New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind

K. Parand,M. Nikarya

doi:10.1515/nleng-2018-0095

Abstract

Abstract Fractional calculus and fractional differential equations (FDE) have many applications in different branches of sciences. But, often a real nonlinear FDE has not the exact or analytical solution and must be solved numerically. Therefore, we aim to introduce a new numerical algorithm based on generalized Bessel function of the first kind (GBF), spectral methods and Newton–Krylov subspace method to solve nonlinear FDEs. In this paper, we use the GBFs as the basis functions. Then, we introduce explicit formulas to calculate Riemann–Liouville fractional integral and derivative of GBFs that are very helpful in computation and saving time. In the presented method, a nonlinear FDE will be converted to a nonlinear system of algebraic equations using collocation method based on GBF, then the solution of this nonlinear algebraic system will be achieved by using Newton-generalized minimum residual (Newton–Krylov) method. To illustrate the reliability and efficiency of the proposed method, we apply it to solve some examples of nonlinear Abel FDE.

Highlights

Fractional calculus is an ancient mathematical topic, in the last few decades fractional calculus and fractional di erential equations (FDEs) have foundIn this paper, we attempt to introduce a new method, based on a new class of Bessel functions namely generalized Bessel function (GBF) and spectral collocation method for solving nonlinear Abel FDE of the rst kind
We introduce explicit formulas to calculate Riemann–Liouville fractional integral and derivative of GBFs that are very helpful in computation and saving time
Real and practical FDEs often have not exact or analytical solution, numerical solving of the fractional differential equations have become an attractive eld of applied mathematics and computer science

Summary

Introduction

We attempt to introduce a new method, based on a new class of Bessel functions namely GBFs and spectral collocation method for solving nonlinear Abel FDE of the rst kind. The classical Bessel function of the rst kind has been used to solved nonlinear FDEs and fractional intrgro–di erential equation, in their work the calculation of the fractional derivative of basis functions was not easy, for this reason they had to use few bases in spectral expansion [32]. New numerical method based on Generalized Bessel function derivative matrix of the Bessel function and GBF Another novelty in this paper is using Newton–Krylov sub–space method to solve nonlinear system of algebraic equations that be obtained by spectral GBF method

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Methods

Conclusion