Study of 2m-th order parabolic equation in non-symmetric conical domains

Cherfaoui, Saïda; Kessab, Amor; Kheloufı, Arezkı

doi:10.15672/hujms.546340

Cited by 2 publications

(1 citation statement)

References 12 publications

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“…In most papers, the domain in which the solution of the boundary value problem is sought does not degenerate into a point at the initial moment of time. In [1][2][3][4][5][6] authors for solving such problems used a technique which consists in reducing a non-cylindrical domain to a cylindrical one. There are a number of works devoted to numerical methods for solving such problems [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Solution of a Two-Dimensional Boundary Value Problem of Heat Conduction in a Degenerating Domain

Ramazanov

Gulmanov

2021

JMMCS

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In the paper we consider the boundary value problem of heat conduction outside the cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary condition contain a derivative with respect to the time variable. The peculiarity of the problem under consideration consists precisely in the presence of a moving boundary and the degeneration of the solution domain into a point at the initial moment of time. The well-known classical methods are generally not applicable to this type of problems. By the method of heat potentials, such boundary value problems of heat conduction are reduced to the solution of singular Volterra type integral equations of the second kind A singular Volterra type equation is understood as an equation whose kernel has the following property: the integral of the kernel of the equation does not tend to zero as the upper limit tends to the lower one. Such integral equations cannot be solved by the method of successive approximations, and in most cases the corresponding homogeneous integral equations have nonzero solutions. We prove a theorem on the solvability of the considered boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We found a nonzero solution of this singular integral equation.

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Section: Introductionmentioning

confidence: 99%

Solution of a Two-Dimensional Boundary Value Problem of Heat Conduction in a Degenerating Domain

Ramazanov

Gulmanov

2021

JMMCS

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Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains

Boudjeriou

Kheloufı

2019

Journal of Siberian Federal University. Mathematics &Amp; Physics

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This article deals with the heat equation @tu @2x u = f in D; D = {(t; x) 2 R2 : a < t < b; (t) < x < +1} with the function satisfying some conditions and the problem is supplemented with boundary conditions of Robin-Neumann type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f 2 L2(D) there exists a unique solution u such that u; @tu; @jx u 2 L2 (D) ; j = 1; 2: The proof is based on the domain decomposition method. This work complements the results obtained in [10].

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Study of 2m-th order parabolic equation in non-symmetric conical domains

Cited by 2 publications

References 12 publications

Solution of a Two-Dimensional Boundary Value Problem of Heat Conduction in a Degenerating Domain

Solution of a Two-Dimensional Boundary Value Problem of Heat Conduction in a Degenerating Domain

Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains

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