Spectral solutions for a class of nonlinear wave equations with Riesz fractional based on Legendre collocation technique

Abstract

A numerical investigation is presented in this work for a class of Riesz space-fractional nonlinear wave equations (MD-RSFN-WEs). The presence of a spatial Laplacian of fractional order, stated by fractional Riesz derivatives, is taken into consideration by the model. The fractional wave equation governs mechanical diffusive wave propagation in viscoelastic medium with power-law creep and, as a result, gives a physical understanding of this equation within the context of dynamic viscoelasticity. To deal with the independent variables, a totally spectral collocation approach is used. Our approach has shown to be more precise, efficient, and practical for the present model. The findings demonstrated that the spectral scheme is exponentially convergent.

Introduction

Because it is more suited for modeling various real life situations than the classical derivative [1], [2], [3], The concept of fractional derivatives [4], [5], [6], [7] has evolved into one of the most major aspects in applied mathematics. The chief factor is because it is widely used in areas like chemistry [8], biology [9], physics [10], engineering [11], and finance [12]. Fractional derivatives have been mentioned in various strategies in the mathematical literature, in which Riemann–Liouville and Caputo fractional senses are included.

Fractional convection–diffusion equations are frequently used to design a broad variety of physical processes, including how oil reservoir simulations work, global weather production, energy transportation, and mass and chemical spread in reactors. Time-fractional convection–diffusion equations can be employed to model time-related anomalous diffusion processes. It is an outgrowth of conventional convection diffusion in which a fractional order time derivative is substituted for the integer order [13], [14], [15]. Non-linear FCDEs have been solved using a variety of methods, including the Galerkin method [16], direct meshless scheme in addition to the Gaussian radial basis function [17], mixed finite element and H¹-Galerkin [18], cubic B-spline basis functions and Galerkin methods [19] and finite difference method [20], [21], [22].

By applying Riesz fractional problems, the anomalous diffusion and relaxation procedures [23], [24] have been fully described. The problem in which the Riesz derivative indicates nonlocality and is used to represent the diffusion concentration’s dependence on route. To represent the above type of phenomenon, fractional differential equations with the Riesz derivative are needed. Riesz derivatives are fractional operators with two sides (left and right derivatives). This property is incredibly beneficial for fractional modeling on finite domains. There is only a modest number of studies on Riesz fractional differential equations. The existence of Riesz–Caputo fractional differential equations has been studied by Chen et al. [25]. The finite difference method was employed to fix the time–space Riesz fractional advection–diffusion equations in [26]. Variational optimal problems including Riesz–Caputo fractional derivative have been discussed in [27], [28], [29]. Many mathematical technologies have been implemented to deal with Riesz Riemann–Liouville derivatives fractional issues [30], [31], [32], [33], [34]. The fractional problems with both Riesz and distributed operators were treated by second-order finite difference [35], finite element [36] and finite volume techniques [37], [38]. In [12], [39], [40], [41], [42], spectral solutions for numerous issues including the Riesz fractional operator are presented.

Spectral techniques [43], [44], [45], [46] are effective instruments for dealing with many sorts of differential [47], [48] and integral issues [49], [50], [51], [52]. Given the difficulty of obtaining analytical solutions to space and/or time-fractional differential equations, finding effective numerical approaches is a major priority. In comparison to the effort expended on the literature to analyze finite difference schemes for tackling fractional problems, only a small amount of effort has been expended in designing and studying spectral strategies [53], [54], [55].

The objective of this research is to solve a class of Riesz fractional nonlinear wave equations numerically. The wave equation is a basic second-order partial differential equation that is used in physics to describe waves such as light waves, sound waves, and water waves. This equation is used in many fields, including acoustics, electromagnetics, and fluid dynamics. For variable-order fractional wave problem, we used a collocation approach [56] to achieve more accurate numerical solutions. The current work presents a numerical investigation for a class of MD-RSFN-WEs using the Legendre collocation technique. The foremost purpose of this research is to create the Gauss–Lobatto shifted Legendre collocation technique (GLSLCT) and the Gauss–Radau shifted Legendre collocation technique (GRSLCT) for dealing with spatial and temporal variables, respectively.

The structure of the paper is outlined below. Section 2 discusses some preliminary matters. Section 3 outlines the numerical method for solving the MD-RSFN-WEs. Section 4 solves and analyzes three cases to demonstrate the efficiency and correctness of the method. Section 5 discusses the major conclusions.