Numerical and analytical solutions of new generalized fractional diffusion equation
Our current paper is devoted to studying the numerical and analytical solutions for a class of Generalized Fractional Diffusion Equations (GFDEs) with new Generalized Time-Fractional Derivative (GTFD). The GTFD we propose here is defined in the Caputo sense. We consider the GFDEs on a bounded domain. The numerical solutions are obtained by using the Finite Difference Method (FDM) of full discretization. The stability of FDM is discussed and the order of convergence is evaluated numerically. Numerical experiments are given, which illustrate that the FDM is simple and effective for solving the GFDEs with different coefficients and source functions. An interesting phenomenon is that we can observe the period-like solution in GFDEs with a particular positive periodic weight function. Using the method of separation of variables, we convert the homogeneous GFDE into two ordinary differential equations, and solve them via the help of solutions of the initial value problem with Caputo derivative. In the analytical solution, we observe that the weight function in the denominator, and scale function mapping the response domain differently. Since the derivative considered in this article is new, many existing results of FDEs are generalized.
fractional diffusion equation generalized time-fractional derivative numerical solutions finite difference method fractional calculus