On a class of coupled fractional nonlinear singular boundary value problems arising in dusty fluid models

TL;DR

This paper introduces a new numerical method combining fractional Haar wavelet collocation and Newton-Raphson techniques to solve coupled fractional nonlinear boundary value problems relevant in dusty fluid models, demonstrating high accuracy and convergence.

Contribution

The paper presents a novel numerical approach for solving coupled fractional boundary value problems using fractional Haar wavelet collocation combined with Newton-Raphson, validated through convergence analysis and numerical experiments.

Findings

Residual error decreases with higher resolution levels

Method effectively solves nonlinear singular fractional problems

Mathematica software successfully implements the approach

Abstract

In this article, we introduce a new class of coupled fractional Lane-Emden boundary value problems. We employ a novel approach, the fractional Haar wavelet collocation method with the Newton-Raphson method. We analyze the conditions in two cases to present numerical experiments related to the defined system of fractional differential equations. To validate the accuracy of the proposed method we present the convergence of the method, and we demonstrate the method's effectiveness through five numerical experiments, highlighting real-world applications of fractional differential equations. Using figures and tables, we show that the residual error decreases as we increase the value of the maximum level of resolution $J$ while keeping the order of derivatives fixed, and similar trends also observe when $J$ is fixed and vary the order of fractional derivatives. We demonstrate that Mathematica software can be used effectively to solve such nonlinear singular fractional boundary value problems.