Meshless method with ridge basis functions for time fractional two-flow domain model
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ORIGINAL RESEARCH
Meshless method with ridge basis functions for time fractional two‑flow domain model Xinqiang Qin1 · Keyuan Li1 · Gang Hu1 Received: 14 September 2019 / Accepted: 31 July 2020 © Islamic Azad University 2020
Abstract In this paper, a meshless method with ridge basis functions for solving the time fractional two-flow domain model problem is proposed. The method uses the L1 approximation formula based on piecewise linear interpolation to discretize the Caputo time fractional derivative (0 < 𝛼 < 1) , and by means of the ridge basis function to construct the approximation function, and uses the collocation method to discretize the governing equation. The existence and uniqueness of the numerical solution are analyzed. The error between the proposed method and the finite difference method is compared by numerical examples; then the affecting factors of the calculation accuracy are discussed. The results show that the proposed method is feasible and simple. Keywords Time fractional two-flow domain model · Caputo derivative · Ridge basis functions · Meshless method · Collocation method
Introduction In recent years, fractional calculation and fractional differential equation theories have been widely used in mechanics, physics, biomathematics, engineering, automatic control, fractals and other fields. Among them, the fractional derivative is a quasi-differential operator with weak singular integrals, which has historical memory and dependence. Generally speaking, the analytical solutions of differential equations with this derivative are difficult to express, and the calculation amount of numerical solutions is also quite large, which poses new challenges for practical applications and researches. The time fractional two-flow domain model is a generalized form of the two-flow domain model. The model divides regions according to the size of the soil pore water flow velocity, which can better describe the emergence of preferential flow in the soil and the bimodal phenomenon in * Xinqiang Qin [email protected] Keyuan Li [email protected] Gang Hu [email protected] 1
Department of Applied Mathematics, Xi’an University of Technology, Xi’an 710048, China
the solute breakthrough curve. After years of research and practice, the time fractional two-flow domain model problem is often a partial differential equation system, and it is generally difficult to obtain the corresponding analytical solution. Therefore, the numerical method is currently the most effective method to solve it. The meshless method does not require mesh recombination when solving large deformations of the mesh and many discontinuous problems. It is an efficient method for solving numerical solutions of partial differential equations. The basic idea is that nodes are arranged according to certain rules in the solution domain, the unknown field variables of the local region are represented by form functions, and finally the stiffness matrix equations are formed for solving [3]. Hengmu et al. [5] proposed a weighted least squares meshle
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