Solution of the Problem of Initiating the Heat Wave for a Nonlinear Heat Conduction Equation Using the Boundary Element Method

Казаков, А. Л.; Нефедова, О. А.; Спевак, Л. Ф.

doi:10.1134/s0965542519060083

Cited by 15 publications

(12 citation statements)

References 10 publications

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“…However, the radius of convergence of the series is usually small, and it can be estimated only in some spacial cases. To solve this problem and obtain an approximate solution to the problem of heat wave initiating at a given time interval [0, t * ], we usually use a step-by-step method based on the boundary element approach [5]. We also note that presence of additional term (source) requires significant modification of the previously developed approach.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Since a(t k ) is unknown in advance, the solution domain is also unknown at the moment t = t k . That is why, we interchange the desired function and the spatial variable ρ [5]. Equation (4) takes the following form…”

Section: Solution Algorithm Based On Bemmentioning

confidence: 99%

“…The use of the simplest functions ϕ k = r k as RBFs [5,18] results in stable convergence of iterative processes and good accuracy of solutions. However, it is rather complicated to use these functions to solve problems with the source term because convergence of iterative processes is unstable and depends on the parameters of the problem.…”

Section: Solution Algorithm Based On Bemmentioning

confidence: 99%

“…Previously, problem (2), (3) was already considered in the case α = 0, i.e., without a source (sink). Solutions were constructed both in the form of special series [4] and with the use of the boundary element method (BEM) [5]. In this paper, we propose an approximate method for constructing solutions to the problem of heat wave initiation based on the BEM.…”

Section: Introductionmentioning

confidence: 99%

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On the Construction of Solutions to a Problem with a Free Boundary for the Non-linear Heat Equation

Alexander

Lev

Lee

2020

Journal of Siberian Federal University. Mathematics &Amp; Physics

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The construction of solutions to the problem with a free boundary for the non-linear heat equation which have the heat wave type is considered in the paper. The feature of such solutions is that the degeneration occurs on the front of the heat wave which separates the domain of positive values of the unknown function and the cold (zero) background. A numerical algorithm based on the boundary element method is proposed. Since it is difficult to prove the convergence of the algorithm due to the non-linearity of the problem and the presence of degeneracy the comparison with exact solutions is used to verify numerical results. The construction of exact solutions is reduced to integrating the Cauchy problem for ODE. A qualitative analysis of the exact solutions is carried out. Several computational experiments were performed to verify the proposed method

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Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Solution Algorithm Based On Bemmentioning

confidence: 99%

Section: Solution Algorithm Based On Bemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Construction of Solutions to a Problem with a Free Boundary for the Non-linear Heat Equation

Alexander

Lev

Lee

2020

Journal of Siberian Federal University. Mathematics &Amp; Physics

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“…Поскольку рекуррентная процедура, использованная в ходе доказательства, вполне конструктивна, теорема не только обеспечивает существование аналитического решения рассмотренной задачи, но и доставляет алгоритм построения приближенных решений в виде отрезков ряда(7), которые, в частности, можно использовать в качестве тестовых примеров при проверке корректности результатов расчетов, выполненных другими методами[20].…”

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On the Analytical Construction of a Heat Wave for the Nonlinear Heat Equation With a Source in Polar Coordinates

Казаков¹

2019

DREAM

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The paper is devoted to the study of a nonlinear second-order parabolic equation, which in the literature is called the heat equation with a source or the generalized porous medium equation. We construct specialized solutions that describe disturbances propagating over the zero background at a finite velocity (heat waves). Previously, we studied such problems without a source. In this paper, we extend the known results to a more general case. The theorem of the existence and uniqueness of a solution having the form of a heat wave in polar coordinates is proved. A heat wave is constructed in the form of a convergent multiple power series, the c oefficients of which are determined when solving systems of linear algebraic equations. We give an example where the conditions of the theorem are not satisfied. It shows that the solution, in this case, has the form of a stable heat wave.

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The elastoplastic analysis of functionally graded materials using a meshfree RRKPM

Liu

Wei

Qin

et al. 2022

Applied Mathematics and Computation

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Solution of the Problem of Initiating the Heat Wave for a Nonlinear Heat Conduction Equation Using the Boundary Element Method

Cited by 15 publications

References 10 publications

On the Construction of Solutions to a Problem with a Free Boundary for the Non-linear Heat Equation

On the Construction of Solutions to a Problem with a Free Boundary for the Non-linear Heat Equation

On the Analytical Construction of a Heat Wave for the Nonlinear Heat Equation With a Source in Polar Coordinates

The elastoplastic analysis of functionally graded materials using a meshfree RRKPM

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