A Dimension-Unified Homotopy Analysis Framework for Twoand Three-Dimensional Wave Equations

Source abstract

This paper presents a unified application of the Homotopy Analysis Method (HAM) to two- and three-dimensional linear and nonlinear partial differential wave equations. A common solution framework is formulated for both dimensions, while the nonlinear terms are incorporated into the successive deformation equations. The method constructs a series solution without requiring a small perturbation parameter. An auxiliary convergence-control parameter provides a means to examine and adjust the convergence of the resulting series. Representative initialvalue problems are used to illustrate the formulation and the calculation of successive solution terms. Where exact solutions are available, they provide benchmarks for evaluating the approximations; for nonlinear problems, residual errors can be used to assess solution quality. The study outlines how the same HAM procedure can be adapted across different dimensions and equation types, offering a systematic basis for analysing multidimensional wave propagation. The numerical accuracy and convergence range depend on the specific problem and chosen auxiliary parameters.