• New material on Volterra, Fredholm, singular, hypersingular, dual, and nonlinear integral equations, integral transforms, and special functions

• New chapters on mixed multidimensional equations and methods of integral equations for ODEs and PDEs

To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the material in increasing order of complexity. The book can be used as a database of test problems for numerical and approximate methods for solving linear and nonlinear integral equations.

∞ f ( x) = √ g( t) e– ixt dt. 2 π – ∞ 2 ◦. Making an elastic string acquire a given shape under the action of a distributed force. Suppose there is a weightless elastic string of length l that resists tension but does not resist changing its shape. Assume that the string obeys Hooke’s law in tension, so that the force required to extend the string by ∆ l is equal to γ ∆ l, where γ is some constant. Let the ends of the string be fixed at points A and B (Fig. 4) and let the string position

< 0, we must also impose the solvability conditions ∞ f( x) dx = 0, k = 1, 2, . . . , – ν. (31) – ∞ Z ( x) ( x + i) k For the solution of Eq. (25) in the class of functions bounded at infinity, see F. D. Gakhov (1977, 1990). The analog of the characteristic equation on the real axis is the equation of the form b( x) ∞ x – z ϕ( τ) a( x) ϕ( x) + 0 dτ = f ( x), (32) πi – ∞ τ – z 0 τ – x where z 0 is a point that does not belong to the contour. For this equation, all qualitative

+ b( t) a( t) + b( t) By assumption, we have n K( t, τ ) A+( t, τ ) = , Π+( t) = ( t – zk) mk , (13) a( t) + b( t) Π+( t) k=1 where zk ∈ Ω+ and mk are positive integers and the function A+( t, τ ) is analytic with respect to t and with respect to τ on Ω+. 772 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS Relation (12) becomes Π+( t) ϕ+( t) + A+[ ϕ–( t)] = Π+( t)[ D( t) ϕ–( t) + H( t)], (14) where A+ is the integral operator with kernel A+( t, τ ). Since the

Corollary. The number of linearly independent solutions of characteristic equations is minimal among all singular equations with given index ν. 778 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS 15.4-6. Carleman–Vekua Approach to the Regularization. Let us transfer the regular part of a singular equation to the right-hand side and rewrite the equation as follows: b( t) ϕ( τ ) a( t) ϕ( t) + dτ = f ( t) – K( t, τ ) ϕ( τ ) dτ , (34) πi L τ – t L or, in the operator

Problems in Solid and Rock Mechanics. Collection of Papers Devoted to the 75th Birthday of E. I. Shemyakin [in Russian], pp. 411–422, Fizmatlit, Moscow, 2006. McLachlan, N. W., Bessel Functions for Engineers, Clarendon Press, Oxford, 1955. McLean, W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univ. Press, Cambridge, 2000. Mikhailov, L. G., Integral Equations With Homogeneous Kernel of Degree –1 [in Russian], Donish, Dushanbe, 1966. Mikhlin, S. G. (Editor),