On the solution of some axisymmetric boundary value problems by means of integral equations, II: further problems for a circular disc and a spherical cap

Mathematika ◽  
1959 ◽  
Vol 6 (2) ◽  
pp. 120-133 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Value ◽  
Circular Disc ◽  
Spherical Cap

scholarly journals On the Solution of some Axisymmetric Boundary Value Problems by Means of Integral Equations. IV. The Electrostatic Potential due to a Spherical Cap between Two Infinite Conducting Planes

1960 ◽  
Vol 12 (2) ◽  
pp. 95-106 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Electrostatic Potential ◽  
Potential Problem ◽  
Boundary Value ◽  
Circular Disk ◽  
Spherical Cap ◽  
Potential Problems ◽  
Spherical Caps ◽  
Circular Disks

This paper is a sequel to previous papers (1, 2, 3) on the solution of axisymmetric potential problems for circular disks and spherical caps by means of integral equations and applies the methods developed in these papers to the electrostatic potential problem for a perfectly conducting thin spherical cap or circular disk between two infinite earthed conducting planes.

scholarly journals On the Solution of some Axisymmetric Boundary Value Problems by means of Integral Equations

1962 ◽  
Vol 13 (1) ◽  
pp. 13-23 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Circular Cylinder ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Electrostatic Potential ◽  
Potential Problem ◽  
Boundary Value ◽  
Coaxial Cylinder ◽  
Spherical Cap ◽  
Potential Problems ◽  
Circular Disks

This paper is a sequel to a previous paper (1) on axisymmetric potential problems for one or more circular disks situated inside a coaxial cylinder and applies the method used for these problems to the electrostatic potential problem for a perfectly conducting thin spherical cap situated inside an earthed coaxial infinitely long circular cylinder.

Note on the reduction of dual and triple series equations to dual and triple integral equations

1963 ◽  
Vol 59 (4) ◽  
pp. 731-734 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Theory ◽  
Bessel Functions ◽  
Circular Disc ◽  
Legendre Functions ◽  
Dual Integral Equations ◽  
Spherical Cap ◽  
Triple Integral Equations ◽  
Dual Series Equations

Dual integral equations involving Bessel functions occur in the solution of some boundary-value problems in potential theory with conditions prescribed on a circular disc and a considerable amount of attention has been given to the solution of such equations (cf. (1)). The method of solving these dual integral equations is very similar to that employed in the solution of certain dual series equations involving Legendre functions. Equations of this type occur in problems in potential theory with conditions prescribed on a spherical cap and their solution has been obtained by Collins (2). No definite mathematical connexion has, however, been established between these dual series and dual integral equations and the object of this note is to establish such a connexion.

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

Sign in / Sign up

Export Citation Format

Share Document