Modified Semiseparation of Variables Methods for Solving the System of Nonlinear Fractional Partial Differential Equations

The system of nonlinear fractional partial differential equations (SNFPDEs) are widely used in modeling various phenomena in applied sciences. Consequently, finding the solutions to SNFPDEs has become paramount. Recently, an analytic method known as the Semiseparation of Variables Method (S-SVM) has been applied to obtain the exact solution of the SNFPDEs by employing the Mittag–Leffler function method. The Salim function is a generalization of the Mittag–Leffler function with two additional parameters. These two additional parameters in the Salim function offer the flexibility and easier use of the Salim function as compared to the Mittag–Leffler function in one parameter. Similarly, the Salim and Faraj function extends the Salim function. In this paper, we introduce two modifications to the S-SVM, namely, the S-SVM via the Salim function (S-SVSFM) and the S-SVM via the Salim and Faraj function (S-SVSFFM). Interestingly, applying the S-SVSFM and the S-SVSFFM independently to solve the SNFPDEs yields an infinite series that converges easily to the exact solution of the SNFPDEs. In addition, the uniqueness of the exact solution of the SNFPDEs is established therein the paper. Choosing the value of α = 1, confirms the integer solution of the SNFPDEs which is depicted in tables. However, we observed that as the values of the α changes continuously from 0.1 to 1.0, the memory and hereditary properties are observed. The findings clearly show that the S-SVSFM and the S-SVSFFM are reliable and powerful analytic methods for finding the exact solution to the SNFPDEs.