Scattering problem for a multidimensional system of first-order partial differential equations

We construct transformation operators, which enables us to study a scattering problem and investigate the properties of a scattering operator for a multidimensional system of first-order partial differential equations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev; Zhitomir Pedagogical Institute, Zhitomir. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1065–1076, August, 1999.     

A nondispersive and nondissipative numerical method for first-order linear hyperbolic partial differential equations

We propose a new numerical method for a solution of first-order linear hyperbolic equations. The leap-frog scheme is converted to a nondispersive scheme by introducing an adjustable constant in a fictitious absorption term. Then the erroneous decrease in th solution is eliminated by solving two equations equivalent to the original equation. The new scheme perfectly preserves the form of a discontinuous solution.     

The FD method for first-order linear hyperbolic differential equations with piecewise smooth coefficients

A justification of the differential-discrete FD method for solving one class of first-order linear partial differential equations with piecewise smooth coefficients, a right-hand side, and initial conditions is presented. The convergence rate of the method is shown to be limited exclusively by the smoothness of the input information. High accuracy, ease of algorithmic implementation, and feasibility of representation of results in an analytical form point to the advantage of the FD method over the other methods for the given class of problems. Bibliography:6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 1–11     

Asymptotics of the solution to a singularly perturbed system of first-order partial differential equations with small nonlinearity in the critical case

A formal asymptotic representation of the solution to an initial-boundary value problem for a singularly perturbed system of first-order partial differential equations with small nonlinearity is constructed.     

Numerical solution of fuzzy differential equations under generalized differentiability

In this paper, we interpret a fuzzy differential equation by using the strongly generalized differentiability concept. Utilizing the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. Then we show that any suitable numerical method for ODEs can be applied to solve numerically fuzzy differential equations under generalized differentiability. The generalized Euler approximation method is implemented and its error analysis, which guarantees pointwise convergence, is given. The method’s applicability is illustrated by solving a linear first-order fuzzy differential equation.     

On the oscillation properties of first-order impulsive differential equations with a deviating argument

For first order linear impulsive differential equations with a deviating argument a number of oscillation theorems are proved (of the type of the Sturmian Theorem). Various criteria for oscillation or non-oscillation of the solutions of these equations are found. The oscillatory properties of some concrete equations of the type considered are investigated. Supported by the Bulgarian Ministry of Education, Science and Technologies under Grant MM-422.