Solution of Volterra partial integral equations with the use of differential equations

We consider Volterra partial differential equations with two and three independent variables and reduce them to Goursat problems, whose solutions are constructed by the Riemann method. We single out cases in which the corresponding Riemann functions (and hence the solutions of the original equations) can be written out in closed form. © 2008 MAIK Nauka

An addition to the cases of solvability of the Goursat problem in quadratures

© 2017, Pleiades Publishing, Ltd.Six new versions of solvability conditions for the Goursat problem in quadratures are formulated in terms of the coefficients of a linear inhomogeneous hyperbolic equation

Version of the elimination method for a system of partial differential equations

© 2017, Pleiades Publishing, Ltd. For a system of three first-order partial differential equations with three independent variables, we obtain sufficient conditions for one component of the solution to satisfy a third-order Bianchi equation. We also obtain conditions for the solvability of this system by quadratures

Solvability of the Goursat problem in quadratures

We find new variants of Goursat problem solution in quadratures on the basis of the combination of the Riemann method and the cascade integration. The results are applied to two Volterra equations with particular integrals. © 2013 Allerton Press, Inc

Factorization of equations with dominating higher partial derivative

We present necessary and sufficient conditions for the factorization of equations with dominating partial derivatives and illustrate the possibility of using our results by examples of two forms of equations of the second and third orders. © 2014 Pleiades Publishing, Ltd

Conditions for the solvability of a system of integral equations by quadratures

© 2015, Pleiades Publishing, Ltd. By reduction to Goursat problems, we obtain various versions of conditions ensuring that the solution of the system in question can be constructed in closed form

One approach to the solution of volterra integral equations with degenerate kernels

We propose a technique that in certain cases allows one to solve the considered equations in an explicit form. © 2011 Allerton Press, Inc

A remark on spatial boundary value problems for hyperbolic equations

We derive sufficient conditions for the unique solvability of two boundary value problems for factorized hyperbolic equations. © Pleiades Publishing, Ltd., 2010

On Conditions of Solvability of the Goursat Problem for Generalized Aller Equation

© 2018, Allerton Press, Inc. In terms of coefficients of an equation under consideration, we derive 16 variants of collections of conditions of resolvability of this equation in quadratures. Every collection consists of three identities that interconnect five coefficients appearing in left-hand side of an equation

Solution of Volterra partial integral equations with the use of differential equations

We consider Volterra partial differential equations with two and three independent variables and reduce them to Goursat problems, whose solutions are constructed by the Riemann method. We single out cases in which the corresponding Riemann functions (and hence the solutions of the original equations) can be written out in closed form. © 2008 MAIK Nauka