scholarly journals A Difference Scheme and Its Error Analysis for a Poisson Equation with Nonlocal Boundary Conditions

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Chunsheng Feng ◽  
Cunyun Nie ◽  
Haiyuan Yu ◽  
Liping Zhou ◽  
Karthikeyan Rajagopal
Keyword(s):  
Difference Scheme ◽  
Elliptic Problem ◽  
Nonlocal Condition ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Discrete Fourier Transformation ◽  
Optimal Error Estimate ◽  
Taylor Formula ◽  
Accuracy Difference

The elliptic problem with a nonlocal boundary condition is widely applied in the field of science and engineering, such as the chaotic system. Firstly, we construct one high-accuracy difference scheme for a kind of elliptic problem by tactfully introducing an equivalent relation for one nonlocal condition. Then, we obtain the local truncation error equation by the Taylor formula and, initially, prove that the new scheme can reach the asymptotic optimal error estimate O h 2 ln   h in the maximum norm through ingeniously transforming a two-dimensional problem to a one-dimensional one through bringing in the discrete Fourier transformation. Numerical experiments demonstrate the correctness of theoretical results.

scholarly journals Finite difference method for Bitsadze-Samarskii type overdetermined elliptic problem with Dirichlet conditions

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 859-872
Author(s):  
Charyyar Ashyralyyev ◽  
Gulzipa Akyuz
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Elliptic Problem ◽  
Difference Method ◽  
Nonlocal Boundary ◽  
Three Dimensional ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Dirichlet Conditions ◽  
Accuracy Difference

In this paper, we apply finite difference method to Bitsadze-Samarskii type overdetermined elliptic problem with Dirichlet conditions. Stability, coercive stability inequalities for solution of the first and second order of accuracy difference schemes (ADSs) are proved. Then, established abstract results are applied to get stable difference schemes for Bitsadze-Samarskii type overdetermined elliptic multidimensional differential problems with multipoint nonlocal boundary conditions. Finally, numerical results with explanation on the realization in two dimensional and three dimensional cases are presented.

scholarly journals SPECTRUM CURVES FOR STURM–LIOUVILLE PROBLEM WITH INTEGRAL BOUNDARY CONDITION

2015 ◽  
Vol 20 (6) ◽  
pp. 802-818 ◽  
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Critical Points ◽  
Nonlocal Condition ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Integral Boundary ◽  
Sturm Liouville

We consider Sturm–Liouville problem with one integral type nonlocal boundary condition depending on three parameters γ (multiplier in nonlocal condition), ξ1, ξ2 ([ξ1, ξ2] is a domain of integration). The distribution of zeroes, poles, and constant eigenvalue points of Complex Characteristic Function is presented. We investigate how Spectrum Curves depend on the parameters of nonlocal boundary conditions. In this paper we describe the behaviour of Spectrum Curves and classify critical points of Complex-Real Characteristic function. Phase Trajectories of critical points in Phase Space of the parameters ξ1, ξ2 are investigated. We present the results of modelling and computational analysis and illustrate those results with graphs.

scholarly journals Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Dengming Liu ◽  
Chunlai Mu
Keyword(s):  
Boundary Condition ◽  
Global Existence ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Nonlocal Boundary Conditions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Subsolution And Supersolution

We investigate the blowup properties of the positive solutions for a semilinear reaction-diffusion system with nonlinear nonlocal boundary condition. We obtain some sufficient conditions for global existence and blowup by utilizing the method of subsolution and supersolution.

scholarly journals On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Allaberen Ashyralyev ◽  
Okan Gercek
Keyword(s):  
Difference Scheme ◽  
Parabolic Equations ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Second Order ◽  
Well Posedness ◽  
Parabolic Differential Equations ◽  
Implicit Difference Scheme ◽  
Order Of Accuracy

We are interested in studying a second order of accuracy implicit difference scheme for the solution of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this difference scheme is established. In an application, coercivity estimates in Hölder norms for approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic differential equations are obtained.

A Meshfree Computational Approach Based on Multiple-Scale Pascal Polynomials for Numerical Solution of a 2D Elliptic Problem with Nonlocal Boundary Conditions

2019 ◽  
Vol 17 (10) ◽  
pp. 1950080 ◽  
Author(s):  
Ömer Oruç
Keyword(s):  
Boundary Conditions ◽  
Elliptic Problem ◽  
Numerical Solutions ◽  
Nonlocal Boundary ◽  
Meshfree Method ◽  
Multiple Scale ◽  
Nonlocal Boundary Conditions ◽  
Condition Numbers ◽  
Test Problems ◽  
Numerical Integrations

A two-dimensional (2D) elliptic problem with nonlocal boundary conditions on both regular and irregular domains is solved numerically by Pascal polynomial basis unified with multiple-scale technique. Very accurate numerical solutions and quite reasonable condition numbers are obtained with the proposed method which is also a truly meshfree method since difficult meshing processes or numerical integrations over domains are not needed for considered problems. Four test problems are solved to show the accuracy and efficiency of the proposed method. Also stability of the method is studied against large noise effect.

Existence and controllability for nonlinear fractional differential inclusions with nonlocal boundary conditions and time-varying delay

2018 ◽  
Vol 21 (4) ◽  
pp. 960-980 ◽  
Author(s):  
Yi Cheng ◽  
Ravi P. Agarwal ◽  
Donal O’ Regan
Keyword(s):  
Nonlocal Condition ◽  
Nonlocal Boundary ◽  
Reaction Diffusion ◽  
Nonlocal Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Time Varying ◽  
Time Varying Delay ◽  
Fractional Differential ◽  
Varying Delay ◽  
Ky Fan

Abstract This paper discusses the existence and controllability of a class of fractional order evolution inclusions with time-varying delay. In the weak topology setting we establish the existence of solutions. Then the controllability of this system with a nonlocal condition is established by applying the Glicksberg-Ky Fan fixed point theorem. As an application, nonlocal problems of a fractional reaction-diffusion equation with a discontinuous nonlinear term is examined.

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