A rapidly convergent approximation scheme for nonlinear autonomous and non-autonomous wave-like equations

TL;DR

This paper introduces a new approximation scheme for nonlinear wave-like equations that converges rapidly, providing accurate solutions with fewer terms, especially effective for integrable cases, and demonstrated through various examples.

Contribution

The paper proposes a novel exponential series-based approximation method that converges faster than existing methods for nonlinear PDEs with constant or variable coefficients.

Findings

Faster convergence compared to other methods

Exact solutions obtainable from initial terms in integrable cases

Error estimates and convergence proof provided

Abstract

In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of exponential instead of an algebraic function of independent variables. As a consequence: i) the convergence of the series found to be faster than the same obtained by few other methods and ii) the exact analytic solution can be obtained from the first few terms of the series of the approximate solution, in cases the equation is integrable. The convergence of the sum of the successive correction terms has been established and an estimate of the error in the approximation has also been presented. The efficiency of the present method has been illustrated through some examples with a variety of nonlinear terms present in the equation.