scholarly journals Quasi-boundary value method for non-well posed problem for a parabolic equation with integral boundary condition

2001 ◽  
Vol 7 (2) ◽  
pp. 129-145 ◽  
Author(s):  
M. Denche ◽  
K. Bessila
Keyword(s):  
Boundary Condition ◽  
Convergence Rates ◽  
Initial Conditions ◽  
Nonlocal Problem ◽  
Convergence Result ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Boundary Value Method ◽  
Ill Posed ◽  
Well Posed

In this paper we study the problem of control by the initial conditions of the heat equation with an integral boundary condition. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

An a posteriori truncation regularization method for an ill-posed problem for the heat equation with an integral boundary condition

2019 ◽  
Vol 26 (1) ◽  
pp. 35-45
Author(s):  
Mohamed Denche ◽  
Abdelali Benchikha
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Regularization Method ◽  
Initial Conditions ◽  
Integral Boundary Condition ◽  
A Posteriori ◽  
Parameter Choice ◽  
Integral Boundary ◽  
A Posteriori Parameter Choice ◽  
Ill Posed

Abstract The aim of this paper is to investigate the problem of control by the initial conditions of the heat equation with an integral boundary condition. Using the truncation method with an a posteriori parameter choice rule, we give the error estimate between the exact and the regularized solutions. A numerical implementation shows the efficiency of the proposed method.

scholarly journals An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Nihed Teniou ◽  
Salah Djezzar
Keyword(s):  
Nonlocal Problem ◽  
Cauchy Problems ◽  
Convergence Results ◽  
Fixed Sign ◽  
The Cauchy Problem ◽  
Boundary Value Method ◽  
Ill Posed ◽  
The Given ◽  
Differential Operator Equation ◽  
Well Posed

In this paper, we consider a nonhomogeneous differential operator equation of first order u ′ t + A u t = f t . The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u 0 = Φ  or  u T = Φ . We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.

scholarly journals On Periodic Fractional (p, q)-Integral Boundary Value Problems for Sequential Fractional (p, q)-Integrodifference Equations

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 264
Author(s):  
Jarunee Soontharanon ◽  
Thanin Sitthiwirattham
Keyword(s):  
Boundary Condition ◽  
Fixed Point ◽  
Boundary Value Problems ◽  
Fixed Point Theorems ◽  
Existence Results ◽  
Integral Boundary Condition ◽  
Integrodifference Equations ◽  
Integrodifference Equation ◽  
Integral Boundary ◽  
Integral Boundary Value Problems

We study the existence results of a fractional (p, q)-integrodifference equation with periodic fractional (p, q)-integral boundary condition by using Banach and Schauder’s fixed point theorems. Some properties of (p, q)-integral are also presented in this paper as a tool for our calculations.

scholarly journals An Approximate Grid Solution of a Nonlocal Boundary Value Problem with Integral Boundary Condition for Laplace's Equation

2018 ◽  
Vol 22 ◽  
pp. 01016 ◽  
Author(s):  
Adıgüzel A. Dosiyev ◽  
Rifat Reis
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Condition ◽  
Unknown Boundary ◽  
Laplace’S Equation ◽  
Nonlocal Boundary Value Problem ◽  
Integral Boundary ◽  
Laplace's Equation

A new method for the solution of a nonlocal boundary value problem with integral boundary condition for Laplace's equation on a rectangular domain is proposed and justified. The solution of the given problem is defined as a solution of the Dirichlet problem by constructing the approximate value of the unknown boundary function on the side of the rectangle where the integral boundary condition was given. Further, the five point approximation of the Laplace operator is used on the way of finding the uniform estimation of the error of the solution which is order of 0(h2), where hi s the mesh size. Numerical experiments are given to support the theoretical analysis made.

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