New approximations for solving the Caputo-type fractional partial differential equations

Partial differential equations with the Caputo-type fractional derivative have been used in many engineering applications. Because the Caputo-type fractional derivative is an integral of the solution with respect to time, the numerical scheme for solving this type of fractional differential equations requires using the values of all previous time steps. This needs a large size of memory to store the necessary data when computing, which may lead to a memory problem in computer, particularly when solving systems of multi-dimensional fractional differential equations. For this purpose, this article presents a new approximation for solving the fractional differential equations using the Laplace transform method. The obtained differential equations will then be solved using some conventional two or three-level in time finite difference schemes, which reduce the computational cost significantly. For simplicity, the method is presented in 1D and is illustrated by several 1D and 2D examples. In practical, this method can be readily used for solving more complex cases within a reasonable accuracy.