Hybrid Fractional Diffusion Problem with Dirichlet Boundary Conditions

Abstract

In this research, we discuss the construction of the analytic solution of homogenous initial boundary value problem including partial differential equations of fractional order. Since homogenous initial boundary value problem  involves Hybrid fractional order derivative, it  has classical initial and boundary  conditions. By means of separation of the variables method and the inner product defined on $L^2\left[0,l\right]$, the solution is constructed in the form of a Fourier series including the bivariate Mittag-Leffler function. An illustrative example presents the applicability and influence of the separation of variables method on time fractional diffusion problems. Moreover, as the fractional order $\alpha$ tends to $1$, the solution of the fractional diffusion problem tends to the solution of the diffusion problem which proves the accuracy of the solution.