Abstract
It is well known that the appearance of the delay in the fractional delay differential equation (FDDE) makes the convergence analysis very difficult. Dealing with the problem with the traditional reproducing kernel method (RKM) is very tricky. The feature of this paper is to gain a more credible approximate solution via fractional Taylor’s series (FTS). We use the FTS to deal with the delay for improving the accuracy of the approximate solutions. Compared with other methods, the five numerical examples demonstrate the accuracy and efficiency of the proposed method in this paper.
Highlights
For the past few years, FDDE have been applied in various fields of neoteric science and engineering such as economics, physics, dynamics, hydrology, finance, signal processing, neural network, and control theory [1]. ere has been a growing interest in researching numerical methods for this equation
We estimated the parameter m 1, k 1, α 1.5, by Mathematical 7.0, N 14; Figure 1 shows the absolute errors by two methods, the left one is given by the traditional RKM, and the right one is given by the proposed method in this paper
By Mathematical 7.0, N 10, Figure 2 presents the absolute errors by two methods, the left one is given by the traditional RKM, and the right one is given by the proposed method in this paper
Summary
For the past few years, FDDE have been applied in various fields of neoteric science and engineering such as economics, physics, dynamics, hydrology, finance, signal processing, neural network, and control theory [1]. ere has been a growing interest in researching numerical methods for this equation. In [13], Laiq uses the optimal auxiliary function method for solving the general partial differential equations. In [15], Phang uses an operational matrix method for solving the FDDEs. In [16], Salem gives the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations (FLE) with multipoint and nonlocal integral boundary conditions (NIBCs). In [19], Salem and Alghamdi give the existence and uniqueness of solutions for the Langevin equation (LE) that has Caputo fractional derivatives of two different orders. In [20], Salem et al give the existence results for an infinite system of LEs involving generalized derivatives of two distinct fractional orders with three-point boundary conditions.
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