Singular integral equation

Viscoelastic contact problems have drawn the attention of researchers for some time [2,3,104,105]. The mathematical peculiarity of these problems is their time-dependent boundaries. This has limited the ability to quantify the boundary value contact problems by the tools used in elasticity. The normal displacement (u) and pressure (p) fields in the contact region for non-adhesive contact of viscoelastic materials are obtained by a self-consistent solution to the governing singular integral equation given by [106] ... 

Vol. 1549 G. Vainikko, Multidimensional Weakly Singular Integral Equations. XI, 159 pages. 1993. 

Muskelishvili, N. I., Singular Integral Equation, Noordhoff, Groningen, 1953. 

Volterra equations are classified as regular and singular integral equations. If K(x,y) < the equations are called regular otherwise, they are defined as singular. If f(x) = 0, the equations are termed homogeneous. 

Erdogan, F. and Gupta, G. D. (1972), On the numerical solution of singular integral equations, Quarterly of Applied Mathematics 30, 525-534. 

Singular integral equation analysis. A closed form analytical solution can be obtained. We use results from thin airfoil theory (e.g., Ashley and Landahl, 1965) and singular integral equations (Muskhelishvili, 1953 Gakhov, 1966 Carrier, Krook and Pearson, 1966). Now the standard log r source... 

Having demonstrated the power and elegance behind the use of distributed line sources and the use of singular integral equations, we now consider a slightly more complicated example involving incompressible liquids and compressible gases in anisotropic reservoirs under steady-state flow conditions. This second example will illustrate the flexibility of the thin airfoil technique. But it will also reveal the weaknesses inherent in analytical approaches and why a well formulated numerical method is necessary. 

This singular integral equation, with the Cauchy kernel l/(x- ), governs the vortex strength g(Q. The PV indicates that the integral is improper and must be evaluated using the limit process in Example 2-5. 

Equation 5-82 is the governing singular integral equation of interest, satisfied by the source strength m(x). 

Thus, the singular integral equation for the vortex strength g(x) becomes +1... 

Muskhelishvili, N.I., Singular Integral Equations, P. Noordhoff N.V., Holland,... 

Equation (A2.4.1) is a singular integral equation for 0(x) in terms of g(x). The principle value of the integral is understood. It will emerge that the solution of this integral equation is closely related to the solution of the Hilbert problem discussed in the previous section. 

Muskhelishvili, N. I. (1953) Singular Integral Equations, 2nd ed. (Noordhoff, Gro iingen) [English translation by J. R. M. Radokl... 

The null-field method leads to a nonsingular integral equation of the first kind. However, in the framework of the surface integral equation method, the transmission boundary-value problem can be reduced to a pair of singular integral equations of the second kind [97]. These equations are formulated in terms of two surface fields which are treated as independent unknowns. In order to elucidate the difference between the null-field method and the surface integral equation method we follow the analysis of Martin and Ola [155] and review the basic boundary integral equations for the transmission boundary-value problem. We consider the vector potential Aa with density a... 

---