BLUES iteration applied to nonlinear ordinary differential equations for wave propagation and heat transfer

TL;DR

This paper applies the BLUES iteration method to nonlinear ordinary differential equations in physics, demonstrating fast convergence and accurate solutions for problems in wave propagation and heat transfer.

Contribution

It introduces the application of the BLUES iteration method to nonlinear ODEs in physics, showing its effectiveness and fast convergence compared to existing methods.

Findings

The method accurately reproduces oscillatory behavior in nonlinear oscillators.

BLUES method shows favorable comparison with Adomian decomposition in heat transfer problems.

Solutions converge exponentially fast to the exact solutions.

Abstract

The iteration sequence based on the BLUES (Beyond Linear Use of Equation Superposition) function method for calculating analytic approximants to solutions of nonlinear ordinary differential equations with sources is elaborated upon. Diverse problems in physics are studied and approximate analytic solutions are found. We first treat a damped driven nonlinear oscillator and show that the method can correctly reproduce oscillatory behaviour. Next, a fractional differential equation describing heat transfer in a semi-infinite rod with Stefan-Boltzmann cooling is handled. In this case, a detailed comparison is made with the Adomian decomposition method, the outcome of which is favourable for the BLUES method. As a final problem, the Fisher equation from population biology is dealt with. For all cases, it is shown that the solutions converge exponentially fast to the numerically exact solution, either globally or, for the Fisher problem, locally.