By using functional integral methods we determine new types of differential equations satisfied by the joint response excitation probability density function associated with the stochastic solution to first-order nonlinear scalar PDEs. The theory is developed for arbitrary fully nonlinear and for quasilinear first-order stochastic PDEs subject to random boundary conditions, random initial condi...

We give the regularization for fractional integral by delta sequence and apply it to obtain existence-uniqueness theorems in Colombeau algebras for nonlinear equations with singularities: nonlinear system of integral equations with polar kernel and nonlinear parabolic equations (of ordinary type, with nonlinear conservative term and with Schrödinger kernel) with strongly singular initial data a...