Legendre Wavelet Based Numerical Solution of One-Dimensional Space-Time Fractional Order Diffusion Equation: An Eigenfunction Approach

An efficient numerical technique has been used in the proposed work to deal with the space-time fractional diffusion model in which the space derivative is considered a Riemann-Liouville (RL) fractional derivative and the time derivative as a Caputo fractional derivative. First, the model is converted to Caputo fractional diffusion model and then the separation of variable technique has been used to transform this problem into a fractional eigenvalue problem. The Legendre wavelet (LW) collocation method is applied to this fractional eigenvalue problem to acquire a matrix eigenvalue problem to compute both real and complex eigenvalues. In the current work, an eigenfunction technique has also been used to produce a good approximation of the solution to the given fractional diffusion equation (FDE). The dilation parameter of the Legendre wavelet is increased in the present work to refine the eigenvalues and their corresponding eigenfunctions and also to see their effect on the approximate solution.