Abstract
In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions.
Highlights
multistep differential transform method (MsDTM), we found that the differential transform method (DTM) is an upgraded method of the Taylor series method, which requests extra computational work for large orders, and it decreases the size of the computational domain [27]
The unique functions of mathematical physics are found to be very useful for finding solutions of initial- and boundary-value problems governed by partial differential equations and fractional differential equations, and they play a significant and exciting role as solutions of fractional-order differential equations [30]
We can notice that the lower the fractional order, the more the approximate solutions move away from the exact solution of nonfractional order, and their value increases with the value of the variable t being constant
Summary
The unique functions of mathematical physics are found to be very useful for finding solutions of initial- and boundary-value problems governed by partial differential equations and fractional differential equations, and they play a significant and exciting role as solutions of fractional-order differential equations [30]. Many special functions have attracted the attention of researchers, such as the Wright function, the error function, and the Millin–Ross function. Our attention is focused on only two types of these special functions: the Mittag–Leffler function and the Gamma function. We used the Mittag–Leffler function since after finding the solution in a compact form, we can write the exact solution by using the definition of the Mittag–Leffler function, while the Gamma function is an essential part of the definition of fractional derivatives
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