A Wavelet Numerical Method for Solving Nonlinear Fractional Vibration, Diffusion and Wave Equations

In this paper, we present an efficient wavelet-based algorithm for solving a class of fractional vibration, diffusion and wave equations with strong nonlinearities. For this purpose, we first suggest a wavelet approximation for a function defined on a bounded interval, in which expansion coefficients are just the function samplings at each nodal point. As the fractional differential equations containing strong nonlinear terms and singular integral kernels, we then use Laplace transform to convert them into the second type Voltera integral equations with non-singular kernels. Certain property of the integral kernel and the ability of explicit wavelet approximation to the nonlinear terms of the unknown function in the equations enable us to numerically decouple complex spatial and temporal dependencies during solution of these equations, and eventually get a stable, high accuracy and efficient numerical method without involving any matrix inversions for numerically solving the nonlinear fractional vibration, diffusion and wave differential equations. Efficiency and accuracy of the proposed method are justified by numerical examples.

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