+ \psi(u) = g(x, t) \end{equation}$

(3.1)

with the following conditions:

$\begin{align} u(x, 0)& = g_1(x), \; \; u_t(x, 0) = g_2(x), \; \; a\leq x \leq b, \end{align}$

(3.2)

$\begin{align} u(a, t)& = f_{1}(t), \; \; u(b, t) = f_2(t), \; \; t > 0. \end{align}$

(3.3)

First, re-arrange Eq (3.1)

$\begin{equation} u_{tt} = g(x, t)-\alpha \Delta u- \psi(u). \end{equation}$

(3.4)

Now, using Störmer's method for the time discretization of Eq (3.4), we have the following:

$\begin{equation} u^{n+1} = 2u^{n}-u^{n-1}+{dt}^2(g(x, t)-\alpha \Delta u^{n}- \psi(u)^{n}), \end{equation}$

(3.5)

where

$u^{n+1} = u(t^{n+1}, x),$

$dt$ is time stepsize for $t\in[0, T]$ .

Then, let us discretize the spacial domain $[a, b]$ into the $M+1$ number of the nodes $x_i, \; i = 0, 1, 2, \ldots, M$ , where $x_0, \; x_M$ are the boundary points while the remaining (i.e., $x_i, \; i = 1, 2, \ldots, M-1$ ) are interior points of the domain, which are computed as follows:

$x_i = a+i*h,$

where

$h = (b-a)/M$

is spatial step size. Here, we approximate the unknown function $u(x)$ by Lucas polynomials at the $nth$ time level, which is denoted as $u^{n}(x)$ :

$\begin{equation} u^{n}(x) = \sum\limits_{j = 0}^{M} c_{j}^{n}(t)L_{j}(x) = {\bf L}^{T}(x){\bf C}, \end{equation}$

(3.6)

where ${\bf C}$ is the vector of unknown coefficients dependent on time, and ${\bf L}$ is the $(M+1)\times 1$ vector of Lucas polynomials, such that

$\begin{equation} {\bf L}(x) = [L_{0}(x), L_{1}(x), ..., L_{M}(x)]^{T}. \end{equation}$

(3.7)

Now, for the collocation points $x_{i}$ ,

$\begin{equation} u^{n}(x_i) = \sum\limits_{j = 0}^{M} c_{j}^{n}(t)L_{j}(x_i) = {\bf L}(x_i){\bf C}. \end{equation}$

(3.8)

Let

$u^{n}(x_i) = u_{i};$

then, in the matrix notation,

$\begin{equation} {\bf U} = {\bf AC}, \end{equation}$

(3.9)

where

${\bf U} = [u_{0}, u_{1}, ..., u_{M}]^{T}, \; \; \; {\bf C} = [c_{0}, c_{1}, ..., c_{M}]^{T}$

and

$\begin{align} {\bf A} & = \begin{bmatrix} L_{0}(x) \\ L_{1}(x) \\ \vdots \\ L_{M}(x) \end{bmatrix} = \begin{bmatrix} L_{0}(x_{0}) & L_{0}(x_{1}) & \cdots & L_{0}(x_{M}) \\ L_{1}(x_{0}) & L_{1}(x_{1}) & \cdots & L_{1}(x_{M}) \\ \vdots & \vdots & \ddots & \vdots \\ L_{M}(x_{0}) & L_{M}(x_{1}) & \cdots & L_{M}(x_{M}) \end{bmatrix}. \end{align}$

(3.10)

Similarly, the $kth$ order derivative of unknown functions are approximated by the $kth$ order derivatives of Lucas polynomials

$\begin{equation} u^{(k)}(x_i) = \sum\limits_{j = 0}^{M} c_{j}^{n}(t)L_{j}^{(k)}(x_i) = {\bf L}^{(k)}(x_i){\bf C}. \end{equation}$

(3.11)

In the matrix form,

$\begin{equation} {\bf U}^{(k)} = {\bf BC}, \end{equation}$

(3.12)

where

${\bf U}^{(k)} = [u_{0}^{(k)}, u_{1}^{(k)}, ..., u_{M}^{(k)}]^{T}, \; \; \; {\bf C} = [c_{0}, c_{1}, ..., c_{M}]^{T}$

and

$\begin{align} {\bf B} & = \begin{bmatrix} L_{0}^{(k)}(x) \\ L_{1}^{(k)}(x) \\ \vdots \\ L_{M}^{(k)}(x) \end{bmatrix} = \begin{bmatrix} L_{0}^{(k)}(x_{0}) & L_{0}^{(k)}(x_{1}) & \cdots & L_{0}^{(k)}(x_{M}) \\ L_{1}^{(k)}(x_{0}) & L_{1}^{(k)}(x_{1}) & \cdots & L_{1}^{(k)}(x_{M}) \\ \vdots & \vdots & \ddots & \vdots \\ L_{M}^{(k)}(x_{0}) & L_{M}^{(k)}(x_{1}) & \cdots & L_{M}^{(k)}(x_{M}) \end{bmatrix}. \end{align}$

(3.13)

Using the above approximation in Eq (3.5), we have the following:

$\begin{equation} {{\bf u}^{n+1} = 2{\bf u}^{n}-{\bf u}^{n-1}+{dt}^2\left({\bf g}^{n}-\alpha {\bf D}{\bf u}^{n}- \psi({\bf u}^{n})\right)}, \end{equation}$

(3.14)

where

${\bf g}^{n} = g(x_i, t^{n})\ \; (i = 0, 1, \cdots, M)$

and

${\bf D} = {\bf B}{\bf A}^{-1}.$

For $n$ = 1, ${\bf u}^{1}$ and ${\bf u}^{0}$ can be obtained from the initial conditions. To satisfy the boundary conditions we collocated the boundary conditions that corresponded to each of the following:

$x_i \in \Gamma = \{a, b\}.$

That is, for

$\begin{equation*} u(x_i, t) = f(x_i, t), \; \forall x_i \in \Gamma, \end{equation*}$

we then update the solution vector at boundary points to have

$\begin{equation*} {\bf u}^{n+1}_{\Gamma} = {\bf f}(t^{n+1}). \end{equation*}$

Remark 3.1. The stability condition for the second order Störmer method is as follows:

$\begin{equation*} \alpha \left(\frac{dt}{dx} \right)\leq 1, \end{equation*}$

where $\alpha$ is the coefficient of the Laplace operator. Then the method is then said to be stable, by the rule of thumb, if all the eigenvalues, $\lambda$ , of the discretized operator,

${\bf D} = {\bf B}{\bf A}^{-1},$

scaled by ${dt}^2$ lies within the stability region of the time-stepping method ^[53], i.e.,

$\begin{equation*} -4\leq \Re(\lambda {dt}^2 )\leq 0. \end{equation*}$

4. Numerical results and discussion

This section demonstrates the numerical solution of one dimension Klien-Gordon and two dimension sinh-Gordon equations utilizing the proposed scheme. All the computations are performed using MATLAB (R2012a) on Dell PC Laptop with an Intel(R) Core(TM)i5-2450M CPU 2.50 GHz 2.50 GHz 8 GB RAM. To check the accuracy and convergence, we used the following error norms:

$\begin{align*} L_{\infty}& = max{\mid E_{k}-u_{k}}\mid_{k = 1}^{M+1}, \\ L_{2}& = \left(dx{\sum\limits_{k = 1}^{M+1} \left(E_{k}-u_{k}\right)^{2}}\right)^{1/2}, \\ L_{rms}& = \left(1/M{\sum\limits_{k = 1}^{M+1} \left(E_{k}-u_{k}\right)^{2}}\right)^{1/2}, \\ C-rate& = \frac{L_{\infty}({dt}_{\ell})}{L_{\infty}({dt}_{\ell+1})}, \end{align*}$

where $E$ is the exact solution.

4.1. (1+1)-dimensional case

Test Problem 4.1. Consider Eq (1.2) with $\alpha = \beta = -1$ , $\gamma = 0$ , $g(x, t) = 0$ ; we have

$\begin{align} u_{tt}-\Delta u-u = 0, \end{align}$

(4.1)

with the following initial and boundary conditions:

$\begin{align*} u(x, 0)& = 1+\sin(x), \ \ u_{t}(x, 0) = 0, \ \ {x \in [0, 1]}, \\ u(0, t) & = \sin(0)+\cosh(t), \ u(1, t) = \sin(1)+\cosh(t), \ t > 0, \end{align*}$

and the actual solution is as follows:

$\begin{align*} u(x, t) = sin(x)+cosh(t). \end{align*}$

Using the proposed methodology for this particular equation the iterative scheme (3.14) becomes the following:

$\begin{equation} {\bf u}^{n+1} = 2{\bf u}^{n}-{\bf u}^{n-1}+{dt}^2({\bf g}^{n}+\mathit{\boldsymbol{D}}{\bf u}^{n}+{\bf u}^{n}). \end{equation}$

(4.2)

The above equation is used to generate numerical results, which are provided in in terms of the $L_{\infty}$ and $L_2$ error norms. These findings are obtained using nodal points $M = 10$ and various values of the time step size $dt$ while preserving the final time $T = 0.1$ , and for various values of $T$ and fixing $dt = 0.0001$ . The accuracy increases as the value of $dt$ decreases, as shown in this table. Based on the comparison with ^[29], it is obvious that the suggested scheme is more accurate than those reported in the recent literature. The numerical results in terms of $L_{\infty}$ , $L_2$ , and $L_{rms}$ , as well as central processing unit (CPU) times in seconds, are computed using the suggested approach for various values of $M$ in Table 2. The table illustrates that as the number of collocation points rises, the current approach provides an improved accuracy. Furthermore, the proposed technique may be seen to be quite efficient, whereas the comparison between the exact and approximate solutions are visualized in Figure 1.

Table 1. Results of the proposed method for Test Problem 4.1 with

$M = 10$ and

$T = 0.1$ .

	Present method		MQ-RBF ^[29]		GA-RBF ^[29]
dt	$L_\infty$	$L_2$	$L_\infty$	$L_2$	$L_\infty$	$L_2$
0.01	5.65E-04	4.92E-04	5.06E-04	4.46E-04	1.36E-03	1.26E-03
0.001	5.50E-05	4.86E-05	5.55E-05	4.74E-05	5.08E-05	4.43E-05
0.0005	2.75E-05	2.43E-05	4.04E-05	2.61E-05	2.57E-05	2.21E-05
0.0001	5.49E-06	4.85E-06	2.83E-05	1.12E-05	6.80E-06	5.05E-06
T	$L_\infty$	$L_2$	$L_\infty$		$L_\infty$
0.01	5.00E-07	4.78E-07	7.57E-07	-	5.35E-07	-
0.02	1.00E-06	9.49E-07	2.02E-06	-	1.13E-06	-
0.1	5.49E-06	4.85E-06	2.83E-05	-	6.80E-06	-
0.5	4.19E-05	3.42E-05	8.61E-05	-	4.63E-05	-
1	4.37E-04	2.14E-04	3.95E-05	-	4.20E-04	-

| Show Table

DownLoad: CSV

Table 2. Results of the proposed method for Test Problem 4.1 with

$dt = 0.0001$ and

$T = 0.1$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	5.03E-06	4.45E-06	3.15E-07	0.037078
7	4.32E-06	3.73E-06	2.95E-07	0.045391
10	2.49E-06	1.85E-06	1.85E-07	0.051433
15	9.03E-07	8.56E-07	9.81E-08	0.071921
18	7.12E-07	6.42E-07	7.88E-08	0.109775
20	5.31E-07	5.13E-07	6.36E-08	0.155763
22	2.90E-07	1.65E-07	3.38E-08	0.330819
25	2.35E-07	1.09E-07	1.47E-08	1.366762

| Show Table

DownLoad: CSV

Figure 1. Comparison of exact and approximate solutions for Test Problem 4.1.

DownLoad: Full-Size Img PowerPoint

Test Problem 4.2. Now assume

$\alpha = -1, \; \beta = -2, \; \gamma = 0$

and

$g(x, t) = -2\sin{x} \sin{t}$

in Eq (1.2) with the following conditions:

$\begin{align*} u(x, 0)& = 0, \quad u_{t}(x, 0) = \sin{x}, \quad {x \in [0, \pi/2], }\\ u(0, t)& = 0, \quad u(\pi/2, t) = \sin t, \quad t > 0. \end{align*}$

The exact solution of Eq (1.2) for the above parameter is given in ^[29], which is

$u(x, t) = \sin{x}\sin{t}.$

In Test Problem 4.2, the numerical results are computed utilizing the proposed technique. These results are shown in Table 3 using nodal points

$M = 10$

and various values of $dt$ while preserving the final time

$T = 0.1,$

and for various values of $T$ and fixing

$dt = 0.0001.$

In this situation, as indicated in the table, the method's accuracy improves as the value of $dt$ declines. Furthermore, in comparison with ^[29], we came to know that the current method is more accurate than other articles published in the recent literature. Similarly to the previous test problem, the numerical results are computed in the form of $L_\infty$ , $L_2$ , and $L_{rms}$ , as well as the CPU times in seconds, using the suggested approach for various values of $M$ in Table 4. As the number of collocation points increases, the current approach efficiently improves the accuracy, as seen in the table whereas profiles of the exact and approximate solutions are given in Figure 2.

Table 3. Results of the proposed method for Test Problem 4.2 with

$M = 10$ and

$T = 0.1$ .

	Present method		MQ-RBF ^[29]		GA-RBF ^[29]
dt	$L_\infty$	$L_2$	$L_\infty$	$L_2$	$L_\infty$	$L_2$
0.01	1.59E-06	1.23E-03	2.50E-03	1.04E-03	2.44E-03	9.86E-04
0.001	1.59E-08	1.23E-04	1.53E-04	6.58E-05	5.05E-04	2.09E-04
0.0005	3.97E-09	6.17E-05	7.65E-05	3.24E-05	3.87E-04	1.63E-04
0.0001	1.57E-10	1.23E-05	1.50E-05	2.59E-05	2.92E-04	1.26E-04
T	$L_\infty$	$L_2$	$L_\infty$		$L_\infty$
0.01	1.65E-11	3.91E-06	1.58E-07	-	5.45E-07	-
0.02	3.29E-11	5.53E-06	6.29E-07	-	3.32E-06	-
0.1	1.57E-10	1.23E-05	1.50E-05	-	2.92E-04	-
0.5	6.47E-10	2.27E-05	8.84E-05	-	1.07E-02	-
1	1.16E-09	1.39E-05	5.20E-05	-	2.34E-02	-

| Show Table

DownLoad: CSV

Table 4. Results of the proposed method for Test Problem 4.2 with

$dt = 0.0001$ and

$T = 1$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	4.23E-04	3.32E-03	1.87E-04	0.051816
7	2.99E-05	2.90E-04	3.65E-05	0.051183
10	9.76E-06	8.90E-05	7.09E-06	0.082319
15	6.51E-06	2.40E-05	2.34E-06	0.09778
18	1.64E-06	2.73E-05	6.92E-07	0.143512
20	8.01E-07	8.93E-06	1.01E-07	0.184772
22	9.55E-08	5.80E-06	8.68E-08	0.338581
25	7.12E-08	7.42E-07	7.70E-08	1.288726

| Show Table

DownLoad: CSV

Figure 2. Exact versus approximate solution for Test Problem 4.2.

DownLoad: Full-Size Img PowerPoint

Test Problem 4.3. Consider Eq (1.2) with the exact solution $u(x, t) = x^3t^3$ , when

$\alpha = -1, \; \beta = 0, \; \gamma = 1$

and

$g(x, t) = 6xt(x^2-t^2)+x^6t^6,$

with associated conditions

$\begin{align*} & u(x, 0) = 0, \quad u_{t}(x, 0) = 0, \quad {x \in [0, 1]}, \\ & u(0, t) = 0, \quad u(1, t) = 0, \quad t > 0. \end{align*}$

and show the numerical results for Test Problem 4.3 for various values of $dt$ and $M$ , respectively. We compared our results with the methods in ^[29] for

$M = 10\; \; \mathit{\text{and}}\; \; T = 1$

in . We can see from this table that the method's accuracy improves as the number of iterations increases, with a very good accuracy of $10^{-8}$ for

$dt = 0.0001.$

In comparison to the method in ^[29], the proposed method produces more accurate results. In , the results are computed in terms of $L_{\infty}$ , $L_2$ , and $L_{rms}$ for

$dt = 0.001$

and time $T = 1$ , as well as the CPU time in seconds. The suggested procedure is accurate and efficient, as shown in this table. The profile of the exact and approximate solutions are shown in Figures 3 and 4, respectively.

Table 5. Results of the proposed method for Test Problem 4.3 with

$M = 10$ and

$T = 1$ .

	Present method		MQ-RBF ^[29]		GA-RBF ^[29]
dt	$L_\infty$	$L_2$	$L_\infty$	$L_2$	$L_\infty$	$L_2$
0.01	3.35E-04	1.78E-04	2.24E-02	9.59E-03	4.66E-02	1.96E-02
0.001	3.24E-06	1.64E-06	2.46E-03	1.05E-03	2.27E-03	9.71E-04
0.0005	8.08E-07	4.09E-07	1.36E-03	5.79E-04	1.17E-03	5.00E-04
0.0001	3.17E-08	1.66E-08	4.85E-04	2.07E-04	2.94E-04	1.25E-04

| Show Table

DownLoad: CSV

Table 6. Results of the proposed method for Test Problem 4.3 with

$T = 1$ and

$dt = 0.001$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	2.19E-04	1.37E-04	9.65E-05	0.050013
10	3.24E-05	6.36E-06	1.64E-06	0.102678
15	4.84E-06	4.88E-06	6.82E-07	0.088209
20	1.07E-06	1.3606E-06	1.92E-07	0.185567
22	8.11E-07	5.05E-07	7.13E-08	0.370034
25	6.08E-07	2.72E-07	4.21E-08	1.372918

| Show Table

DownLoad: CSV

Figure 3. Comparison of exact and approximate solutions for Test Problem 4.3.

DownLoad: Full-Size Img PowerPoint

Figure 4. Exact versus approximate solution for Test Problem 4.3.

DownLoad: Full-Size Img PowerPoint

Test Problem 4.4. Consider

$\alpha = -1, \ \beta = -1, \ \gamma = 0$

and

$g(x, t) = -\cos x \sin t$

in Eq (1.2) with the following conditions:

$\begin{align*} u(x, 0)& = 0, \quad u_{t}(x, 0) = \cos x, \quad {x \in [0, 1], }\\ u(0, t)& = \sin t, \quad u(1, t) = \cos 1 \sin t, \quad t > 0. \end{align*}$

The exact solution of Eq (1.2) for the above parameter given as follows:

$u(x, t) = \cos x \sin t.$

The solution is computed for the fixed point $T = 0.1$ and $M = 20,$ while decreasing the time step. The results are reported in , which shows that the accuracy improves as the number of iteration increases. In , the results are recorded for various numbers of nodal points $M$ , and notice that as number of collocation point increases, the numerical convergence accuracy improves. Along with, maximum error, the CPU time in seconds is also reported in the tables. The solution profiles of the exact and approximate solutions are illustrated in Figure 5, which shows that both the solutions are well matched.

Table 7. Results of the proposed method for Test Problem 4.4 with

$M = 15$ and

$T = 0.1$ .

dt	$L_\infty$	$L_2$	$L_{rms}$	CPU
0.01	4.28E-05	3.55E-05	4.79E-05	2.03E-01
0.001	4.88E-06	4.05E-06	1.80E-06	2.04E-01
0.0005	2.46E-06	2.04E-06	6.45E-07	1.77E-01
0.0001	4.97E-07	4.23E-07	5.98E-08	1.92E-01

| Show Table

DownLoad: CSV

Table 8. Results of the proposed method for Test Problem 4.4 with

$dt = 0.0001$ and

$T = 0.1$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	4.97E-07	4.88E-07	3.74E-08	0.04553
7	4.80E-07	3.94E-07	3.53E-08	0.048321
10	4.90E-07	3.76E-07	3.40E-08	0.063045
15	4.64E-07	3.66E-07	3.39E-08	0.081496
18	4.54E-07	3.68E-07	3.36E-08	0.115411
20	4.40E-07	3.43E-07	3.33E-08	0.167235
22	4.36E-07	3.36E-07	3.32E-08	0.372988
25	4.24E-07	3.18E-07	2.82E-08	1.378855

| Show Table

DownLoad: CSV

Figure 5. Exact versus approximate solution for Test Problem 4.4.

DownLoad: Full-Size Img PowerPoint

Test Problem 4.5. (1+2)-dimensional case: Finally, consider the two dimensional sinh-Gordon equation

$\begin{equation} u_{tt}-\Delta u+\sinh(u) = g(x, y, t) \end{equation}$

(4.3)

with initial conditions

$\begin{align*} & u(x, y, 0) = u_{t}(x, y, 0) = \sin(x+y), \quad {(x, y) \in [0, 1]^2, } \end{align*}$

and boundary conditions

$\begin{align*} & u(0, y, t) = e^t\sin(0+y), \quad u(1, y, t) = e^t\sin(1+y), \quad t > 0, \\ & u(x, 0, t) = e^t\sin(x+0), \quad u(x, 1, t) = e^t\sin(x+1). \end{align*}$

The exact solution

$u(x, y, t) = e^t\sin(x+y)$

is considered, as given in ^[47]. Additionally the source term is as follows:

$\begin{equation} g(x, y, t) = 3e^t\sin(x+y)+\sinh(e^t\sin(x+y)). \end{equation}$

(4.4)

The results of the suggested method for Test Problem 4.5 are shown in for various values of $dt$ with $M = 4$ and $T = 1$ . This table shows that the method delivers a good accuracy for very coarse nodes, and that the method's accuracy improves to some extent as the number of iterations increases. Additionally in this table, we also compared the computed results to the approach in ^[47]. The results of the proposed approach in contrast to the method presented in ^[47] are shown in for various values of $M$ for the same test problem. This comparison reveals that the proposed approach is more accurate than ^[47]. The approximate solution of the suggested approach is compared to the exact solution in Figure 6. Although the absolute errors are indicated in Figure 7, the numerical solution is in a good agreement with the exact solution, as demonstrated in Figure 6.

Table 9. Results of the proposed method for Test Problem 4.5 with

$M = 4$ and

$T = 1$ .

	Present method			^[47]
dt	$L_\infty$	$L_2$	C-rate( $L_\infty$ )	$L_\infty$	$L_2$
0.100	1.91E-02	8.66E-03	-	2.20E-02	4.19E-02
0.050	9.19E-03	3.93E-03	2.08E+00	1.15E-02	1.47E-02
0.025	4.52E-03	1.93E-03	2.03E+00	7.53E-03	1.23E-02
0.013	2.24E-03	9.51E-04	2.02E+00	4.88E-03	9.57E-03
0.006	1.11E-03	4.68E-04	2.02E+00	2.90E-03	6.04E-03
0.003	5.45E-04	2.27E-04	2.04E+00	1.60E-03	3.40E-03
0.002	2.63E-04	1.08E-04	2.07E+00	8.49E-04	1.80E-03
0.001	1.22E-04	5.13E-05	2.15E+00	4.43E-04	9.30E-04

| Show Table

DownLoad: CSV

Table 10. Results of the proposed method for Test Problem 4.5 with

$dt = 0.001$ and

$T = 1$ .

	Present method		^[47]
M	$L_\infty$	$L_2$	$L_\infty$	$L_2$
2	5.85E-03	2.93E-03	5.53E-03	5.53E-03
3	2.22E-03	1.47E-03	1.22E-03	1.74E-03
4	1.62E-04	6.66E-05	5.59E-04	1.18E-03
5	1.63E-04	6.75E-05	5.38E-04	9.99E-04
6	1.23E-04	6.37E-05	-	-
7	1.32E-04	7.41E-05	-	-

| Show Table

DownLoad: CSV

Figure 6. Exact versus approximate solution at

$M = 15$ ,

$T = 1$ , for Test Problem 4.5.

DownLoad: Full-Size Img PowerPoint

Figure 7. Error for Test Problem 4.5.

DownLoad: Full-Size Img PowerPoint

5. Conclusions

A numerical method based on Fibonacci polynomials combined with Störmer's method was developed to solve the nonlinear Klein/Sinh-Gordon equation. On a variety of test cases, numerical evaluation exhibited an improved performance of the proposed scheme against the available results in the literature. As evidenced by the comparison, the proposed technique is simple to execute and faster in convergence. Furthermore, it was observed that by reducing the spatial and time steps size, the errors were reduced. With slight modifications, the proposed method can be applied to a variety of PDEs. This modification is mainly dictated by the problem at hand with the associated initial and boundary conditions that are dictated by the particular problem of interest. As such, we are interested in applying the proposed method to time-dependent Neumann and/or mixed boundary value problems with possible discontinuous initial condition(s). The limitations, as is the case in spectral methods, was the high-condition numbers of the collocated/system matrices. However, this can be alleviated by the state-of-the-art pre-conditioned algorithms developed for spectral methods. In the future, the proposed method can be expanded for applications to complex fractal-fractional problems.

Conflict of interest

The authors declare that there are no conflicts of interest in this paper.

References

[1]	W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. Math., 55 (1952), 468–519. https://doi.org/10.2307/1969644 doi: 10.2307/1969644
[2]	P. Cannarsa, G. Fragnelli, D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Networks Heterogen. Media, 2 (2007), 695–715. https://doi.org/10.1137/04062062X doi: 10.1137/04062062X
[3]	G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and carleman estimates, J. Differ. Equations, 260 (2016), 1314–1371. https://doi.org/10.1016/j.jde.2015.09.019 doi: 10.1016/j.jde.2015.09.019
[4]	M. Badii, J. I. Díaz, Time periodic solutions for a diffusive energy balance model in climatology, J. Math. Anal. Appl., 233 (1999), 713–729. https://doi.org/10.1006/jmaa.1999.6335 doi: 10.1006/jmaa.1999.6335
[5]	I. H. Biswas, A. K. Majee, G. Vallet, On the cauchy problem of a degenerate parabolic-hyperbolic pde with lévy noise, Adv. Nonlinear Anal., 8 (2019), 809–844. https://doi.org/10.1515/anona-2017-0113 doi: 10.1515/anona-2017-0113
[6]	A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51 (2009), 3–33. https://doi.org/10.1137/080716827 doi: 10.1137/080716827
[7]	O. Nikan, Z. Avazzadeh, J. A. T. Machado, Numerical simulation of a degenerate parabolic problem occurring in the spatial diffusion of biological population, Chaos, Solitons Fractals, 151 (2021), 111220. https://doi.org/10.1016/j.chaos.2021.111220 doi: 10.1016/j.chaos.2021.111220
[8]	F. Alabau-Boussouira, P. Cannarsa, G. Leugering, Control and stabilization of degenerate wave equations, SIAM J. Control Optim., 55 (2017), 2052–2087. https://doi.org/10.1137/15M1020538 doi: 10.1137/15M1020538
[9]	M. Gueye, Exact boundary controllability of 1-d parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037–2054. https://doi.org/10.1137/120901374 doi: 10.1137/120901374
[10]	F. Chouaou, C. Aichi, A. Benaissa, Decay estimates for a degenerate wave equation with a dynamic fractional feedback acting on the degenerate boundary, Filomat, 35 (2021), 3219–3239. https://doi.org/10.2298/FIL2110219C doi: 10.2298/FIL2110219C
[11]	I. Boutaayamou, G. Fragnelli, D. Mugnai, Boundary controllability for a degenerate wave equation in nondivergence form with drift, SIAM J. Control Optim., 61 (2023), 1934–1954. https://doi.org/10.1137/22M151491X doi: 10.1137/22M151491X
[12]	M. Akil, G. Fragnelli, I. Issa, Stability for degenerate wave equations with drift under simultaneous degenerate damping, preprint, arXiv: 2308.08645. https://doi.org/10.48550/arXiv.2308.08645
[13]	M. Akil, G. Fragnelli, I. Issa, Energy decay rate of a transmission system governed by degenerate wave equation with drift and under heat conduction with memory effect, preprint, arXiv: 2311.16296. https://doi.org/10.48550/arXiv.2311.16296
[14]	B. Allal, A. Hajjaj, J. Salhi, A. Sbai, Boundary controllability for a coupled system of degenerate/singular parabolic equations, Evol. Equations Control Theory, 11 (2022), 1579–1604. https://doi.org/10.3934/eect.2021055 doi: 10.3934/eect.2021055
[15]	I. Boutaayamou, G. Fragnelli, A degenerate population system: Carleman estimates and controllability, Nonlinear Anal., 195 (2020), 111742. https://doi.org/10.1016/j.na.2019.111742 doi: 10.1016/j.na.2019.111742
[16]	G. Fragnelli, Carleman estimates and null controllability for a degenerate population model, J. Math. Pures Appl., 115 (2018), 74–126. https://doi.org/10.1016/j.matpur.2018.01.003 doi: 10.1016/j.matpur.2018.01.003
[17]	I. Boutaayamou, G. Fragnelli, L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and neumann boundary conditions, J. Anal. Math., 135 (2018), 1–35. https://doi.org/10.1007/s11854-018-0030-2 doi: 10.1007/s11854-018-0030-2
[18]	G. Fragnelli, Null controllability for a degenerate population model in divergence form via carleman estimates, Adv. Nonlinear Anal., 9 (2019), 1102–1129. https://doi.org/10.1515/anona-2020-0034 doi: 10.1515/anona-2020-0034
[19]	G. Fragnelli, M. Yamamoto, Carleman estimates and controllability for a degenerate structured population model, Appl. Math. Optim., 84 (2020), 999–1044. https://doi.org/10.1007/s00245-020-09669-0 doi: 10.1007/s00245-020-09669-0
[20]	C. L. Epstein, R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Princeton University Press, 2013. https://doi.org/10.1365/s13291-015-0131-0
[21]	J. Vancostenoble, E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864–1902. https://doi.org/10.1016/j.jfa.2007.12.015 doi: 10.1016/j.jfa.2007.12.015
[22]	P. S. Hagan, D. E. Woodward, Equivalent black volatilities, Appl. Math. Finance, 6 (1999), 147–157. https://doi.org/10.1080/135048699334500 doi: 10.1080/135048699334500
[23]	F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equations, 6 (2006), 161–204. https://doi.org/10.1007/s00028-006-0222-6 doi: 10.1007/s00028-006-0222-6
[24]	P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1–19. https://doi.org/10.1137/04062062X doi: 10.1137/04062062X
[25]	G. Fragnelli, D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal., 2 (2013). https://doi.org/10.1515/anona-2013-0015
[26]	G. Fragnelli, D. Mugnai, Carleman Estimates, Observability Inequalities and Null Controllability for Interior Degenerate Non Smooth Parabolic Equations, American Mathematical Society, 242 (2016). https://doi.org/10.1090/MEMO/1146
[27]	M. Fotouhi, L. Salimi, Controllability results for a class of one dimensional degenerate/singular parabolic equations, Commun. Pure Appl. Anal., 12 (2013), 1415–1430. https://doi.org/10.3934/cpaa.2013.12.1415 doi: 10.3934/cpaa.2013.12.1415
[28]	G. Fragnelli, D. Mugnai, Singular parabolic equations with interior degeneracy and non smooth coefficients: The neumann case, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1495–1511. https://doi.org/10.3934/dcdss.2020084 doi: 10.3934/dcdss.2020084
[29]	Z. J. Han, Z. Liu, J. Wang, Sharper and finer energy decay rate for an elastic string with localized kelvin-voigt damping, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 1455–1467. https://doi.org/10.3934/dcdss.2022031 doi: 10.3934/dcdss.2022031
[30]	Z. J. Han, Z. Liu, Q. Zhang, Sharp stability of a string with local degenerate kelvin–voigt damping, ZAMM, 102 (2022). https://doi.org/10.1002/zamm.202100602
[31]	Z. J. Han, Z. Liu, K. i Yu, Stabilization for wave equation with localized kelvin–voigt damping on cuboidal domain: A degenerate case, SIAM J. Control Optim., 62 (2024), 441–465. https://doi.org/10.1137/22M153210X doi: 10.1137/22M153210X
[32]	B. Allal, A. Moumni, J. Salhi, Boundary controllability for a degenerate and singular wave equation, Math. Methods Appl. Sci., 45 (2022), 11526–11544. https://doi.org/10.1002/mma.8464 doi: 10.1002/mma.8464
[33]	G. Fragnelli, D. Mugnai, A. Sbai, Boundary controllability for degenerate/singular hyperbolic equations in nondivergence form with drift, preprint, arXiv: 2402.18247. https://doi.org/10.48550/arXiv.2402.18247
[34]	G. Fragnelli, D. Mugnai, A. Sbai, Stabilization for degenerate equations with drift and small singular term, preprint, arXiv: 2403.17802. https://doi.org/10.48550/arXiv.2403.17802
[35]	M. Akil, G. Fragnelli, I. Issa, Stability of degenerate wave equation with a singular potential and local damping, (2024), hal-04539408f, In press.
[36]	D. Matignon, Asymptotic stability of webster-lokshin equation, Math. Control Relat. Fields, 4 (2014). https://doi.org/10.3934/mcrf.2014.4.481
[37]	R. L. Bagley, P. J. Torvik, Fractional calculus–-a different approach to the analysis of viscoelastically damped structures, AIAA J., 21 (1983), 741–748. https://doi.org/10.2514/3.8142 doi: 10.2514/3.8142
[38]	M. Mainardi, E. Bonetti, The application of real-order derivatives in linear viscoelasticity, in Progress and Trends in Rheology II, Steinkopff, Heidelberg, (1988), 64–67. https://doi.org/10.1007/978-3-642-49337-9_11
[39]	P. Torvik, R. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984). https://doi.org/10.1115/1.3167615
[40]	M. Akil, Y. Chitour, M. Ghader, A. Wehbe. Stability and exact controllability of a timoshenko system with only one fractional damping on the boundary, Asymptotic Anal., 119 (2020), 221–280. https://doi.org/10.3233/ASY-191574 doi: 10.3233/ASY-191574
[41]	M. Akil, I. Issa, A. Wehbe, Energy decay of some boundary coupled systems involving wave euler-bernoulli beam with one locally singular fractional kelvin-voigt damping, Math. Control Relat. Fields, 13 (2023), 330–381. https://doi.org/10.3934/mcrf.2021059 doi: 10.3934/mcrf.2021059
[42]	M. Akil, A. Wehbe, Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions, Math. Control Relat. Fields, 9 (2019), 97–116. https://doi.org/10.3934/mcrf.2019005 doi: 10.3934/mcrf.2019005
[43]	A. Benaissa, S. Gaouar, Asymptotic stability for the lamé system with fractional boundary damping, Comput. Math. Appl., 77 (2019), 1331–1346. https://doi.org/10.1016/j.camwa.2018.11.011 doi: 10.1016/j.camwa.2018.11.011
[44]	B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inf., 23 (2006), 237–257. https://doi.org/10.1093/imamci/dni056 doi: 10.1093/imamci/dni056
[45]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., 44 (1983). https://doi.org/10.1007/978-1-4612-5561-1
[46]	G. Fragnelli, D. Mugnai, Linear stabilization for a degenerate wave equation in non divergence form with drift, preprint arXiv: 2212.05264. https://doi.org/10.48550/arXiv.2212.05264
[47]	W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Am. Math. Soc., 306 (1988), 837–852. https://doi.org/10.2307/2000826 doi: 10.2307/2000826
[48]	M. Akil, Stability of piezoelectric beam with magnetic effect under (coleman or pipkin)–gurtin thermal law, Z. Angew. Math. Phys., 73 (2022). https://doi.org/10.1007/s00033-022-01867-w
[49]	Z. Liu, S. Zheng, Semigroups Associated with Dissipative Systems, CRC Press, 1999.
[50]	A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455–478. https://doi.org/10.1007/s00208-009-0439-0 doi: 10.1007/s00208-009-0439-0
[51]	Z. Liu, B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630–644. https://doi.org/10.1007/s00033-004-3073-4 doi: 10.1007/s00033-004-3073-4
[52]	C. J. K. Batty, T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equations, 8 (2008), 765–780. https://doi.org/10.1007/s00028-008-0424-1 doi: 10.1007/s00028-008-0424-1

This article has been cited by:

1.	Saleh Alshammari, Othman Abdullah Almatroud, Mohammad Alshammari, Hamzeh Zureigat, M. Mossa Al-Sawalha, Exploring Kink Solitons in the Context of Klein–Gordon Equations via the Extended Direct Algebraic Method, 2024, 12, 2227-7390, 3433, 10.3390/math12213433
2.	Imtiaz Ahmad, Muhammad Nawaz Khan, Rashid Jan, Normy Norfiza Abdul Razak, Efficient numerical method for pricing multi-asset options with the time-fractional Black-Scholes model: focus on American and digital options, 2025, 5, 2767-8946, 147, 10.3934/mmc.2025011

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Electronic Research Archive

1 1.7

Metrics

Article views(962) PDF downloads(49) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Electronic Research Archive

Stabilization for a degenerate wave equation with drift and potential term with boundary fractional derivative control

Related Papers:

Abstract

1. Introduction

2. Fundamental concepts

2.1. Fibonacci polynomials

2.2. Lucas polynomials

2.3. Function approximation

3. Proposed methodology

4. Numerical results and discussion

4.1. (1+1)-dimensional case

5. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

Electronic Research Archive

Stabilization for a degenerate wave equation with drift and potential term with boundary fractional derivative control

Related Papers:

Abstract

1. Introduction

2. Fundamental concepts

2.1. Fibonacci polynomials

2.2. Lucas polynomials

2.3. Function approximation

3. Proposed methodology

4. Numerical results and discussion

4.1. (1+1)-dimensional case

5. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog