Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations
Vipul K Baranwal1, Ram K Pandey2, Om P Singh3
1Department of Applied Mathematics, Maharaja Agrasen Institute of Technology, Rohini, Delhi 110086, India.
International Scholarly Research Notices
|July 21, 2016
Summary
We introduce an optimal variational asymptotic method to solve complex nonlinear partial differential equations with fractional time derivatives. This efficient technique provides accurate solutions in few iterations for fractional advection-diffusion and Swift-Hohenberg equations.
Area of Science:
- Numerical Analysis
- Applied Mathematics
- Computational Physics
Background:
- Fractional partial differential equations (FPDEs) model complex phenomena but are challenging to solve.
- Existing methods may lack efficiency or accuracy for nonlinear FPDEs.
Purpose of the Study:
- To develop an efficient and accurate numerical method for time fractional nonlinear partial differential equations.
- To introduce an enhanced variational iteration method with optimal parameter selection.
Main Methods:
- The optimal variational asymptotic method is proposed, extending the standard variational iteration method.
- Auxiliary parameters and functions are introduced into the correction functional.
- Optimal parameter values are determined by minimizing the square residual error.
Main Results:
- The method is successfully applied to nonlinear fractional advection-diffusion and Swift-Hohenberg equations.
- Fairly accurate solutions are achieved with only a few iterations.
- The proposed technique demonstrates high efficiency and accuracy.
Conclusions:
- The optimal variational asymptotic method is a powerful tool for solving time fractional nonlinear PDEs.
- The method offers a balance of accuracy, efficiency, and simplicity.
- It provides a reliable approach for analyzing complex fractional dynamics.
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