Numerical technique for fractional variable-order differential equation of fourth-order with delay

Abstract

In this paper, we construct a new efficient numerical scheme for variable fractional order differential equation of fourth order with delay. Here, we propose to use parametric quintic spline in the spatial dimension and $L2-1_{\sigma }$ formula for time dimension. The stability, convergence, and solvability are rigorously proved using discrete energy method. Our proposed scheme improves convergence in both aspects (spatial-dimension and time-dimension) in comparison to those earlier work. Numerical simulation is carried out using the MATLAB software to demonstrate the effectiveness of our scheme.

Introduction

Recently, researchers have found that fractional-order behavior is encountered in many dynamical processes. As most of the dynamical phenomena evolve continuously; therefore, it becomes natural to broaden the integer-order derivatives concept to fractional-order derivatives. Broadening integer-order derivatives field leads to a well-known field known as fractional calculus. It possesses an old history and has critical applications in various areas such as (see [18], [24], [30], [31]). Fractional differential equations applications list has grown at a vast rate in multiple areas such as the study of creep or relaxation visco-elastoplastic materials, control problems, plasma Physics, diffusion process models, etc. [38], [41]. Although delay differential equations (DDEs) have an important place in various areas of real-life based practical applications. These equations are also employed to model time-delayed systems such as in control systems, high-speed machining, power-systems, communication [40]. Fractional differential equations including delays are used to model dynamical systems or natural phenomenon in a more accurate way [5], [6], [45].

Nowadays, variable-order-calculus have many vital applications mainly in viscoelastic mechanics [12], geographic data [13], signal confirmation [44], and diffusion process [39]. The kernel of a variable-order operator has a variable exponent, therefore, it's very tedious to obtain a solution for the variable-order fractional differential equation. Thus, it becomes vital to construct numerical techniques to solve variable-order fractional differential equations accurately and efficiently. Many researchers have studied various numerical methods extensively for solving variable-order fractional differential equations which mainly cover finite difference methods [3], [4], [8], [15], [29], [37], [42], [46], [47], [48], spectral methods [10], [11], matrix methods [27], reproducing kernel methods [28], and so on.

Zhuang et al. [49] considered a fractional advection-diffusion equation of variable-order comprising a nonlinear source term on a finite domain and implemented explicit and implicit Euler approximations for its numerical solution.

Chen et al. [9] studied the variable-order fractional sub-diffusion equation of two-dimensional and solved using explicit and implicit difference scheme. Shen et al. [39] adopted Coimbra Variable Order time-fractional derivative, considering it is having more desirable properties for physical modeling. They developed a difference approximation scheme and analyzed it using Fourier analysis.

Zhao et al. [48] derived an approximation formula for the variable-order fractional time derivatives involved with anomalous diffusion and wave propagation. The authors analyzed that variable-order fractional derivative can be employed to model the anomalous transport with spatiotemporal variability. They constructed a new efficient numerical technique that can be used for long-standing issues of monotonicity of integer-order PDEs and outflow boundary conditions.

Li and Wu [27] proposed a numerical scheme based on the reproducing kernel method for variable fractional functional boundary value problems and discussed error estimates for the same. Chen et al. [11] proposed a numerical approach to solve a class of nonlinear variable-order fractional differential equations using Legendre wavelet functions and operational matrices which converts FDEs into a system of algebraic equations.

Cao and Qiu [7] derived a numerical technique of high order for variable-order fractional differential equations with the help of second-order numerical approximation using shifted Grunwald and proved stability and convergence of the scheme.

Authors in [3] proposed accurate and robust approach to approximate the solution of the functional Dirichlet boundary value problem based on the shifted Chebyshev polynomials. The representation the variable-order fractional derivative was reduced to a system algebraic equations, which significantly simplifies the solution process.

Tayebi et al. [43] developed an accurate mesh-less numerical scheme based on the moving least squares (MLS) approximation and the finite difference scheme for variable-order time-fractional advection-diffusion equation. They considered the finite difference technique with a θ-weighted approach, and in the spatial dimension, the MLS approximation is employed to obtain the semi-discrete solution. Jia et al. [20] presented an efficient numerical scheme for variable-order fractional functional boundary value problems which relies on the simplified reproducing kernel method. They also proved stability, convergence, and demonstrated their numerical schemes significance through a few examples. Authors in [33], [34], [35], [36] have studied various types of delay differential equation of fourth-order. The authors constructed efficient numerical schemes being focused on increasing the order of convergence in both spatial and time dimensions. They analyzed the stability and convergence of numerical schemes constructed and also presented numerical experimentation to demonstrated the effectiveness of numerical schemes developed. Moghaddam and Machado [32] derived a stable three-level explicit scheme for a class of nonlinear time variable-order fractional partial differential equations based on the linear B-spline approximation of the time-variable order fractional derivative in the Caputo sense and the Du Fort-Frankel algorithm. Further, they discussed unconditional stability and the convergence of the established scheme.

Bhawry and Zaky [4] considered a natural generalization of the fractional Schrödinger equation to variable-order fractional Schrödinger equation to study fractional quantum phenomena. They developed an exponentially accurate Jacobi-Gauss-Lobatto collocation scheme for one dimension and two dimensions. In this method, the problem, as mentioned above, is reduced to a system of ordinary differential equations (ODEs) in the time variable. They proposed two efficient schemes (one based on the implicit Runge-Kutta method of fourth-order and second is based on Jacobi-Gauss-Radau collocation method) to deal with initial value problems for a nonlinear system of ordinary differential equations.

Gámez-Aguilar [16] analyzed a model of alcoholism, which involves the impact of Twitter via Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives of constant and variable-order. They considered two fractional mathematical models with and without delay and obtained solutions using an iterative scheme via Laplace and Sumudu transform and solved the generalized model numerically via the Adams method and the Adams-Bashforth-Moulton scheme.

The study discussed above so far was not precisely concerned with increasing convergence order in both spatial and temporal dimensions. In this paper, we would like to augment the convergence order with the help of parametric quintic spline operator and $L2-1_{\sigma }$ formula (an approximation of Caputo fractional derivative). Additionally, we will prove the stability and convergence of numerical scheme introduced in this paper along with numerical experimentation. Analysis of numerical methods for solving fractional differential equations with and without a single delay and studying linear FDDEs with multiple delays has been studied in [5], [6], [40], [45]. There is no study available for a non-linear fourth-order fractional differential equation with delay by the use of parametric quintic spline to the best of the author's knowledge. All previous studies available for variable-order fractional differential equations have been developed using the finite difference (Implicit and Explicit), Meshless method, using Chebyshev polynomials, Grunwald-Letinkov approximation, etc. [3], [7], [9], [11], [16], [20], [27], [32], [33], [34], [35], [36], [39], [43], [48], [49]. Most of the ways as mentioned earlier are arduous and time-consuming and exhibit first-order convergence for sub-diffusion variable-order differential equations. Moreover, the topic of variable-order fractional differential equations has not been explored extensively using delay, non-linearity, and enhancing the order of convergence. In the current work, we shall discuss the numerical treatment of a fourth-order fractional diffusion-wave problem. Noting that $O(h^4)$ and $O({\tau }^{2-\alpha })$ are the best convergence orders achieved in the spatial and temporal dimensions respectively, our aims are to improve the spatial convergence order to 4.5 and temporal order to 2, and to provide rigorous proof of solvability, convergence and stability of the proposed method. In the existing literature, a few work is available to construct higher-order numerical methods for the variable-order fractional differential equations with delay. Construction of a mathematical approach for such equations involves the more numerical analysis.

In this paper, we consider the following variable-order time-fractional sub-diffusion equation of fourth-order with delay

$${}_0{}^{C}D_t^{\alpha (x,t)}U(x,t)+\frac{{\partial }^{4}U(x,t)}{\partial x^4}=F(x,t,U(x,t),U(x,t-s)),s>0,(x,t)\in (0,L)\times (0,T),$$

$$\begin{gathered} \text{Following are the initial and boundary conditions:} \\ U(x,t)=\varphi (x,t),(x,t)\in [0,L]\times [-s,0], \end{gathered}$$

$$U(0,t)={\alpha }_1(t),U(L,t)={\alpha }_2(t),t\in [0,T],$$

$$\frac{{\partial }^{2}U(0,t)}{\partial x^2}={\beta }_1(t),\frac{{\partial }^{2}U(L,t)}{\partial x^2}={\beta }_2(t),t\in [0,T].$$

Here, ${}_0{}^{C}D_t^{\alpha (x,t)}U(x,t)$ is the variable-order time fractional derivative of order $\alpha (x,t)\in (0,1)$ defined in the sense of Caputo, $s>0$ is the delay, right hand side gives the non-linear source function $F(x,t,U(x,t),U(x,t-s))$ with delay.

Here, $s>0$ is the time delay constant, $\varphi (x,t)$, ${\alpha }_1(t)$, ${\alpha }_2(t)$, ${\beta }_1(t)$, ${\beta }_2(t)$ are all given sufficiently smooth functions. $\varphi (0,t)={\alpha }_1(t)$, $\varphi (L,t)={\alpha }_2(t)$, ${\varphi }^{″}(0,t)={\beta }_1(t)$, and ${\varphi }^{″}(L,t)={\beta }_2(t)$.