The Fibonacci wavelets approach for the fractional Rosenau–Hyman equations

Abstract

In this study, a novel approach, called the Fibonacci wavelet collocation technique (FWCT), is presented for the numerical solution of nonlinear fractional order partial differential equations (FPDEs). The chosen numerical strategy is based on the collocation method with the Fibonacci wavelet and its functional integration matrix. The newly developed numerical approach for the nonlinear time-fractional Rosenau–Hyman equation is the focus of this paper. The corresponding fractional order derivative is taken in the Caputo sense. This effective numerical approach converts the nonlinear problem into a set of algebraic-type nonlinear equations. This nonlinear system of equations is analyzed using the Newton–Raphson method to find the unknown coefficients. We provide the different types of error analysis of the present technique. The exact solution and the results obtained by the other literary methods are compared to the results achieved. The numerical outcomes demonstrate that the current analysis offers a more accurate approximation than the prior approaches. The result derived with the aid of this technique is in good agreement and indicates that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally attractive, efficient, effective, reliable, and robust for solving various physical models in science and engineering. We made all calculations and plotted the graphs using Mathematica software.

Keywords

• Fibonacci wavelets
• Collocation method
• Functional matrix of integration
• Rosenau–Hyman equation
• Caputo fractional derivative
• Newton–Raphson technique