scholarly journals Initial-boundary value problem for a time-fractional subdiffusion equation on the torus

2021 ◽  
Vol 65 (3) ◽  
pp. 17-24
Author(s):  
Ravshan Ashurov ◽  
◽  
Oqila Muhiddinova
Keyword(s):  
Boundary Value Problem ◽  
Initial Function ◽  
Boundary Value ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Subdiffusion Equation ◽  
Initial Boundary Value ◽  
The Right

An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. It should be noted, that the condition on the initial function found in this paper is less restrictive than the analogous condition in the case of an equation with derivatives in the sense of Caputo.

scholarly journals INITIAL-BOUNDARY VALUE PROBLEM FOR HYPERBOLIC EQUATIONS WITH AN ARBITRARY ORDER ELLIPTIC OPERATOR

2020 ◽  
pp. 8-19
Author(s):  
Р.Р. Ашуров ◽  
А.Т. Мухиддинова
Keyword(s):  
Boundary Value Problem ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Order Elliptic Operator ◽  
Initial Boundary Value

В настоящей работе исследуется начально-краевые задачи для гиперболических уравнений, эллиптическая часть которых имеет наиболее общий вид и определена в произвольной многомерной области (с достаточно гладкой границей). Установливаются требования на правую часть уравнения и начальные функции, при которых к рассматрываемую задачу применим классический метод Фурье. Другими словами, доказывается методом Фурье существование и единственность решения смешанной задачи и показана устойчивость найденного решения от данных задачи: от начальных функций и правой части уравнения. Введено понятие обобщенного решения и доказана теорема о его существования. Аналогичные результаты справедливы и для параболических уравнений. An initial-boundary value problem for a hyperbolic equation with the most general elliptic differential operator, defined on an arbitrary bounded domain, is considered. Uniqueness, existence and stability of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. The notion of a generalized solution is introduced and existence theorem is proved. Similar results are formulated for parabolic equations too.

scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Initial Value ◽  
Initial Boundary Value

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

scholarly journals On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

2019 ◽  
Vol 21 (4) ◽  
pp. 413-429
Author(s):  
Pavel N. Burago ◽  
Albert I. Egamov
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Phase Constraint ◽  
Initial Boundary Value

We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.

scholarly journals Development and correctness analysis of the mathematical model of transport and suspension sedimentation depending on bottom relief variation

2018 ◽  
Vol 18 (4) ◽  
pp. 350-361
Author(s):  
A. I. Sukhinov ◽  
V. V. Sidoryakina
Keyword(s):  
Boundary Value Problem ◽  
Coastal Area ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quadratic Functional ◽  
Bottom Relief ◽  
Aquatic Medium ◽  
Initial Boundary Value ◽  
The Right

Introduction. The paper is devoted to the study on the three-dimensional model of transport and suspension sedimentation in the coastal area due to changes in the bottom relief. The model considers the following processes: advective transfer caused by the aquatic medium motion, micro-turbulent diffusion, and gravity sedimentation of suspended particles, as well as the bottom geometry variation caused by the particle settling or bottom sediment rising. The work objective was to conduct an analytical study of the correctness of the initial-boundary value problem corresponding to the constructed model.Materials and Methods. The change in the bottom relief aids in solution to the initial-boundary value problem for a parabolic equation with the lowest derivatives in a domain whose geometry depends on the desired function of the solution, which in general leads to a nonlinear formulation of the problem. The model is linearized on the time grid due to the “freezing” of the bottom relief within a single step in time and the subsequent recalculation of the bottom surface function on the basis of the changed function of the suspension concentration, as well as a possible change in the velocity vector of the aquatic medium.Research Results. For the linearized problem, a quadratic functional is constructed, and the uniqueness of the solution to the corresponding initial boundary value problem is proved within the limits of an unspecified time step. On the basis of the quadratic functional transformation, we obtain a prior estimate of the solution norm in the functional space L2 as a function of the integral time estimates of the right side, and the initial condition. Thus, the stability of the solution to the initial problem from the change of the initial and boundary conditions, the right-hand side function, is established.Discussion and Conclusions. The model can be of value for predicting the spread of contaminants and changes in the bottom topography, both under an anthropogenic impact and due to the natural processes in the coastal area.

scholarly journals Critical convective-type equations on a half-line

2006 ◽  
Vol 2006 ◽  
pp. 1-24
Author(s):  
Elena I. Kaikina
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Behavior ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equationsut+ℕ(u,ux)+(an∂xn+am∂xm)u=0,(x,t)∈ℝ+×ℝ+,u(x,0)=u0(x),x∈ℝ+,∂xj−1u(0,t)=0forj=1,…,m/2, where the constantsan,am∈ℝ,n,mare integers, the nonlinear termℕ(u,ux)depends on the unknown functionuand its derivativeuxand satisfies the estimate|ℕ(u,v)|≤C|u|ρ|v|σwithσ≥0,ρ≥1, such that((n+2)/2n)(σ+ρ−1)=1,ρ≥1,σ∈[0,m). Also we suppose that∫ℝ+xn/2ℕdx=0. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem above-mentioned. We find the main term of the asymptotic representation of solutions in critical case. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in critical convective case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.

scholarly journals Initial-Boundary Value Problem for Fractional Partial Differential Equations of Higher Order

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Djumaklych Amanov ◽  
Allaberen Ashyralyev
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Partial Differential ◽  
Initial Boundary Value ◽  
The Right

The initial-boundary value problem for partial differential equations of higher-order involving the Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established.

scholarly journals SOLVABILITY OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE QUASILINEAR EQUATION OF HEAT CONDUCTIVITY IN DOMAINS THAT CAN BE TRANSFORMED INTO RECTANGLES

2020 ◽  
Vol 70 (2) ◽  
pp. 36-46
Author(s):  
S.E. Aytzhanov ◽  
◽  
S.Z. Saidalimov ◽  
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Quasilinear Equation ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Existence Of A Solution ◽  
Initial Boundary Value

In this paper, we study the initial-boundary-value problem for the quasilinear heat equation in regions that are reduced to rectangular. Mathematical modeling of many processes taking place in the real world leads to the study of the problems of equations of mathematical physics, when the areas are not rectangular. The theory of nonlinear problems is an actively developing section of the theory of modern differential equations. In the theory of nonlinear equations, a special place is occupied by the study of unbounded solutions or, in other words, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally unsolvable: solutions grow unlimitedly over a finite period of time. In this paper, the initial-boundary-value problem for the quasilinear heat equation in regions that can be reduced to rectangular ones, the existence of a solution is proved by the Galerkin method. The uniqueness of the solution was proved by the obtained a priori estimates. Sufficient conditions for the destruction of the solution in a finite time in a bounded domain are obtained. The exponential decay of the solution with an infinite increase in time is proved. In the final time, it was proved that the solution is localized, i.e. disappears (nullifies).

COMPLETENESS OF ROOT FUNCTIONS AND ELEMENTARY SOLUTIONS OF THE THERMOELASTICITY SYSTEM

1995 ◽  
Vol 05 (05) ◽  
pp. 587-598
Author(s):  
YAKOV YAKUBOV
Keyword(s):  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Root Functions ◽  
Associated Functions ◽  
Initial Boundary Value

In this paper we prove the completeness of the root functions (eigenfunctions and associated functions) of an elliptic system (in the sense of Douglis-Nirenberg) corresponding to the thermoelasticity system with the Dirichlet boundary value condition. The problem is considered in a domain with a non-smooth boundary. Then an initial boundary value problem corresponding to the thermoelasticity system with the Dirichlet boundary value condition is considered. We find sufficient conditions that guarantee an approximation of a solution to the initial boundary value problem by linear combinations of some “elementary solutions” to the thermoelasticity system.

Sign in / Sign up

Export Citation Format

Share Document