Cauchy Integral Method Research Articles

On the reversion m ethod for the approxim ate solution of non-linear ordinary differential equations

In this paper we attempt to revive interest in the reversion method for solving non-linear differential equations by combining the reversion technique with the Cauchy integral method for solving differential equations by direct quadratures. The resultant synthesis is easy to implement on any standard symbolic manipulation package and various standard problems are solved showing the superiority of the current method over previous attempts to employ the reversion technique using, for example, Laplace transforms.

Electro-elastic fundamental solutions of anisotropic piezoelectric materials with an elliptical hole

By using Stroh' complex formalism and Cauchy's integral method, the electro-elastic fundamental solutions of an infinite anisotropic piezoelectric solid containing an elliptic hole or a crack subjected to a line force and a line charge are presented in closed form. Particular attention is paid to analyzing the characteristics of the stress and electric displacement intensity factors. When a line force-charge acts on the crack surface, the real form expression of intensity factors is obtained. It is shown through a special example that the present work is correct.

Solution of differential equations by quadratures: the Cauchy integral method

The Cauchy method for finding a particular solution to second‐order linear ODE with constant coefficients is derived within the ‘factorization’ approach to solving differential equations discussed recently by Powles. The Cauchy method reduces the problem of finding particular solutions to the inhomogeneous equation to the evaluation of an integral involving the product of a particular solution of the homogeneous equation and the ‘forcing function’.

A Cauchy integral element for power-type singularities

A boundary element formulation is derived to take into account power-type singularities of a harmonic function. The element, formulated by using complex variables, can be used in conjunction with a Cauchy integral equation method to solve two-dimensional boundary value problems for Laplace's equation defined in polygonal domains. All the integrals in the element formulation are performed analytically, and an asymptotic expansion is presented for evaluation of the element contribution in the far field. The method is implemented to solve potential flow problems involving equipotential and streamline boundary conditions. Examples of flow nets are presented to illustrate graphically the element capabilities.

A cauchy integral equation method for analytic solutions of half-space convolution equations

This paper is devoted to a method of Cauchy integral equation for the solution of half-space convolution equations. It was introduced by Frisch and Frisch to solve Wiener-Hopf integral equations with algebraically decreasing kernels, arising in non-coherent transfer with complete frequency redistribution. The standard Wiener-Hopf technique, which requires exponentially decreasing kernels, is not directly applicable to those equations. We show here that coherent transfer may also be treated by the Cauchy integral method, which draws its name from the fact that the first step is to perform an inverse Laplace transformation on the convolution equation, in order to recast it into a singular integral equation of the Cauchy-type. For complete redistribution, this Cauchy integral equation may be solved by the classical reduction technique to a boundary value problem in the complex plane. For monochromatic transfer, the situation is somewhat more complicated, because of the presence of discrete eigenvalues in the spectrum of the transport operator. Dirac distributions localized at the isolated eigenvalues must be introduced before the complex plane method of solution becomes applicable again. The analytic solution readily yields the behavior of the radiation field at infinity. The emergent radiation field is also easily obtained. Various examples chosen among standard problems are used to illustrate the method.

Analytical solution of the velocity-slip and diffuslon-slip problems by a cauchy integral method

Abstract Some one-dimensional boundary value problems of the kinetic theory of gases can be solved analytically in closed form when the Boltzmann collision operator is replaced a simple kinetic model such as the linear BGK (Bhatnagar-Gross-Krook) model. Such are the slip-flow and the diffusion slip problems. Analytical solutions were obtained by Wiener-Hopf or singular eigenfunctions expansions methods in the sixties and seventies. Here, a Cauchy integral method of solution, developed for radiative transfer problems, is applied to the slip-flow problem and to the diffusion slip problem for a binary gas mixture (with A. Latyshev1). In the CI method, which can handle Wiener-Hopf integral equations with exponentially or non-exponentially decreasing kernels, the Wiener-Hopf equation is recast into a singular integral equation of the Cauchy type, by an inverse Laplace transformation. The latter equation is solved by a classical method of reduction to a boundary value problem in the complex plane (i.e. a Riemann-Hilbert problem). The equivalence between our solutions and other other analytical representations is discussed.

Cauchy integral method for two-dimensional solidification interface shapes

A method is developed to determine the shape of steady state solidification interfaces formed when liquid above its freezing point circulates over a cold surface. The solidification interface, which is at uniform temperature, will form in a shape such that the non-uniform energy convected to it is locally balanced by conduction into the solid. The interface shape is of interest relative to the crystal structure formed during solidification; regulating the crystal structure has application in casting naturally strengthened metallic composites. The results also pertain to phase-change energy storage devices, where the solidified configuration and overall heat transfer are needed. The analysis uses a conformal mapping technique to relate the desired interface coordinates to the components of the temperature gradient at the interface. These components are unknown because the interface shape is unknown. A Cauchy integral formulation provides a second relation involving the components, and a simultaneous solution yields the interface shape.