A Novel Numerical Approach in Solving Fractional Neutral Pantograph Equations via the ARA Integral Transform

In this article, a new, attractive method is used to solve fractional neutral pantograph equations (FNPEs). The proposed method, the ARA-Residual Power Series Method (ARA-RPSM), is a combination of the ARA transform and the residual power series method and is implemented to construct series solutions for dispersive fractional differential equations. The convergence analysis of the new method is proven and shown theoretically. To validate the simplicity and applicability of this method, we introduce some examples. For measuring the accuracy of the method, we make a comparison with other methods, such as the Runge–Kutta, Chebyshev polynomial, and variational iterative methods. Finally, the numerical results are demonstrated graphically.

Approximate and Exact Solutions to Fractional Order Cauchy Reaction-Diffusion Equations by New Combine Techniques

In this paper, we present a simple and efficient novel semianalytic method to acquire approximate and exact solutions for the fractional order Cauchy reaction-diffusion equations (CRDEs). The fractional order derivative operator is measured in the Caputo sense. This novel method is based on the combinations of Elzaki transform method (ETM) and residual power series method (RPSM). The proposed method is called Elzaki residual power series method (ERPSM). The proposed method is based on the new form of fractional Taylor’s series, which constructs solution in the form of a convergent series. As in the RPSM, during establishing the coefficients for a series, it is required to compute the fractional derivatives every time. While ERPSM only requires the concept of the limit at zero in establishing the coefficients for the series, consequently scarce calculations give us the coefficients. The recommended method resolves nonlinear problems deprived of utilizing Adomian polynomials or He’s polynomials which is the advantage of this method over Adomain decomposition method (ADM) and homotopy-perturbation method (HTM). To study the effectiveness and reliability of ERPSM for partial differential equations (PDEs), absolute errors of three problems are inspected. In addition, numerical and graphical consequences are also recognized at diverse values of fractional order derivatives. Outcomes demonstrate that our novel method is simple, precise, applicable, and effectual.

Residual Power Series Method for Solving Klein-Gordon Schrödinger Equation

In this work, the Residual Power Series Method(RPSM) is used to find the approximate solutions of Klein Gordon Schrödinger (KGS) Equation. Furthermore, to show the accuracy and the efficiency of the presented method, we compare the obtained approximate solution of Klein Gordon Schrödinger equation by Residual Power Series Method(RPSM) numerically and graphically with the exact solution.

An Approximate Analytical Approach for Systems of Fredholm Integro-Differential Equations of Fractional Order

A new technique for solving a system of fractional Fredholm integro-differential equations (IDEs) is introduced in this manuscript. Furthermore, we present a review for the derivation of the residual power series method (RPSM) to solve fractional Fredholm IDEs in the paper done by Syam, as well as, corrections to the examples mentioned in that paper. The numerical results demonstrated the new technique’s applicability, efficacy, and high accuracy in dealing with these systems. On the other hand, a comparison has been done between the two schemes using the two corrected examples in addition to a problem that had been solved in many previous studies, and the results of these studies were compared with the new technique and RPSM. The comparison demonstrated clear superiority of our method over the RPSM for solving this class of equations. Moreover, they dispel the misconception that the RPSM works effectively on fractional Fredholm IDEs as mentioned in the paper done by Syam, whereas two problems solved by the RPSM produced an unaccepted error. Also, the comparison with the previous studies indicates the importance of the new method in dealing with the fractional Fredholm IDEs despite its simplicity, ease of use, and negligible computational time.