scholarly journals A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients

1980 ◽  
Vol 22 (2) ◽  
pp. 218-228 ◽  
Author(s):  
D. L. Clements
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Equation ◽  
Elliptic Partial Differential Equation ◽  
Integral Equation Method ◽  
Variable Coefficients ◽  
Boundary Integral ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation

AbstractA method is derived for the solution of boundary value problems governed by a second-order elliptic partial differential equation with variable coefficients. The method is obtained by expressing the solution to a particular problem in terms of an integral taken round the boundary of the region under consideration.

scholarly journals High-order blended compact difference schemes for the 3D elliptic partial differential equation with mixed derivatives and variable coefficients

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Tingfu Ma ◽  
Yongbin Ge
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Elliptic Partial Differential Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Sixth Order ◽  
Test Problems ◽  
Partial Differential ◽  
Compact Difference ◽  
Mixed Derivatives

Abstract Blended compact difference (BCD) schemes with fourth- and sixth-order accuracy are proposed for approximating the three-dimensional (3D) variable coefficients elliptic partial differential equation (PDE) with mixed derivatives. With truncation error analyses, the proposed BCD schemes can reach their theoretical accuracy, respectively, for the interior gird points and require 19 points compact stencil. They fully blend the implicit compact difference (CD) scheme and the explicit CD scheme together to make the derivation method and programming easier. The BCD schemes are also decoupled, which means the unknown function and its derivatives are separately resolved by different finite difference equations. Moreover, the sixth-order schemes are developed to solve the first-order derivatives, the second-order derivatives and the second-order mixed derivatives on boundaries. Several test problems are applied to show that the present BCD schemes are more accurate than those in the literature.

scholarly journals A note on the boundary integral equation method for the solutions of second order elliptic equations

1985 ◽  
Vol 26 (4) ◽  
pp. 415-421 ◽  
Author(s):  
D. L. Clements ◽  
M. Haselgrove ◽  
D. M. Barnett
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Equation ◽  
Boundary Integral Equation ◽  
Elliptic Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Boundary Integral Equation Method ◽  
Equation Method ◽  
Second Order Elliptic Equations

AbstractThe boundary integral equation method is obtained by expressing a solution to a particular partial differential equation in terms of an integral taken round the boundary of the region under consideration. Various methods exist for the numerical solution of this integral equation and the purpose of this note is to outline an improvement to one of these procedures.

scholarly journals Integral Equation-Wavelet Collocation Method for Geometric Transformation and Application to Image Processing

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Lina Yang ◽  
Yuan Yan Tang ◽  
Xiang Chu Feng ◽  
Lu Sun
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Transformation Model ◽  
Geometric Transformation ◽  
Mathematical Form ◽  
Partial Differential ◽  
Distorted Image

Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet-based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.

scholarly journals A unified boundary integral equation method for a class of second order elliptic boundary value problems

1984 ◽  
Vol 25 (4) ◽  
pp. 501-517 ◽  
Author(s):  
M. Rezayat ◽  
F. J. Rizzo ◽  
D. J. Shippy
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Second Order ◽  
Elliptic Type ◽  
Integral Equation Formulation

AbstractA generalized integral equation formulation and a systematic numerical solution procedure are presented for a class of boundary value problems governed by a general second-order differential equation of elliptic type. Diverse numerical examples include problems of plane-wave scattering, three-dimensional fluid flow, and plane heat transfer for a body with a moving flame boundary. The last example employs certain representation functions useful to increase solution effectiveness in problems with an isolated integrable singularity.

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