Rational approximation solution of the fractional Sharma–Tasso–Olever equation

AbstractIn the paper, we implement relatively new analytical techniques, the variational iteration method, the Adomian decomposition method and the homotopy perturbation method, for obtaining a rational approximation solution of the fractional Sharma–Tasso–Olever equation. The three methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The numerical results demonstrate the significant features, efficiency and reliability of the three approaches

A Method to Solve One-dimensional Nonlinear Fractional Differential Equation Using B-Polynomials

In this article, the fractional Bhatti-Polynomial bases are applied to solve one-dimensional nonlinear fractional differential equations (NFDEs). We derive a semi-analytical solution from a matrix equation using an operational matrix which is constructed from the terms of the NFDE using Caputo’s fractional derivative of fractional B-polynomials (B-polys). The results obtained using the prescribed method agree well with the analytical and numerical solutions presented by other authors. The legitimacy of this method is demonstrated by using it to calculate the approximate solutions to four NFDEs. The estimated solutions to the differential equations have also been compared with other known numerical and exact solutions. It is also noted that for solving the NFDEs, the present method provides a higher order of precision compared to the various finite difference methods. The current technique could be effortlessly extended to solving complex linear, nonlinear, partial, and fractional differential equations in multivariable problems

Symbolic-numeric interface: A review

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach

Numerical study of time-fractional fourth-order differential equations with variable coefficients

AbstractIn this article, we study numerical solutions of time-fractional fourth-order partial differential equations with variable coefficients by introducing the fractional derivative in the sense of Caputo. We implement reliable series solution techniques namely Adomian decomposition method (ADM) and He’s variational iteration method (HVIM). Some applications are presented to highlight the significant features of these techniques. The comparison shows that the solutions obtained are in good agreement with each other and with their respective exact solutions. Some of these types of differential equations arise practically in the theory of transverse vibrations

HYPERBOLIC TYPE SOLUTIONS FOR THE COUPLE BOITI-LEON-PEMPINELLI SYSTEM

In this paper, the (1/G')-expansion method is used to solve the coupled Boiti-Leon-Pempinelli (CBLP) system. The proposed method was used to construct hyperbolic type solutions of the nonlinear evolution equations. To asses the applicability and effectiveness of this method, some nonlinear evolution equations have been investigated in this study. It is shown that with the help of symbolic computation, the (1/G')-expansion method provides a powerful and straightforward mathematical tool for solving nonlinear partial differential equations