Abstract
We introduce two algorithms in order to find the exact solution of the nonlinear Time-fractional Partial differential equation, in this research work. Those algorithms are proposed in the following structure: The Modified Homotopy Perturbation Method (MHPM), The Homotopy Perturbation and Sumudu Transform Method. The results achieved using the both methods are the same. However, we calculate the approached theoretical solution of the Black-Scholes model in the form of a convergent power series with a regularly calculated element. Finally, we propose a descriptive example to demonstrate the efficiency and the simplicity of the methods.
Highlights
In recent years, fractional calculus has been increasingly used for numerous applications in many scientific and technical fields such as medical sciences, biological research, as well as various chemical, biochemical and physical fields
We introduce two algorithms in order to find the exact solution of the nonlinear Time-fractional Partial differential equation, in this research work
Those algorithms are proposed in the following structure: The Modified Homotopy Perturbation Method (MHPM), The Homotopy Perturbation and Sumudu Transform Method
Summary
Fractional calculus has been increasingly used for numerous applications in many scientific and technical fields such as medical sciences, biological research, as well as various chemical, biochemical and physical fields. Examples of such a numerical method for solving FPDEs are the Homotopy Perturbation Method (HPM) [7] [8] [9] [10] [11], the Differential Transform Method (DTM) [12], the Variational Iteration Method (VIM) [13], the New Iterative Method (NIM) [3] [14], the Homotopy Analysis Method (HAM) [1] [6] [15] and the Adomian Decomposition Method (ADM) [16] Among these numerical methods, the VIM and the ADM are the most popular ones that are used to solve differential and integral equations of integer and fractional order.
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