Finite integration method for solving multi-dimensional partial differential equations

Abstract

a b s t r a c tBased on the recently developed Finite Integration Method (FIM) for solving one-dimensional ordinary and partial differential equations, this paper extends the technique to higher dimensional partial differential equations. The main idea is to extend the first order finite integration matrices constructed by using either Ordinary Linear Approach (OLA) (uniform distribution of nodes) or Radial Basis Function (RBF) interpolation (uniform/random distributions of nodes) to higher order integration matrices. Using standard time integration techniques, such as Laplace transform, we have shown that the FIM is capable for solving time-dependent partial differential equations. Illustrative numerical examples are given in two-dimension to compare the FIM (FIM-OLA and FIM-RBF) with the finite difference method and point collocation method to demonstrate its superior accuracy and efficiency.

Large deformations of tapered beam with finite integration method

“…The integration matrix of the first order can be obtained by direct integration with Trapezoidal rule, Simpson rules, Cotes formula and Lagrange formula introduced in [13]. It has been demonstrated that the Lagrange formula gives the highest accuracy results.…”

“…Thereafter, Li et al [12] extended this method to solve a nonlocal elasticity for static and dynamic problems. Subsequently, FIM was applied to multi-dimensional partial differential equations for practical problems in engineering by Li et al [13]. With higher order numerical quadratic formula by Simpson's algorithm and Chebyshev polynomial, the higher accurate solutions for general PDEs can be obtained, see [13,14] by Li et al and [15] by Boonklurb et al More recently, Yun et al [16], Li et al [17] and Li and Hon [18] have -3 -demonstrated the applications of FIM to solve various kinds of stiff PDEs problems with its unconditional stability and distinct advantage in smoothing stiffness in terms of singularities, discontinuities and stiff boundary layers.…”

Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations

“…Consequently, for k ∈ {1, 2, 3, ..., M} by hiring (16) and the idea of Boonklurb et al [21], we can convert (14) into the matrix form as…”

“…Originally, the FIM has been firstly proposed by Wen et al [15]. They constructed the integration matrices based on trapezoidal rule and radial basis functions for solving one-dimensional linear PDEs and then Li et al [16] continued to develop it in order to overcome the two-dimensional problems. After that, the FIM was improved using three numerical quadratures, including Simpson's rule, Newton-Cotes, and Lagrange interpolation, presented by Li et al [17].…”

Numerical Solution of Direct and Inverse Problems for Time-Dependent Volterra Integro-Differential Equation Using Finite Integration Method with Shifted Chebyshev Polynomials

“…In recent years, the finite integration method (FIM) has been developed to find approximate solutions of linear boundary value problems for partial differential equations (PDEs). The concept of FIM is to transform a given PDE into an equivalent integral equation, following which a numerical integration method, such as the trapezoid, Simpson, or Newton-Cotes methods (see References [14][15][16]), are applied. In 2018, Boonklurb et al [17] modified the traditional FIM by using Chebyshev polynomials to solve one-and two-dimensional linear PDEs and obtained a more accurate result compared to the traditional FIMs and FDM.…”