Numerical investigation of fractional Fisher partial differential equation via natural transform decomposition method

Abstract

In modern Science, a Fisher non-linear differential equation plays a significant role due to its diverse applications in fisher hypothesis, mathematical biology, engineering, physics and ecology. In this regard, the authors utilized the Natural transform decomposition method for the approximate solution of the well known non-linear Fisher’s partial differential equation. Since the fractional differential equations has memory effect, therefore, a fractional Fisher’s partial differential equation is taken for analyzing the behavior of the model. For the evaluation, we applied the iterative method of Natural transform decomposition method, that is the combination of Adomian decomposition method and Natural transform. The Natural transform decomposition method is a relatively new method, but it has quickly become one of the most popular methods for solving fractional-order differential equations. Since dealing non-linearity is pivotal step in finding the solution of non-linear differential equation. Therefore, the non-linearity of the model is decomposed through Adomian polynomial. Thus, from the first few iterations, we obtained the required numerical solution of the said model. Further, The method is verified through several examples that shows the authenticity and simplicity of the method. Finally, the graphical representation has been provided to illustrate the dynamics of the model.

Keywords

• Caputo’s derivative
• Approximate solution
• Adomian polynomial
• Matlab