scholarly journals Initial data problem for an equation related to a peridynamic model in a two-dimensional domain

2021 ◽  
pp. 45-52
Author(s):  
А.В. Юлдашева
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Initial Data ◽  
Solid Mechanics ◽  
Two Dimensional ◽  
Existence Of A Solution ◽  
Peridynamic Model ◽  
Data Problem ◽  
Uniqueness And Existence ◽  
Dimensional Domain

В настоящей работе доказывается единственность и существование решения задачи Коши для интегро-дифференциального уравнения, связанного с перидинамической моделью механики твёрдого тела с двумя пространственными переменными. In this paper the uniqueness and existence of a solution of Cauchy problem for an integro-differential equation associated with a peridynamic model of solid mechanics in a two-dimensional domain are proved.

scholarly journals Regularization and Error Estimate for the Poisson Equation with Discrete Data

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 422
Author(s):  
Nguyen Anh Triet ◽  
Nguyen Duc Phuong ◽  
Van Thinh Nguyen ◽  
Can Nguyen-Huu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Poisson Equation ◽  
Regularization Method ◽  
Random Noise ◽  
Two Dimensional ◽  
The Poisson Equation ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Dimensional Domain

In this work, we focus on the Cauchy problem for the Poisson equation in the two dimensional domain, where the initial data is disturbed by random noise. In general, the problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. To regularize the instable solution of the problem, we have applied a nonparametric regression associated with the truncation method. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged; and the error has been enhanced once the number of observation points is increased.

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

scholarly journals Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces

2002 ◽  
Vol 29 (9) ◽  
pp. 501-516
Author(s):  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Initial Data ◽  
Asymptotic Representation ◽  
Analytic Solutions ◽  
Two Dimensional ◽  
Nonlinear Schrödinger ◽  
Scattering States ◽  
Suitable Norm

We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spacesi∂tu+(1/2)Δu=𝒩(u),(t,x)∈ℝ×ℝ2;u(0,x)=φ(x),x∈ℝ2, where𝒩(u)=Σj,k=12(λjk(∂xju)(∂xku)+μjk(∂xju¯)(∂xku¯)), whereλjk,μjk∈ℂ. We prove that if the initial dataφsatisfy some analyticity and smallness conditions in a suitable norm, then the solution of the above Cauchy problem has the asymptotic representation in the neighborhood of the scattering states.

On the existence of a continuously differentiable solution to the Cauchy problem for implicit differential equations

2019 ◽  
pp. 376-383
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Initial Point ◽  
Local Solvability ◽  
Time Interval ◽  
Phase Variable ◽  
Existence Of A Solution ◽  
Differentiable Functions ◽  
The Cauchy Problem

We study the question of the existence of a solution to the Cauchy problem for a differential equation unsolved with respect to the derivative of the unknown function. Differential equations generated by twice continuously differentiable mappings are considered. We give an example showing that the assumption of regularity of the mapping at each point of the domain is not enough for the solvability of the Cauchy problem. The concept of uniform regularity for the considered mappings is introduced. It is shown that the assumption of uniform regularity is sufficient for the local solvability of the Cauchy problem for any initial point in the class of continuously differentiable functions. It is shown that if the mapping defining the differential equation is majorized by mappings of a special form, then the solution of the Cauchy problem under consideration can be extended to a given time interval. The case of the Lipschitz dependence of the mapping defining the equation on the phase variable is considered. For this case, estimates of non-extendable solutions of the Cauchy problem are found. The results are compared with known ones. It is shown that under the assumptions of the proved existence theorem, the uniqueness of a solution may fail to hold. We provide examples llustrating the importance of the assumption of uniform regularity.

scholarly journals ON THE ALGORITHM FOR SOLVING OF A LINEAR BOUNDARY VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATION WITH PARAMETER

2020 ◽  
Vol 70 (2) ◽  
pp. 71-76
Author(s):  
N.B. Iskakova ◽  
◽  
Zh. Kubanychbekkyzy ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Linear Boundary ◽  
Existence Of A Solution ◽  
The Cauchy Problem ◽  
Linear Boundary Value Problem

A linear boundary value problem for a system of ordinary differential equations containing a parameter is considered on a bounded segment. For a fixed parameter value, the Cauchy problem for an ordinary differential equation is solved. Using the fundamental matrix of differential part and assuming uniqueness solvability of the Cauchy problem an origin boundary value problem is reduced to the system of linear algebraic equation with respect to unknown parameter. The existence of a solution to this system ensures the existence of a solution to the boundary value problem under study. The algorithm of finding of solution for initial problem is offered based on a construction and solving of the linear algebraic equation. The basic auxiliary problem of algorithm is: the Cauchy problem for ordinary differential equations. The numerical implementation of algorithm offered in the article uses the method of Runge-Kutta of fourth order to solve the Cauchy problem for ordinary differential equations.

scholarly journals Accuracy Investigation of FDM, FEM and MoM for a Numerical Solution of the 2D Laplace’s Differential Equation for Electrostatic Problems

2021 ◽  
Vol 1 (2) ◽  
pp. 26-30
Author(s):  
Bojan Glushica ◽  
Andrijana Kuhar ◽  
Vesna Arnautovski Toseva
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Advantages And Disadvantages ◽  
Electrostatic Problems ◽  
Dimensional Domain

Laplace’s differential equation is one of the most important equations which describe the continuity of a system in various fields of engineering. As a system gets more complex, the need for solving this equation numerically rises. In this paper we present an accuracy investigation of three of the most significant numerical methods used in computational electromagnetics by applying them to solve Laplace’s differential equation in a two-dimensional domain with Dirichlet boundary conditions. We investigate the influence of discretization on the relative error obtained by applying each method. We point out advantages and disadvantages of the investigated computational methods with emphasis on the hardware requirements for achieving certain accuracy.

scholarly journals Existence and uniqueness of a finite energy solution for the mixed value problem of porous thermoelastic bodies

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. Marin ◽  
S. Vlase ◽  
C. Carstea
Keyword(s):  
Initial Data ◽  
Mixed Problem ◽  
Existence And Uniqueness ◽  
Finite Energy ◽  
Theory Of Elasticity ◽  
Existence Of A Solution ◽  
Porous Bodies ◽  
Finite Energy Solution ◽  
Uniqueness And Existence ◽  
Dipolar Structure

AbstractWe consider the mixed problem with boundary and initial data in thermoelasticity of porous bodies with dipolar structure. By generalizing some known results developed by Dafermos in a more simple case of the classical theory of elasticity, we prove new theorems in which we address the issues regarding the uniqueness and existence of a solution with finite energy of the respective problem after we define this type of solution.

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