Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions

Abstract

This paper investigates a computational method to find an approximation to the solution of fractional differential equations subject to local and nonlocal m-point boundary conditions. The method that we will employ is a variant of the spectral method which is based on the normalized Bernstein polynomials and its operational matrices. Operational matrices that we will developed in this paper have the ability to convert fractional differential equations together with its nonlocal boundary conditions to a system of easily solvable algebraic equations. Some test problems are presented to illustrate the efficiency, accuracy, and applicability of the proposed method. MSC: 35C11; 65T99

Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations

“…The nonlinear fractional differential equation for the multi-point boundary value problem has been studied extensively. For details, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein. For m = 3, Bai [15] investigated the existence and uniqueness of positive solutions for a nonlocal boundary value problem of the fractional differential equation [16] investigated the existence and uniqueness of a positive solution to nonzero three-point boundary values problem for a coupled system of fractional differential equations.…”

Infinitely many nontrivial solutions for fractional boundary value problems with impulses and perturbation

“…In the past decades, the theory of fractional differential equations has become an important area of investigation because of its wide applicability in many branches of physics, economics and technical sciences, see [18,20,[30][31][32][33][34][35]38] and references therein. The BVPs of fractional differential equations have also attracted more and more attention (for example [2,10,12,17]). The classical approaches to study such problems mainly include fixed point theorems, degree theory, the method of upper and lower solutions and so on.…”

Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

“…Further we can easily establish a simple relationship for convergence of the propped method. The aforesaid procedure has now recently being applied for some nonlinear fractional order differential equations and some fruitful results were obtained; for details see [44]. In our future work we will use operational matrices method to compute approximate solutions of nonlinear FPDEs and their system.…”