An Efficient Numerical Approach for Solving Two-Point Fractional Order Nonlinear Boundary Value Problems with Robin Boundary Conditions

TL;DR

This paper introduces a new efficient numerical approach for solving two-point fractional nonlinear boundary value problems with Robin boundary conditions, utilizing high-order predictor-corrector methods and nonlinear shooting techniques.

Contribution

The paper develops novel numerical schemes that transform fractional boundary value problems into initial value problems and employ high-order predictor-corrector methods with nonlinear shooting for improved accuracy.

Findings

Schemes achieve convergence rates of O(h^2) and O(h^3) for Newton and Halley's methods.

Numerical examples confirm the effectiveness and high accuracy of the proposed methods.

The methods can handle both linear and nonlinear fractional boundary value problems with Robin conditions.

Abstract

This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems(FN-BVPs) with Robin Boundary Conditions(RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems(FIVPs) with unknown Initial Conditions(ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton's method and Halley's method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor-Corrector Methods(HPCMs) with linear interpolation and quadratic interpolation into Volterra integral equations which are equivalent to FIVPs. The advantage of proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems(FBVPs) with RBCs and have uniform convergence rates of HPCMs, $O(h^2)$ and $O(h^3)$ for shooting techniques with Newton's method and Halley's method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.