Comparative Numerical Study of Spline-Based Numerical Techniques for Time Fractional Cattaneo Equation in the Sense of Caputo–Fabrizio

Reny George,Sana Khan,Muhammad Yaseen,Qamar Un Nisa Arif

doi:10.3390/fractalfract6020050

Reny George, Sana Khan + Show 2 more

Open Access

https://doi.org/10.3390/fractalfract6020050

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Abstract

This study focuses on numerically addressing the time fractional Cattaneo equation involving Caputo–Fabrizio derivative using spline-based numerical techniques. The splines used are the cubic B-splines, trigonometric cubic B-splines and extended cubic B-splines. The space derivative is approximated using B-splines basis functions, Caputo–Fabrizio derivative is discretized, using a finite difference approach. The techniques are also put through a stability analysis to verify that the errors do not pile up. The proposed scheme’s convergence analysis is also explored. The key advantage of the schemes is that the approximation solution is produced as a smooth piecewise continuous function, allowing us to approximate a solution at any place in the domain of interest. A numerical study is performed using various splines, and the outcomes are compared to demonstrate the efficiency of the proposed schemes.

Highlights

The time fractional Cattaneo differential equation (TFCDE) under consideration is [1]Academic Editors: Amar Debbouche ∂2 v(s, t) ∂v(s, t) CF α + a Dt v(s, t) = + g(s, t), ∂t∂2 s and Hari Mohan Srivastava (1)Received: 15 December 2021Accepted: 11 January 2022Published: 18 January 2022 with initial conditions v(s, 0) = φ(s), vt (s, 0) = ψ(s), Publisher’s Note: MDPI stays neutral
Inspired by the popularity of spline approaches in finding numerical solutions of fractional partial differential equations, various splines-based numerical techniques have been developed for the numerical solution of the Cattaneo equation involving the Caputo–Fabrizio derivative
By following the same procedure as was done for cubic B-splines and using the extended cubic B-spline (ECuBS) approximation given in (16), we obtain the following approximation to the solution of (1)

Summary

Introduction

The time fractional Cattaneo differential equation (TFCDE) under consideration is [1]. A numerical method must be used to obtain the solution to these partial differential equations To tackle these problems numerically, many approaches have been developed and extended. When compared to the finite difference approach, other spectral methods, such as the operational matrix method, are popular since they provide good accuracy and take less time to compute This method works well with fractional ordinary differential equations (ODEs), fractional partial differential equations (PDEs), and variable order PDEs. Jafari et al [9] gave applications of Legendre wavelets in solving FDEs numerically. Inspired by the popularity of spline approaches in finding numerical solutions of fractional partial differential equations, various splines-based numerical techniques have been developed for the numerical solution of the Cattaneo equation involving the Caputo–Fabrizio derivative. The presented schemes are tested for stability and convergence analysis

Numerical Scheme Based on Cubic B-Splines

Numerical Scheme Based on Extended Cubic B-Splines

Numerical Scheme Based on Trigonometric Cubic B-Splines

Stability Analysis

Convergence Analysis

Numerical Findings and Discussion

Concluding Remarks