### The Analytic Solution of Time-Space Fractional Diffusion Equation via New Inner Product with Weighted Function

#### Abstract

In this research, we determine the analytic solution of initial boundary value problem including time-space fractional differential equation with Dirichlet boundary conditions in one dimension. By using separation of variables the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Caputo sense. A new inner product with weighted function is defined to obtain coefficients in the Fourier series.

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DOI: http://dx.doi.org/10.26713%2Fcma.v10i4.1290

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