The BLUES function method for second-order partial differential equations: application to a nonlinear telegrapher equation

TL;DR

This paper extends the BLUES function method to second-order PDEs, introduces a matrix formalism for initial conditions, and demonstrates its effectiveness on a nonlinear telegrapher equation compared to other methods.

Contribution

The paper develops a matrix-based extension of the BLUES method for second-order PDEs, incorporating initial conditions as a source vector, and applies it to a nonlinear telegrapher equation.

Findings

The matrix BLUES method effectively solves the nonlinear telegrapher equation.

It compares favorably with Adomian, variational iteration, and homotopy perturbation methods.

The method offers a promising alternative for solving second-order PDEs.

Abstract

An analytic iteration sequence based on the extension of the BLUES (Beyond Linear Use of Equation Superposition) function method to partial differential equations (PDEs) with second-order time derivatives is studied. The original formulation of the BLUES method is modified by introducing a matrix formalism that takes into account the initial conditions for higher-order time derivatives. The initial conditions of both the solution and its derivatives now play the role of a source vector. The method is tested on a nonlinear telegrapher equation, which can be reduced to a nonlinear wave equation by a suitable choice of parameters. In addition, a comparison is made with three other methods: the Adomian decomposition method, the variational iteration method (with Green function) and the homotopy perturbation method. The matrix BLUES function method is shown to be a worthwhile alternative for the other methods.