Numerical Solutions of the Multi-Space Fractional-Order Coupled Korteweg–De Vries Equation with Several Different Kernels

Abstract

In this article, the authors propose to investigate the numerical solutions of several fractional-order models of the multi-space coupled Korteweg–De Vries equation involving many different kernels. In order to transform these models into a set or system of differential equations, various properties of the first-kind Chebyshev polynomial are used in this study. The main objective of the present study is to apply the spectral collocation approach for the multi-space fractional-order coupled Korteweg–De Vries equation with different kernels. We use finite differences to numerically solve these differential equations by reducing them to algebraic equations. The Newton (or, more precisely, the Newton–Raphson) method is then used to solve these resulting algebraic equations. By calculating the error involved in our approach, the precision of the numerical solution is verified. The use of spectral methods, which provide excellent accuracy and exponential convergence for issues with smooth solutions, is shown to be a benefit of the current study.

Keywords

• multi-space fractional-order coupled Korteweg–De Vries equation
• Chebyshev polynomials of the first kind
• Chebyshev spectral collocation method
• Newton–Raphson method
• operators of fractional calculus
• Riemann–Liouville and Liouville–Caputo fractional derivatives