Complementary variational principles for a boundary-value problem of the helmholtz equation with mixed boundary conditions

1971 ◽  
Vol 1 (11) ◽  
pp. 453-456 ◽  
Author(s):  
N. Anderson ◽  
A. M. Arthurs
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Variational Principles ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Conditions

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

scholarly journals A new application of Schrödinger-type identity to singular boundary value problem for the Schrödinger equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Bo Meng
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Conditions ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Type Identity

Abstract In this paper, we present a modified Schrödinger-type identity related to the Schrödinger-type boundary value problem with mixed boundary conditions and spatial heterogeneities. This identity can be regarded as an $L^{1}$ L 1 -version of Fisher–Riesz’s theorem and has a broad range of applications. Using it and fixed point theory in $L^{1}$ L 1 -metric spaces, we prove that there exists a unique solution for the singular boundary value problem with mixed boundary conditions and spatial heterogeneities. We finally provide two examples, which show the effectiveness of the Schrödinger-type identity method.

scholarly journals Numerical Results for Linear Sequential Caputo Fractional Boundary Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 910
Author(s):  
Bhuvaneswari Sambandham ◽  
Aghalaya S. Vatsala ◽  
Vinodh K. Chellamuthu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Conditions ◽  
Numerical Code ◽  
Fractional Boundary Value Problem ◽  
Monotone Method ◽  
Fractional Boundary Value Problems

The generalized monotone iterative technique for sequential 2 q order Caputo fractional boundary value problems, which is sequential of order q, with mixed boundary conditions have been developed in our earlier paper. We used Green’s function representation form to obtain the linear iterates as well as the existence of the solution of the nonlinear problem. In this work, the numerical simulations for a linear nonhomogeneous sequential Caputo fractional boundary value problem for a few specific nonhomogeneous terms with mixed boundary conditions have been developed. This in turn will be used as a tool to develop the accurate numerical code for the linear nonhomogeneous sequential Caputo fractional boundary value problem for any nonhomogeneous terms with mixed boundary conditions. This numerical result will be essential to solving a nonlinear sequential boundary value problem, which arises from applications of the generalized monotone method.

scholarly journals A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation

2010 ◽  
Vol 2010 ◽  
pp. 1-20
Author(s):  
Zheyan Zhou ◽  
Jianhe Shen
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Degree Theory ◽  
Mixed Boundary Conditions ◽  
Second Order ◽  
Order Differential Operator ◽  
Uniqueness Of Solutions

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered,Lx=f(t,x,x′),t∈(a,b),g(x(a),x(b),x′(a),x′(b))=0,x(b)=x(a)in whichLis a formally self-adjoint second-order differential operator. Under appropriate assumptions onL,f, andg, existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.

Sign in / Sign up

Export Citation Format

Share Document