Abstract
In this work, we study and extend the one-dimensional fractional derivative to the multidimensional space-time fractional derivative, determine the exact solution via the Laplace transform, and develop mathematical foundations of the respective operators. The multidimensional space-time fractional derivative is developed to augment the equations. The results show that this method is effective, convenient, and easily executable. The main advantage of the proposed approach is that it is an accurate analytical method (the Laplace transform method) that can be implemented for both space and time discretizations of the fractional derivatives and allows us to present new solutions to problems by certain applications for solving space-time fractional derivatives.
Highlights
The integrals of integer order have clear physical interpretations in engineering and geometry and are features that help to solve applied problems in various scientific fields
Mainardi et al [ ] discussed the fundamental solutions of space-time diffusion equation of fractional order using the Fourier and Laplace integral transforms and Mittag-Leffler functions, in which the fundamental solution can only be expressed as a convolution form of a Green function and the initial value function, which is computed
We provide some basic definitions and examples where the method is applied for solving linear space-time fractional differential equations
Summary
The integrals of integer order have clear physical interpretations in engineering and geometry and are features that help to solve applied problems in various scientific fields. They presented a new and efficient algorithm for solving a time fractional subdiffusion equation on a semiinfinite domain. Bhrawy [ ] adapted an operational matrix formulation of the collocation method for one- and two-dimensional nonlinear fractional subdiffusion equations.
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