Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations

Abstract

In this research, systems of linear and nonlinear differential equations of fractional order are solved analytically using the novel interesting method: ARA- Residual Power Series (ARA-RPS) technique. This approach technique is based on the combination of the residual power series scheme with the ARA transform to establish analytical approximate solutions in a fast convergent series representation using the concept of the limit. The proposed method needs less time and effort compared with the residual power series technique. To prove the simplicity, applicability, and reliability of the presented method, three numerical examples are proposed and simulated. The obtained results show that the ARA-RPS technique is applicable, simple, and effective to get solutions to linear and nonlinear engineering and physical problems.

On the Analytical Solution of Fractional SIR Epidemic Model

“…Te 20th terms solution of systems ( 33) and ( 34) at ϑ � 1 is given by the following equation: Te following fgures illustrate the 20th solution of systems (33) and (34) with various values of (ϑ � 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4) and 0.3.…”

“…Te following tables, Tables 1-3 present the LRPSM solution of systems (33) and (34) at various values of ϑ(ϑ � 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3). In Table 1, we propose diferent values of φ 20 (τ) with diferent values of ϑ, in Table 2, we propose diferent values of ψ 20 (τ) with diferent values of ϑ, and in Table 3, we propose diferent values of ϕ 20 (τ) with diferent values of ϑ.…”

“…Tis research presents the LRPSM, which is a new analytical method that combines the Laplace transform with the residual power series method; it was frst introduced in [26], and it is implemented by researchers to solve several models of fractional ordinary and partial diferential equations and systems. Tis method shows its efciency and applicability in solving similar problems [27][28][29][30][31][32][33][34][35].…”

ARA-residual power series method for solving partial fractional differential equations

“…In other development, the Laplace residual power series method (LRPSM) is established in 2020 [26,27] by combining the Laplace transform with RPSM. In this article, another promotion of the RPSM is constructed by adapting the ARA transform [28,29] in the methodology of RPSM [30,31]. This powerful technique, is a new scheme in finding approximate solutions of nonlinear partial differential equations of fractional order in a series form.…”

A New Computational Technique for Analytic Treatment of Time-Fractional Nonlinear Equations Arising in Magneto-Acoustic Waves

“…Tere are various analytical and numerical methods available for handling various forms of ffth-order KdV-type equations in the literature. Some of them are the Adomian decomposition technique [35], modifed Adomian decomposition method [36], Laplace decomposition approach [37], diferential transform technique [38,39], Hirota's bilinear techniques [40], inverse scattering algorithm [41], He's semiinverse scheme [42], extended Tanh method [43], homotopy analysis technique [14,44], fractional homotopy analysis transform algorithm [45], modifed homotopy perturbation technique [46], variational iteration technique [47], homotopy perturbation method [48,49], homotopy perturbation transform method [50], hyperbolic and exponential ansatz methods [51], multiple exp-function method [52], and others [53][54][55]. Moreover, many methods are available to solve the fractional-order KdV equations.…”