hybrid fractional diffusion problem with dirichlet boundary conditions

The aim of this research is to establish the analytic solution of partial differential equations with homogenous initial boundary conditions. having hybrid fractional order derivative allows us to have classical boundary and initial conditions. the solution of the problem is obtained in terms of bivariate mittag-leffler function as a fourier series by utilizing separation of variables method (svm) and the inner product defined on l ² [0, l]. the presented examples illustrate the accuracy and effectiveness of the svm for the fractional diffusion problems. the accuracy of the obtained solution can also be seen from the observation that as the fractional order α tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem.