Numerical solution of fractional differential equations by using fractional B-spline

Abstract

Abstract:In this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work. PACS

Solution of Fractional Differential Equation Systems and Computation of Matrix Mittag–Leffler Functions

“…. , n y j (0) = 0, for j = 1 and j = p 1 + p 2 + · · · + p k−1 + 1 where the accessorial zero initial values are required from the composition rule (21).…”

“…Finally, numerical examples were solved using the proposed methods.The existence, uniqueness, and stability of solutions for fractional differential equations were studied [3,[5][6][7][15][16][17]. Analytical and numerical methods are presented in [4,5,11,13,[18][19][20][21]. Solutions of some linear fractional differential equations may be expressed in terms of the Mittag-Leffler functions, which are defined as [5,22] If both of the parameters are 1, the exponential function is reduced: E 1,1 (z) = e z .…”

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

“…Several numerical techniques are available to find approximate solutions to differential equations of mathematical models of engineering problems . One of the techniques available is collocation technique.…”

“…Several numerical techniques are available to find approximate solutions to differential equations of mathematical models of engineering problems. [20][21][22] One of the techniques available is collocation technique. A collocation method involves satisfying a differential equation to some tolerance at a finite number of points, called collocation points.…”

A generalization of the Mittag–Leffler function and solution of system of fractional differential equations

“…Fractional differential equations on scalar functions, including the existence, uniqueness and stability, and the analytic and numeric methods of solutions, were studied by many scholars [3-9, 13, 15, 17, 22-27]. In particular, new numerical schemes were designed [9,23,24,28], and a Lie symmetry analysis was given and the conservation laws for fractional evolution equations were systematically investigated [29][30][31][32]. The solutions of many fractional differential equations involve a class of important special functions-Mittag-Leffler functions (AMS 2000 Mathematics Subject Classification 33E12).…”