THE METHOD OF FUNDAMENTAL SOLUTIONS FOR AN INVERSE INTERNAL BOUNDARY VALUE PROBLEM FOR THE BIHARMONIC EQUATION

2009 ◽  
Vol 06 (04) ◽  
pp. 557-567 ◽  
Author(s):  
D. LESNIC ◽  
A. ZEB
Keyword(s):  
Boundary Value Problem ◽  
Biharmonic Equation ◽  
Numerical Solutions ◽  
Least Squares Method ◽  
Regularization Parameter ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Internal Boundary ◽  
Unknown Boundary

In this paper, an inverse internal boundary value problem associated to the biharmonic equation is considered. The problem consists of determining unknown boundary conditions from extra interior measurements. The method of fundamental solutions (MFS) is used to discretize the problem and the resulting ill-conditioned system of linear equations is solved using the Tikhonov regularization technique. It is shown that, unlike the least-squares method, the MFS-regularization numerical technique produces stable and accurate numerical solutions for an appropriate choice of the regularization parameter given by the L-curve criterion.

AN IMPLEMENTATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS FOR THE DYNAMIC RESPONSE OF VON KARMAN NONLINEAR PLATE MODEL

2013 ◽  
Vol 10 (02) ◽  
pp. 1341005 ◽  
Author(s):  
A. USCILOWSKA ◽  
D. BERENDT
Keyword(s):  
Dynamic Response ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Iteration Step ◽  
Nonlinear Plate ◽  
Von Karman ◽  
Von Kármán ◽  
Radial Basis Function Approximation

The nonlinear plate dynamics is a subject of investigations across many disciplines. Various models have been proposed. In this paper, one of the models derived by von Karman is under consideration. In this case, the initial-boundary value problem of the dynamics of a plate large deflection is described by two nonlinear, coupled partially differential equations of fourth order with two initial conditions and four boundary conditions given at each boundary point. There are various means of simulating the dynamic response of thin nonlinear plate, to various degrees of accuracy. The proposal of this paper is to implement one of the meshfree methods i.e., the method of fundamental solutions (MFS). The problem is solved in discretized time domain. This discretization is done in a conception of the finite difference method. The nonlinearity of the equations, obtained at each time step, is solved by application of Picard iterations. For each iteration step, a boundary value problem is to be solved. Moreover, the equations at each iteration step are inhomogeneous ones. So, the radial basis function approximation is applied and a particular solution of boundary value problems is obtained. The final solution is calculated by implementation of the MFS. The numerical experiment has confirmed that the proposed numerical procedure gives the solutions with demanded accuracy and is a good tool to solve the considered problem.

A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

2018 ◽  
Vol 29 (10) ◽  
pp. 1850069
Author(s):  
Hong Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Manifold With Boundary ◽  
Prescribed Mean Curvature Equation

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

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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