Error estimate of a Legendre-Galerkin Chebyshev collocation method for a class of parabolic inverse problem

Liao, Huiqing; Ma, Heping

doi:10.1016/j.apnum.2021.07.023

Cited by 3 publications

(1 citation statement)

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“…During spectral methods, the Galerkin method is the most accurate and involves the least number of equations in a multidimensional space while the collocation method is much easier to implement even though the problems might be nonlinear or complex, which means that each of the Galerkin method and the collocation method has its strengths. Indeed, a Legendre-Galerkin Chebyshev collocation (LGCC) method, which combines the advantages of the Galerkin method and the collocation method, is proposed and has been successfully used to solve partial differential equations such as in the case of earlier studies [5][6][7][8][9] and so on. The LGCC method is basically in the Legendre-Galerkin form, but the nonlinear term is interpolated through the Chebyshev-Gauss-type points, which not only shares the essential features of the Legendre-Galerkin method but also possesses the convenience of the Chebyshev collocation treatment for the nonlinear term.…”

Section: Introductionmentioning

confidence: 99%

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

Pan

2022

Math Methods in App Sciences

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In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.

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Section: Introductionmentioning

confidence: 99%

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

Pan

2022

Math Methods in App Sciences

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Determination of an unknown coefficient in the Korteweg–de Vries equation

Sang,

Qiao,

2024

Journal of Inverse and Ill-Posed Problems

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In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2 {L^{2}} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.

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Analytical Inverse Analysis Methodological Approach for Thermo-Physical Parameters Estimation of Multilayered Medium Terrain with Homogenized Sampled Measurements

2022

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The current paper presents results of the inverse theory approach utilized for the analytical estimation of thermo-physical properties for a multi-layered medium terrain with homogenized experimental measurements. We demonstrate the derivation steps of the exact solution for the heat transfer problem with third-kind boundary conditions due to natural convection on the outlets posed for the considered experimental domain. There are received analytical expressions. Initially, we illustrate the homogenization of the boundary conditions. We then discuss the process of derivation for the analytical solution of the posed problem with the help of key elements of the Fourier method. We provide an algorithm for applying the contact condition to extend received expressions for multiple layers. After that we demonstrate the major steps for the construction of nonlinear systems of equations to be solved in order to obtain exact values of key thermo-physical and geometrical parameters of the investigated medium with the help of received exact analytical expressions. Along with analytical procedures, we present a posed experimental design and discuss an algorithm of numerical exploitation for a suggested method, outlining its advantages and possible limitations in terms of initial approximations.

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Error estimate of a Legendre-Galerkin Chebyshev collocation method for a class of parabolic inverse problem

Cited by 3 publications

References 20 publications

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

Determination of an unknown coefficient in the Korteweg–de Vries equation

Analytical Inverse Analysis Methodological Approach for Thermo-Physical Parameters Estimation of Multilayered Medium Terrain with Homogenized Sampled Measurements

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