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Approximate analytical solutions of the regularized long wave equation using the optimal homotopy perturbation method
Constantin Bota1, Bogdan Căruntu1
1Department of Mathematics, Politehnica University of Timişoara, P-ta Victoriei 2, 300006 Timişoara, Romania.
Thescientificworldjournal
|July 9, 2014
Summary
A new optimal homotopy perturbation method accelerates convergence for solving nonlinear partial differential equations. This advanced technique demonstrates high accuracy in analytical solutions compared to existing methods.
Area of Science:
- Applied Mathematics
- Numerical Analysis
- Computational Science
Background:
- Nonlinear partial differential equations (NPDEs) are fundamental in modeling complex phenomena across science and engineering.
- Traditional analytical and numerical methods often face challenges in solving NPDEs due to nonlinearity and complexity.
- The homotopy perturbation method (HPM) offers a framework for approximate analytical solutions but can suffer from slow convergence.
Purpose of the Study:
- To introduce and evaluate the optimal homotopy perturbation method (OHPM) for solving NPDEs.
- To demonstrate the accelerated convergence properties of OHPM compared to the standard HPM.
- To validate the high accuracy of OHPM through practical applications and comparisons.
Main Methods:
- The study develops the optimal homotopy perturbation method (OHPM) by enhancing the standard homotopy perturbation method (HPM).
- OHPM incorporates an optimization process to determine optimal parameters within the perturbation equations.
- The method is applied to solve representative nonlinear partial differential equations.
Main Results:
- The optimal homotopy perturbation method (OHPM) achieved significantly faster convergence rates than the traditional homotopy perturbation method (HPM).
- Applications demonstrated the high accuracy of OHPM, yielding results that closely match exact solutions where available.
- Comparisons with previously reported results confirmed the superior performance of OHPM in terms of accuracy and efficiency.
Conclusions:
- The optimal homotopy perturbation method (OHPM) is a powerful and efficient technique for obtaining approximate analytical solutions to nonlinear partial differential equations.
- OHPM offers a notable improvement over the standard HPM, particularly in terms of convergence speed and solution accuracy.
- The method's effectiveness makes it a valuable tool for researchers and practitioners dealing with complex nonlinear systems.
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