Regularization in Marchaud's form and approximate evaluation of a class of integral-differential expressions (Q1333631)

In a number of applied investigations connected with the study of reflection and diffraction of acoustic and elastic waves there occur integral-differential operators acting onto a function \(f\) of two independent variables via the formula \[ Bf(x,y,t)= {\partial\over \partial t} \int^ t_ 0 d\tau\int^ \infty_ 0 d\xi g(x,y+\xi) {\tau H(\tau- c^{-1}\sqrt{x^ 2+ (y+\xi)^ 2})\over \sqrt{\tau^ 2- c^{- 2} (x^ 2+ (y+\xi)^ 2)}} f(\xi,t-\tau),\tag{1} \] where \(H\) is the Heaviside function, \(g(x,\alpha)\) is a sufficiently smooth function of two positive independent variables (playing a role of weight function), \(c\) is a positive constant, \(x>0\), \(y>0\), \(t\in [0,t^*]\) \((0< t^*<\infty)\). Due to instability of the operation of numerical differentiation it seems to be impossible to apply here the known numerical methods in order to get approximate values of the integro-differential expression (1). Therefore, in this article we first regularize expression (1), using a definition of a fractional derivative in Marchaud's form, and then we develop a stable algorithm of its approximation.