Least‐Squares Residual Power Series Method for the Time‐Fractional Differential Equations

Abstract

In this study, an applicable and effective method, which is based on a least-squares residual power series method (LSRPSM), is proposed to solve the time-fractional differential equations. The least-squares residual power series method combines the residual power series method with the least-squares method. These calculations depend on the sense of Caputo. Firstly, using the classic residual power series method, the analytical solution can be solved. Secondly, the concept of fractional Wronskian is introduced, which is applied to validate the linear independence of the functions. Thirdly, a linear combination of the first few terms as an approximate solution is used, which contains unknown coefficients. Finally, the least-squares method is proposed to obtain the unknown coefficients. The approximate solutions are solved by the least-squares residual power series method with the fewer expansion terms than the classic residual power series method. The examples are shown in datum and images.The examples show that the new method has an accelerate convergence than the classic residual power series method.

Efficacious Analytical Technique Applied to Fractional Fornberg–Whitham Model and Two-Dimensional Fractional Population Model

“…In this section, we apply the q-HATM to solve the Caputo time-fractional Fornberg-Whitham equation and compare our results with those obtained from other methods. Recently, in [63], LSRPSM was used to obtain an approximate solution for the fractional Fornberg-Whitham equation. It was established that LSRPSM outperformed other methods such as RPSM and VIM in terms of accuracy and rate of convergence.…”

“…Numerical comparison when α = 1 of the exact solution (Equation ( 21)), and the solutions obtained by the q-homotopy analysis transform method (q-HATM) with = −1.28 , the least-squares residual power series method (LSRPSM) [63], the residual power series method (RPSM) [63] and the variational iteration method (VIM) [26]. Remark 2.…”

“…We present in Table 4 the numerical results (when α = 1) for t = 1 and some values of x and y. The solutions obtained using the exact solution (Equation ( 37)), the q-HATM solution (Equation ( 39)) and absolute errors for the q-HATM and other methods (the LSRPSM [63], the RPSM [63] and the HPM [64]) are tabulated. In Table 4, "itr" means the number of iterations performed for the particular method.…”

“…Table 4. Numerical comparison when α = 1 of the exact solution (Equation ( 21)), and the solutions obtained by the q-HATM with = −1.1146 , the LSRPSM [63], the RPSM [63] and the homotopy perturbation method (HPM) [64]. We take t = 1, λ = 0.5.…”

The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model

“…• Boundary value problems for ordinary differential equations [2,3]. • Fractional partial differential equations [4][5][6].…”