Existence and uniqueness of the solution of the boundary-value problem for a parabolic equation of unsteady filtration

Казаков, А. Л.; Lempert, A. A.

doi:10.1134/s0021894413020107

Cited by 21 publications

(16 citation statements)

References 3 publications

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“…To be able to consider the class of analytical functions for any valid values of the constants σ and δ, we will make the substitution u = T σ , v = S δ in Equation (2). Such transformations are standard [10] and are regularly used by the authors (see [14] and subsequent works).…”

Section: Formulationmentioning

confidence: 99%

“…We now construct a majorant for the series in Equation (14). Note that U 0 , V 0 , U 1 , V 1 , f i , g i , i = 0, 1, 2, 3 are analytical functions.…”

Section: Existence Theoremmentioning

confidence: 99%

“…The authors of this article also belong to the scientific school of A. F. Sidorov. Previously, we studied the analytical solvability of similar problems in various formulations: one-dimensional [14,15], symmetric [16,17] and non-one-dimensional [18].…”

Section: Introductionmentioning

confidence: 99%

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Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

2020

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The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

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Section: Formulationmentioning

confidence: 99%

“…We now construct a majorant for the series in Equation (14). Note that U 0 , V 0 , U 1 , V 1 , f i , g i , i = 0, 1, 2, 3 are analytical functions.…”

Section: Existence Theoremmentioning

confidence: 99%

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Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

2020

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“…В этой части работа развивает идеи и результаты научной школы А.Ф. Сидорова [7,8], включая результаты авторов [9]. Также с использованием метода Кларксона -Крускалла [4,10] найдены новые точные [11] решения рассматриваемого уравнения, имеющие вид тепловых волн, построение которых сводится к интегрированию обыкновенных дифференциальных уравнений второго порядка, наследующих особенность у исходной задачи [12].…”

Section: Introductionunclassified

On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation

Казаков¹

2019

Sib. Elektron. Mat. Izv.

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The paper deals with a nonlinear second order parabolic PDE, which is usually called "the nonlinear heat equation". We construct and study a particular class of solutions having the form of a heat wave that propagates on a cold (zero) background with finite velocity. The equation degenerates on the front of a heat wave and its order decreases. This fact complicates the study. We prove a new existence and uniqueness theorem for a boundary-value problem with a given heatwave front in the class of analytical functions. Also, we are looking for exact heatwave type solutions. The construction of these solutions is reduced to integration of the nonlinear second order ODE with singularity.

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“…Sidorov [5,6]. Using the methodology [6] the existence and uniqueness theorems are proved for solutions of one-dimensional [7,8] and two-dimensional [9] problems. The method of power series [10] for the porous medium equation with special boundary conditions is applied in [11].…”

Section: Introductionmentioning

confidence: 99%

On exact solutions of a heat-wave type with logarithmic front for the porous medium equation

Казаков¹,

Lempert²,

Orlov³

2017

J. Phys.: Conf. Ser.

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Abstract. The paper deals with a nonlinear second-order parabolic equation with partial derivatives, which is usually called "the porous medium equation". It describes the processes of heat and mass transfer as well as filtration of liquids and gases in porous media. In addition, it is used for mathematical modeling of growth and migration of population. Usually this equation is studied numerically like most other nonlinear equations of mathematical physics. So, the construction of exact solution in an explicit form is important to verify the numerical algorithms. The authors deal with a special solutions which are usually called "heat waves". A new class of heat-wave type solutions of one-dimensional (plane-symmetric) porous medium equation is proposed and analyzed. A logarithmic heat wave front is studied in details. Considered equation has a singularity at the heat wave front, because the factor of the highest (second) derivative vanishes. The construction of these exact solutions reduces to the integration of a nonlinear second-order ordinary differential equation (ODE). Moreover, the Cauchy conditions lead us to the fact that this equation has a singularity at the initial point. In other words, the ODE inherits the singularity of the original problem. The qualitative analysis of the solutions of the ODE is carried out. The obtained results are interpreted from the point of view of the corresponding heat waves' behavior. The most interesting is a damped solitary wave, the length of which is constant, and the amplitude decreases.

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Existence and uniqueness of the solution of the boundary-value problem for a parabolic equation of unsteady filtration

Cited by 21 publications

References 3 publications

Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation

On exact solutions of a heat-wave type with logarithmic front for the porous medium equation

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