Abstract
Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect.
Highlights
In recent years, fractional calculus (FC) has drawn a substantial amount of attention in the field of applied mathematics
Symmetric properties contribute to numerous applications that depend on time space fractional differential equations (TSFDEs), such as applications of physics, chemistry, and engineering, so these properties play an important role in knowing the correct way to solve these equations [1,2]
The fast distribution of pollutants in hydrology [3,4], physical phenomena in fluid dynamics [5], biological population and disease models [6], and several other models are well explained within the domain of differential equations of fractional order [7]
Summary
Fractional calculus (FC) has drawn a substantial amount of attention in the field of applied mathematics. The proposed method provides Taylor Series solutions by unifying the linear part and nonlinear part together, and use only calculus of several variables to carry on. This new modification is expected to be more direct and easier to understand, in comparison with the Adomain Decomposition Method. A thorough survey reveals that the Fractional Differential Equation has not been studied as yet using this method Such are the reasons which provided motivation for the present research to discuss solving the Linear and Non-linear Fractional KdV Equations. Novel Analytic Method (NAM) to construct numerical solutions for the Linear and Nonlinear KdV Equation of Fractional Order.
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