Numerical solution of a non-classical parabolic problem: An integro-differential approach

Ang, W.T.

doi:10.1016/j.amc.2005.08.011

Cited by 13 publications

(10 citation statements)

References 7 publications

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“…. , N − 1) are approximated as cubic functions of time t over the interval [τ , τ + 3 t], that is (as in Ang [12]),…”

Section: Methodsmentioning

confidence: 99%

“…Similar integro-differential equations were used to obtain numerical methods for solving one-dimensional heat and wave equations in Ang [12,13].…”

Section: Integro-differential Formulationmentioning

confidence: 99%

“…Thus, it is not necessary to approximate any spatial derivative using finite-difference formulae. For some examples of problems solved using integro-differential formulations, one may refer to Ang [12,13] and Chen and You [14].…”

Section: Introductionmentioning

confidence: 99%

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A numerical method based on integro‐differential formulation for solving a one‐dimensional Stefan problem

Ang

2007

Numerical Methods Partial

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A numerical method based on an integro-differential formulation is proposed for solving a one-dimensional moving boundary Stefan problem involving heat conduction in a solid with phase change. Some specific test problems are solved using the proposed method. The numerical results obtained indicate that it can give accurate solutions and may offer an interesting and viable alternative to existing numerical methods for solving the Stefan problem.

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“…. , N − 1) are approximated as cubic functions of time t over the interval [τ , τ + 3 t], that is (as in Ang [12]),…”

Section: Methodsmentioning

confidence: 99%

“…Similar integro-differential equations were used to obtain numerical methods for solving one-dimensional heat and wave equations in Ang [12,13].…”

Section: Integro-differential Formulationmentioning

confidence: 99%

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A numerical method based on integro‐differential formulation for solving a one‐dimensional Stefan problem

Ang

2007

Numerical Methods Partial

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“…So far, there are many publications about parabolic equation with non‐local or fixed value boundary condition , but to the best knowledge of the authors, there is little research performed for non‐local and time‐dependent boundary value problem presented here. Therefore, it is also significant mathematically.…”

Section: Introductionmentioning

confidence: 99%

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

Zhou

2015

Math Methods in App Sciences

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Communicated by W. SprößigThe paper is devoted to the investigation of a parabolic partial differential equation with non-local and time-dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz-Galerkin method, which is a first attempt at tackling parabolic equation with such non-classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non-local boundary condition, we use a trick of introducing the transition function G.x, t/ to convert non-local boundary to another non-classical boundary, which can be handled with the Ritz-Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper.

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“…The usual numerical methods for PDEs subject to these nonclassical conditions are finite difference methods (FDMs), Galerkin techniques [1], collocation approaches [2], and Tau schemes [3]. Moreover, one can point out to the new methods such as Bernstein Tau technique [4], Sinc collocation method [5], and also [6]. It should be noted that, in [5], two types of one-dimensional parabolic PDEs subject to nonlocal boundary conditions were considered; meanwhile in [4] a different problem was considered for obtaining the solution numerically.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Spectral Approximation for Solving Several Types of Parabolic PDEs with Nonlocal Boundary Conditions

Tohidi

Kılıçman

2014

Mathematical Problems in Engineering

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The problem of solving several types of one-dimensional parabolic partial differential equations (PDEs) subject to the given initial and nonlocal boundary conditions is considered. The main idea is based on direct collocation and transforming the considered PDEs into their associated algebraic equations. After approximating the solution in the Legendre matrix form, we use Legendre operational matrix of differentiation for representing the mentioned algebraic equations clearly. Three numerical illustrations are provided to show the accuracy of the presented scheme. High accurate results with respect to the Bernstein Tau technique and Sinc collocation method confirm this accuracy.

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Numerical solution of a non-classical parabolic problem: An integro-differential approach

Cited by 13 publications

References 7 publications

A numerical method based on integro‐differential formulation for solving a one‐dimensional Stefan problem

A numerical method based on integro‐differential formulation for solving a one‐dimensional Stefan problem

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

An Efficient Spectral Approximation for Solving Several Types of Parabolic PDEs with Nonlocal Boundary Conditions

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