Boundary Determination of the Inverse Heat Conduction Problem in One and Two Dimensions via the Collocation Method Based on the Satisfier Functions

Sarabadan, Saeed; Rashedi, Kamal; Adibi, Hojatollah

doi:10.1007/s40995-017-0240-y

Cited by 7 publications

(5 citation statements)

References 39 publications

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“…[17][18][19] In the present contribution, we use a spectral technique [53][54][55][56][57][58][59] which is a powerful tool to provide more accurate and at the same time stable numerical solution for the inverse problems (1.1)- (1.5). By applying the satisfier function 21,[60][61][62][63][64][65] which fulfills all the initial and boundary conditions and employing this function within a linear transformation, we get an equivalent problem with the homogeneous initial and boundary conditions. We expand the desired unknown function in terms of orthonormal Bernstein basis functions (OBBFs); the operational matrices 21,[66][67][68][69][70][71][72] of integration and differentiation of these polynomials are utilized to formulate the discrete version of the problem.…”

Section: Literature Reviewmentioning

confidence: 99%

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

Rashedi

2022

Math Methods in App Sciences

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In this work, we present a spectral method for recovering an unknown time-dependent lower-order coefficient and unknown wave displacement in a nonlinear Klein-Gordon equation with overdetermination at a boundary condition. We apply the initial and boundary conditions to construct the satisfier function and use this function in a transformation to convert the main problem to a nonclassical hyperbolic equation with homogeneous initial and boundary conditions. Then, we utilize the orthonormal Bernstein basis functions to approximate the solution of the reformulated problem and use a direct technique based on the operational matrices of integration and differentiation of these basis functions together with the collocation technique to reduce the problem to a system of nonlinear algebraic equations. Regarding the perturbed measurements, the method takes advantage of the mollification method in order to derive stable numerical derivatives. Numerical simulations for solving several test examples are presented to show the applicability of the proposed method for obtaining accurate and stable results.

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Section: Literature Reviewmentioning

confidence: 99%

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

Rashedi

2022

Math Methods in App Sciences

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“…To contribute the boundary conditions () in computations, we define the satisfier function 54,55,59,66,79 as s ( x , t ) = g 1 ( t ) + x ( g 2 ( t ) − g 1 ( t ))and consider the approximate solution of u ( x , t )as

u_{K K^{'}} false(x, t false) ≃ x false(x - 1 false) true \sum_{i = 0}^{K} true \sum_{j = 0}^{K^{'}} c_{i j} ϕ_{i} false(x false) ψ_{j} false(t false) + s false(x, t false) = x false(x - 1 false) ϕ^{T} false(x false) C ψ false(t false) + ϕ^{T} false(x false) S ψ false(t false),

where

ϕ^{T} false(x false) = ([], ϕ_{0} false(x false), ϕ_{1} false(x false), \dots, ϕ_{K} false(x false))

and

ψ false(t false) = {([], ψ_{0} false(t false), ψ_{1} false(t false), \dots, ψ_{K^{'}} false(t false))}^{T}

are the vectors including the orthonormal basis functions defined over the intervals [0, 1]and [0, t f ], respectively and

C = ((), \begin{matrix} center center center \\ array c_{00} & array \dots & array c_{0 K^{'}} \\ array ⋮ & array & array ⋮ \\ array c_{K 0} & array \dots & array c_{K K^{'}} \end{matrix}),

is the coefficient matrix including the unknown parameter...…”

Section: Solution Proceduresmentioning

confidence: 99%

“…Operational matrices have been used as an efficient tool in the context of spectral methods such as Tau method and Ritz-Galerkin method. 36,37,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] These techniques provide a general approach based on expanding the approximate solution of the unknown functions in terms of the basis functions with unknown coefficients. Then, the operational matrices of the integration, differentiation, and product regarding the considered basis functions are utilized to tackle integral, differential, and nonlinear terms included in the problem.…”

Section: Literature Reviewmentioning

confidence: 99%

“…To contribute the boundary conditions (1.3) in computations, we define the satisfier function 54,55,59,66,79 as s(x, t) = g 1 (t) + x(g 2 (t) − g 1 (t)) and consider the approximate solution of u(x, t) as…”

Section: Solution Proceduresmentioning

confidence: 99%

“…Operational matrices have been used as an efficient tool in the context of spectral methods such as Tau method and Ritz–Galerkin method 36,37,44–62 . These techniques provide a general approach based on expanding the approximate solution of the unknown functions in terms of the basis functions with unknown coefficients.…”

Section: Introductionmentioning

confidence: 99%

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A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

Rashedi

2021

Math Methods in App Sciences

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The inverse problem of identifying the diffusion coefficient in the one‐dimensional parabolic heat equation is studied. We assume that the information of Dirichlet boundary conditions along with an integral overdetermination condition is available. By applying the given assumptions, the problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. Then we employ the direct technique based on the operational matrices for integration, differentiation, and the product of the orthonormal polynomials together with the Ritz–Galerkin technique to reduce the main problem to the solution of a system of nonlinear algebraic equations. Numerical simulations are presented to show the applicability of the proposed method.

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Numerical study of temperature distribution in an inverse moving boundary problem using a meshless method

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Engineering with Computers

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Boundary Determination of the Inverse Heat Conduction Problem in One and Two Dimensions via the Collocation Method Based on the Satisfier Functions

Cited by 7 publications

References 39 publications

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

Numerical study of temperature distribution in an inverse moving boundary problem using a meshless method

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