scholarly journals Regularization method for parabolic equation with variable operator

2005 ◽  
Vol 2005 (4) ◽  
pp. 383-392
Author(s):  
Valentina Burmistrova
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximation Solution ◽  
Elliptic Boundary Value ◽  
Dirichlet Data ◽  
Ill Posed ◽  
Initial Boundary Value

Consider the initial boundary value problem for the equationut=−L(t)u,u(1)=won an interval[0,1]fort>0, wherew(x)is a given function inL2(Ω)andΩis a bounded domain inℝnwith a smooth boundary∂Ω.Lis the unbounded, nonnegative operator inL2(Ω)corresponding to a selfadjoint, elliptic boundary value problem inΩwith zero Dirichlet data on∂Ω. The coefficients ofLare assumed to be smooth and dependent of time. It is well known that this problem is ill-posed in the sense that the solution does not depend continuously on the data. We impose a bound on the solution att=0and at the same time allow for some imprecision in the data. Thus we are led to the constrained problem. There is built an approximation solution, found error estimate for the applied method, given preliminary error estimates for the approximate method.

A non-homogeneous boundary value problem for the Kuramoto-Sivashinsky equation posed in a finite interval

2020 ◽  
Vol 26 ◽  
pp. 43 ◽  
Author(s):  
Jing Li ◽  
Bing-Yu Zhang ◽  
Zhixiong Zhang
Keyword(s):  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Solution Map ◽  
Ill Posed ◽  
Initial Boundary Value ◽  
Well Posed ◽  
Sivashinsky Equation

This paper studies the initial boundary value problem (IBVP) for the dispersive Kuramoto-Sivashinsky equation posed in a finite interval (0, L) with non-homogeneous boundary conditions. It is shown that the IBVP is globally well-posed in the space Hs(0, L) for any s > −2 with the initial data in Hs(0, L) and the boundary value data belonging to some appropriate spaces. In addition, the IBVP is demonstrated to be ill-posed in the space Hs(0, L) for any s < −2 in the sense that the corresponding solution map fails to be in C2.

scholarly journals Mathematical Modelling and Numerical Simulation of Diffusive Processes in Slow Changing Domains

2020 ◽  
Author(s):  
Dmytro V. Yevdokymov ◽  
Yuri L. Menshikov
Keyword(s):  
Heat Conduction ◽  
Time Scale ◽  
Numerical Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Numerical Approach ◽  
Elliptic Boundary Value ◽  
Initial Boundary Value

Nowadays, diffusion and heat conduction processes in slow changing domains attract great attention. Slow-phase transitions and growth of biological structures can be considered as examples of such processes. The main difficulty in numerical solutions of correspondent problems is connected with the presence of two time scales. The first one is time scale describing diffusion or heat conduction. The second time scale is connected with the mentioned slow domain evolution. If there is sufficient difference in order of the listed time scale, strong computational difficulties in application of time-stepping algorithms are observed. To overcome the mentioned difficulties, it is proposed to apply a small parameter method for obtaining a new mathematical model, in which the starting parabolic initial-boundary-value problem is replaced by a sequence of elliptic boundary-value problems. Application of the boundary element method for numerical solution of the obtained sequence of problems gives an opportunity to solve the whole considered problem in slow time with high accuracy specific to the mentioned algorithm. Besides that, questions about convergence of the obtained asymptotic expansion and correspondence between initial and obtained formulations of the problem are considered separately. The proposed numerical approach is illustrated by several examples of numerical calculations for relevant problems.

scholarly journals Comparative Analysis of Methods for Regularizing an Initial Boundary Value Problem for the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Sergey Igorevich Kabanikhin ◽  
M. A. Shishlenin ◽  
D. B. Nurseitov ◽  
A. T. Nurseitova ◽  
S. E. Kasenov
Keyword(s):  
Comparative Analysis ◽  
Boundary Value Problem ◽  
Helmholtz Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Estimate ◽  
Ill Posed ◽  
Continuation Problem ◽  
Initial Boundary Value

We consider an ill-posed initial boundary value problem for the Helmholtz equation. This problem is reduced to the inverse continuation problem for the Helmholtz equation. We prove the well-posedness of the direct problem and obtain a stability estimate of its solution. We solve numerically the inverse problem using the Tikhonov regularization, Godunov approach, and the Landweber iteration. Comparative analysis of these methods is presented.

scholarly journals Stability estimation of the generalized solution to the direct problem for the acoustic equation

2021 ◽  
Vol 2092 (1) ◽  
pp. 012005
Author(s):  
Sergey Kabanikhin ◽  
Altyn Nurseitova ◽  
Syrym Kasenov
Keyword(s):  
Boundary Value Problem ◽  
Direct Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Timelike Surface ◽  
Acoustic Equation ◽  
Ill Posed ◽  
Initial Boundary Value

Abstract The initial-boundary value problem for the acoustic equation with data on a timelike surface is considered in this paper. Such a problem arises, for example, if it is required to determine the acoustic pressure inside the region from a fixed response to part of the boundary from the source involved at the same boundary. It is assumed that the medium is at rest up to a certain instant of time and the parameters of the medium, for example, acoustic density, are known. The problem is considered in a triangular domain. The advisability of this was shown in the second half of the last century in the works of Romanov V.G. (for example, [1]), where it was proved that the solution to the direct problem of acoustic is representable as the sum of a singular and a continuous terms. The author has written out the form of the singular part, investigated the problem in an integral statement, and also proved conditional well-posedness theorems for three cases: for a small parameter of the domain, for small data, and for the source representability of the sought solution. It is known that the initial-boundary value problem for the acoustic equation with data on a timelike surface is ill-posed. In this paper, the original ill-posed problem is reduced to an inverse problem with respect to some direct (well-posed) problem. The theorem is proved and a stability estimate of the generalized solution to the direct problem is obtained.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

scholarly journals An extension of Gronwall inequality in the theory of bodies with voids

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1161-1167
Author(s):  
Marin Marin ◽  
Praveen Ailawalia ◽  
Ioan Tuns
Keyword(s):  
Boundary Value Problem ◽  
Internal State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Internal Variables ◽  
State Variables ◽  
Gronwall’S Inequality ◽  
Initial Boundary Value

Abstract In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

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