Improved finite integration method for partial differential equations

Li, M.; Tian, Zhaolu; Hon, Y.C.; Chen, C.S.; Wen, P.H.

doi:10.1016/j.enganabound.2015.12.012

Cited by 19 publications

(14 citation statements)

References 5 publications

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“…which can be denoted by Bc m = Ψ m or BS −1 u m = Ψ m . Finally, we can construct the system of m th iterative linear equations from (16) and 17, which has N + n unknowns containing u m and d, as follows:…”

Section: Procedures For Solving the Direct Tvide Problemmentioning

confidence: 99%

“…Now, we remove all spatial derivatives from (19) and use the shifted Chebyshev integration matrix (as explained in Section 2.1). Then, we obtain the following matrix equation, based on the same process as in (16), as…”

Section: Procedures For Solving Inverse Problem Of Tvidementioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2020

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In this article, the direct and inverse problems for the one-dimensional time-dependent Volterra integro-differential equation involving two integration terms of the unknown function (i.e., with respect to time and space) are considered. In order to acquire accurate numerical results, we apply the finite integration method based on shifted Chebyshev polynomials (FIM-SCP) to handle the spatial variable. These shifted Chebyshev polynomials are symmetric (either with respect to the point x = L 2 or the vertical line x = L 2 depending on their degree) over [ 0 , L ] , and their zeros in the interval are distributed symmetrically. We use these zeros to construct the main tool of FIM-SCP: the Chebyshev integration matrix. The forward difference quotient is used to deal with the temporal variable. Then, we obtain efficient numerical algorithms for solving both the direct and inverse problems. However, the ill-posedness of the inverse problem causes instability in the solution and, so, the Tikhonov regularization method is utilized to stabilize the solution. Furthermore, several direct and inverse numerical experiments are illustrated. Evidently, our proposed algorithms for both the direct and inverse problems give a highly accurate result with low computational cost, due to the small number of iterations and discretization.

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Section: Procedures For Solving the Direct Tvide Problemmentioning

confidence: 99%

Section: Procedures For Solving Inverse Problem Of Tvidementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2020

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“…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]. The FIM has been successfully applied to solve various kinds of PDEs and it was verified by comparing with several existing methods that it offers a very stable, highly accurate and efficient approach, see [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

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

2019

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The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented.

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“…Despite the great success of the finite element method, boundary element method and meshless methods, there is still a need for developing numerical schemes for multi-dimensional IHCP. Recently, Wen and his colleagues [16][17][18] developed the Finite Integration Method (FIM) for solving one and two-dimensional, differential equation problems and demonstrated its applications to non-local elasticity problems. It has been shown that the FIM gives much higher degree of accuracy than the Finite Difference Method (FDM) and the Point Collocation Method (PCM).…”

Section: Introductionmentioning

confidence: 99%

Inverse heat conduction in anisotropic and functionally graded media by finite integration method

Jin¹,

Zheng²,

Huang³

et al. 2018

Int. J. CMEM

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Based on the recently developed finite integration method (FIM) for solving one and two dimensional ordinary and partial differential equations, this paper extends FIM to both stationary and transient heat conduction inverse problems for anisotropic and functionally graded materials with high degree of accuracy. Lagrange series approximation is applied to determine the first order of integral and differential matrices, which are used to form the system equation matrix for two and three dimensional problems. Singular Value Decomposition (SVD) is applied to solve the ill-conditioned system of algebraic equations obtained from the integral equation, boundary conditions and scattered temperature measurements. The convergence and accuracy of this method are investigated with two examples for anisotropic media and functionally graded materials.

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Improved finite integration method for partial differential equations

Cited by 19 publications

References 5 publications

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

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

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

Inverse heat conduction in anisotropic and functionally graded media by finite integration method

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