Multiple Solutions of Nonlinear Boundary Value Problems of Fractional Order: A New Analytic Iterative Technique

Abstract

Abstract:The purpose of this paper is to present a new kind of analytical method, the so-called residual power series, to predict and represent the multiplicity of solutions to nonlinear boundary value problems of fractional order. The present method is capable of calculating all branches of solutions simultaneously, even if these multiple solutions are very close and thus rather difficult to distinguish even by numerical techniques. To verify the computational efficiency of the designed proposed technique, two nonlinear models are performed, one of them arises in mixed convection flows and the other one arises in heat transfer, which both admits multiple solutions. Graphical results and tabulate data are presented and discussed quantitatively to illustrate the multiple solutions. The results reveal that the method is very effective, straightforward, and powerful for formulating these multiple solutions.

New computational method for solving fractional Riccati equation

“…Now, we survey some needed definitions and theorems regarded the fractional power series (FPS) also, more information can be found in [1,2,9]. …”

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

“…We state some useful results of Res P (x, t) and Res Q (x, t) from [29,[31][32][33], which are essential in the RPSM.…”

“…In this section, some fundamental definitions and preliminary results of fractional calculus are presented [32,33]. There are different definitions of fractional integration and differentiation, such as Grunwald-Letnikov's definition, Riemann-Liouville's definition and Caputo's definition.…”

“…This method provides a power series solution with rapid convergence. It has been advantageously implemented for the nonlinear fractional Korteweg-de Vries-Burgers equation [30], for the fractional foam drainage equation [31], for the time-fractional two-component evolutionary system of order two [32] and for other equations [33]. It has been proven that the RPSM is a convenient and effective method in its application.…”

Asymptotic Solutions of Time-Space Fractional Coupled Systems by Residual Power Series Method

“…An essential topic is to construct the solutions to fractional differential equations. And there are some effective methods to obtain different kinds of solutions, like Sumudu transform and variational iteration method [4], fractional Taylor vector approximate method [5], iterative method [6][7][8], Residual Power Series (RPS) method [9][10][11][12][13], and so on [14][15][16]. On the other hand, the study of coupled systems which involve fractional differential equations is also important because fractional coupled systems occur in many fields [17][18][19][20][21].…”