Abstract
In this study, a relatively new method to solve partial differential equations (PDEs) called the fractional reduced differential transform method (FRDTM) is used. The implementation of the method is based on an iterative scheme in series form. We test the proposed method to solve nonlinear fractional Burgers equations in one, two coupled, and three dimensions. To show the efficiency and accuracy of this method, we compare the results with the exact solutions, as well as some established methods. Approximate solutions for different values of fractional derivatives together with exact solutions and absolute errors are represented graphically in two and three dimensions. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional partial differential equations over existing methods.
Highlights
Mathematical modeling of nonlinear systems is a major challenge for scientists currently.The study of the exact and approximate solutions helps us to understand the applications of these mathematical models
In the comparison between differential transform method (DTM), reduced differential transform method (RDTM), fractional reduced differential transform method (FRDTM), and multi-step differential transform method (MsDTM), we find that DTM is an improved method of the Taylor series method, which needs additional computational work for large orders, and it decreases the size of the computational domain and is appropriate for numerous problems [14]
Example 1 indicated that the third order approximate solution was accurate in comparison with the exact solution; see Figure 1
Summary
Mathematical modeling of nonlinear systems is a major challenge for scientists currently. In [8,9], we modified the definition of the beta fractional derivative to find exact and approximate solutions of time fractional diffusion equations in different dimensions. Abuasad et al [19] introduced a new modification of the fractional reduced differential transform method (m-FRDTM) to find exact and approximate solutions for multi-term time fractional diffusion equations (MT-TFDEs). Studied eight different cases to obtain the approximate analytical solutions of the Benney–Lin equation with the fractional time derivative by FRDTM and the homotopy perturbation method (HPM). Singh and Srivastava [24] presented an approximate series solution of the multi-dimensional (heat-like) diffusion equation with the time fractional derivative using FRDTM. Singh [25] presented FRDTM to compute an alternative approximate solution of the initial valued autonomous system of linear and nonlinear fractional partial differential equations.
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