Some effective solutions for Prandtl's type integro-differential equation

Prandtl’s type integro-differential equations with a different coefficient are investigated. Using the methods of the theory of analytic functions and integral transformations, the singular integro-differential equation is reduced to boundary value problems of the theory of analytic functions. Effective solutions of this equation and the corresponding asymptotic estimates are obtained

Solutions of a singular integro-differential equation related to the adhesive contact problems of elasticity theory

Exact and approximate solutions of a some type singular integro-differential equation related to problems of adhesive interaction between elastic thin half-infinite or finite homogeneous patch and elastic plate are investigated. For the patch loaded with vertical forces, there holds a standard model in which vertical elastic displacements are assumed to be constant. Using the theory of analytic functions, integral transforms and orthogonal polynomials, the singular integro-differential equation is reduced to a different boundary value problem of the theory of analytic functions or to an infinite system of linear algebraic equations. Exact or approximate solutions of such problems and asymptotic estimates of normal contact stresses are obtained.

Fox H-Functions in Self-Consistent Description of a Free-Electron Laser

A fractional calculus concept is considered in the framework of a Volterra type integro-differential equation, which is employed for the self-consistent description of the high-gain free-electron laser (FEL). It is shown that the Fox H-function is the Laplace image of the kernel of the integro-differential equation, which is also known as a fractional FEL equation with Caputo–Fabrizio type fractional derivative. Asymptotic solutions of the equation are analyzed as well.