On axially symmetric waves

1963 ◽  
Vol 59 (3) ◽  
pp. 637-654 ◽  
Author(s):  
J. W. Craggs
Keyword(s):  
Integral Equation ◽  
Wave Equation ◽  
Analytic Function ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Compressible Flow ◽  
Basic Solution ◽  
Axially Symmetric ◽  
Method Of Solution

AbstractThe wave equation with axial symmetry is reduced, by the assumption of dynamic similarity, to an equation in two variables, r/t and θ A basic solution, having some of the attributes of a source, is introduced, and it is shown that, by use of this solution and a suitable contour integral representation, the solution of any appropriate boundary-value problem may be reduced to the determination of an analytic function, with boundary conditions given by an integral equation.The method of solution is illustrated by reference to some basic problems in the theory of linearized compressible flow.

scholarly journals ABOUT THE APPROXIMATE SOLUTION OF THE USUAL AND GENERALIZED HILBERT BOUNDARY VALUE PROBLEMS FOR ANALYTICAL FUNCTIONS

2000 ◽  
Vol 5 (1) ◽  
pp. 119-126
Author(s):  
V. R. Kristalinskii
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Cauchy Type ◽  
Analytical Functions ◽  
Method Of Solution ◽  
Cauchy Type Integrals ◽  
Hilbert Boundary Value Problem

In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the formproposed by the author. All methods are implemented with the Mathcad and Maple.

On the capacity of the circular disc condenser at small separation

1970 ◽  
Vol 68 (1) ◽  
pp. 235-254 ◽  
Author(s):  
Frank Leppington ◽  
Harold Levine
Keyword(s):  
Integral Equation ◽  
Asymptotic Expansion ◽  
Boundary Value Problem ◽  
Charge Density ◽  
Boundary Value ◽  
Dual Variety ◽  
Long Span ◽  
Infinite Extent ◽  
Crude Estimate

AbstractA pair of identical circular discs, held at equal and opposite potentials, forms a condenser whose capacityCdepends on the ratio ε of separation against diameter. The determination of an asymptotic expansion forCwhen ε is small poses an axisymmetric boundary-value problem for harmonic functions that has engaged the attention of numerous investigators over a long span of time. It is a simple matter to construct a Fredholm integral equation of the first kind for the charge density ± σ on the discs, in terms of which the potential field and the capacity are implicitly determined, but the equation is unsuitable if ε ≪ 1. Integral equations of the second kind and of the dual variety have also been proposed as a means of securing a more manageable formulation of the boundary-value problem. An elementary approximation follows from the hypothesis that the charge density is almost the same as though the discs were of infinite extent, except for a region close to the edges, and leads to the resultC∼ l/8ε as ε → 0. Kirchhoff considerably improved on this crude estimate by suggesting a plausible edge correction which yields two further terms forC, of orders log ∈ and a constant, respectively, and his results have been rigorously established by the more refined analysis of Hutson. In the present work an integral equation of the first kind for the distribution of potential off the discs is derived and utilized to obtain an approximation forCwhen ε is small, reproducing the result of Kirchhoff and Hutson. Furthermore, an estimate of the error provides explicit details regarding the next term in the asymptotic expansion ofC, which is of the order ε(log ε)2.

scholarly journals OBTAINING AND INVESTIGATION OF THE INTEGRAL REPRESENTATION OF SOLUTION AND BOUNDARY INTEGRAL EQUATION FOR THE NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY

2016 ◽  
Vol 6 ◽  
pp. 53-58 ◽  
Author(s):  
Grigoriy Zrazhevsky ◽  
Vera Zrazhevska
Keyword(s):  
Thermal Conductivity ◽  
Integral Equation ◽  
Fundamental Solution ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Boundary Integral Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Integral ◽  
Initial Boundary Value

Technological processes in the energy sector and engineering require the calculation of temperature regime of functioning of different constructions. Mathematical model of thermal loading of constructions is reduced to a non-stationary initial-boundary value problem of thermal conductivity. The article examines the formulation of the non-stationary initial-boundary value problem of thermal conductivity in the form of a boundary integral equation, analyzes the singular equation and builds the fundamental solution. To build the integral representation of the solution the method of weighted residuals is used. The correctness of the obtained integral representation of the solution in Minkowski space is confirmed. Singularity of the fundamental solution is investigated. The boundary integral equation and fundamental solution for axially symmetric domain for internal problem is built. The results of the article can be useful for numerical implementation of boundary element method.

scholarly journals The method for calculating singular integrals in problems of axially symmetric Stokes flows

2019 ◽  
Keyword(s):  
Integral Equation ◽  
Integral Representation ◽  
Integral Equations ◽  
Singular Integral ◽  
Geometric Mean ◽  
Singular Integral Equations ◽  
Elliptic Integrals ◽  
Quadrature Formulas ◽  
Axially Symmetric ◽  
One Dimensional

The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.

scholarly journals On the Modified Jump Problem for the Laplace Equation in the Exterior of Cracks in a Plane

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
P. A. Krutitskii ◽  
A. Sasamoto
Keyword(s):  
Boundary Condition ◽  
Integral Equation ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Classical Solution ◽  
Laplace Equation ◽  
Normal Derivative ◽  
Index Zero ◽  
Jump Problem ◽  
Unique Classical Solution

The boundary value problem for the Laplace equation outside several cracks in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified on the cracks. The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The integral representation and integral equation are essentially simpler than those derived for this problem earlier. The singularities at the ends of the cracks are investigated.

Load Transfer Problem for an Embedded Shaft in Torsion

1970 ◽  
Vol 37 (4) ◽  
pp. 959-964 ◽  
Author(s):  
L. M. Keer ◽  
N. J. Freeman
Keyword(s):  
Integral Equation ◽  
Half Space ◽  
Load Transfer ◽  
Transfer Problem ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Infinite Cylinder ◽  
Axially Symmetric ◽  
Homogeneous Cylinder

This paper deals with the axially symmetric torsion of a semi-infinite cylinder embedded into an elastic half space, where the cylinder is allowed to protrude by a finite amount. The problem is formulated to include the case of the protruding portion of the cylinder when it is a different material and partially bonded to the embedded portion. With the use of integral transforms and Dini series, the problem is reduced to the determination of the solution of an integral equation. Stress singularities of a fractional order are noted and computed at the juncture, when all members are perfectly bonded. A numerical solution of the integral equation is obtained for the case of a homogeneous cylinder. The bond stress on the cylinder—half space interface and the torque-twist (and consequently, strain energy) for the entire system are computed for different values of the elastic constants.

The determination of energies and wavefunctions with full electronic correlation

1969 ◽  
Vol 310 (1500) ◽  
pp. 43-61 ◽  
Keyword(s):  
Correlation Function ◽  
Wave Equation ◽  
Variational Problems ◽  
Variation Theory ◽  
Electronic Correlation ◽  
Convergence Properties

It has been widely thought that the use of wavefunctions with full electronic correlation would involve integrals of 3 N dimensions, where N is the number of electrons. Here it is shown that by a method similar to that of variation theory a set of equations which determine the orbitals and correlation function can be derived so that they only involve integrals of up to nine dimensions. Even these nine-dimensional integrals have some special characteristics which make them equivalent to six-dimensional integrals in some methods of integration. The method is formulated for the particular canonical choice of correlation function that has been previously investigated by the authors and is based on a particular trans-correlated kind of wave equation and on some particular convergence properties recently shown for bi-variational problems. This appears to provide a solution to the problem of including all r ij - quantities in wavefunctions: a problem which has been variously discussed for the last thirty years.

Construction of complex potentials for multiply connected domain

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Pyotr N. Ivanshin
Keyword(s):  
Integral Equation ◽  
Analytic Function ◽  
Linear System ◽  
Unbounded Domain ◽  
Connected Domain ◽  
Complex Potential ◽  
Cauchy Integral ◽  
Multiply Connected Domain ◽  
Multiply Connected ◽  
Impermeable Boundary

AbstractThe method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential – an analytic function in an unbounded multiply connected domain with a simple pole at infinity which maps the domain onto a plane with horizontal slits. We consider a locally sourceless, locally irrotational flow on an arbitrary given 𝑛-connected unbounded domain with impermeable boundary. The complex potential has the form of a Cauchy integral with one linear and 𝑛 logarithmic summands. The method is easily computable.

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