Exact Solutions for Real and Complex Nonlinear Partial Differential Equations in Engineering Using a Modified Homotopy Perturbation Method
Abstract
Nonlinear partial differential equations (PDEs) play a fundamental role in modeling complex phenomena in physics and engineering. However, solving them analytically and numerically remains challenging due to divergence, slow convergence, and computational inefficiency. This study implements a refined variation of the homotopy perturbation method (HPM), termed the modified homotopy perturbation method (MHPM), to improve accuracy and stability in solving nonlinear PDEs. The motivation behind this work stems from the need for a more efficient and reliable approach, particularly for equations where traditional methods fail to yield convergent or computationally efficient solutions. By integrating Padé approximants and Laplace transforms, MHPM enhances convergence, reduces computational effort, and maintains high precision. To validate its effectiveness, the method is applied to several well-known nonlinear equations, including the nonlinear gas dynamics equation, nonlinear wave-like equation, cubic complex Ginzburg-Landau equation, one- and two-dimensional nonlinear Schrödinger equations, and higher-order nonlinear PDEs that are fundamental in fluid dynamics, wave propagation, quantum mechanics, and mathematical modeling. A systematic comparison confirms that MHPM not only improves solution accuracy but also provides a practical, computationally efficient alternative to conventional analytical and numerical techniques. The results underscore its potential for multi-dimensional and time-dependent PDEs, reinforcing its value in nonlinear system analysis across theoretical and applied sciences.