Local RBF Method for Transformed Three Dimensional Sub-Diffusion Equations

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In this article we study the numerical solution of time fractional 3D sub-diffusion equation in Caputo sense. The time fraction sub-diffusion equation is obtained by replacing the time derivative in the standard sub-diffusion equation with time fractional derivative. Numerous physical phenomenon are modeled via sub-diffusion equation such as porous systems, polymers, and turbulent plasma etc. The local RBF method is one of most successful methods for numerical solution of integer and non-integer order PDEs, as it overcome the computational burden of the global RBF method. Our proposed numerical scheme approximates the solution in three main steps. Firstly, we use the Laplace transform to reduce the given time dependent problem to an equivalent time independent problem. Secondly, we approximate the solution of the transformed equation using the local RBF method. Finally, the solution of original problem is retrieved via inverse Laplace transform by representing it as a Bromwich integral. The Bromwich integral is then approximated using the mid point rule. The purpose of using the Laplace transform is avoiding the classical time marching procedure. One of the main benefits of local RBF method is that it only considers the nodes located in the sub-domain surrounding a local node for numerical approximation at this node. This method perform well for large scale problem and can also reduce the ill conditioning issues. The convergence and stability of the proposed numerical method are discussed. The proposed numerical method is tested against three examples. The computational results highlight the accuracy and efficiency of the method.

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International Journal of Applied and Computational Mathematics



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