Boundary value problem for a third-order equation of mixed type in a rectangular domain

For a third-order equation of the parabolic-hyperbolic type, we suggest a method for studying a boundary value problem by solving the inverse problem for a second-order equation of the mixed parabolic-hyperbolic type with unknown right-hand side depending implicitly on time. We prove a criterion for the uniqueness of the solution of the boundary value problem constructed in the form of the sum of a series in the eigenfunctions of the corresponding one-dimensional Sturm-Liouville problem. We prove the stability of the solution with respect to the boundary data in the norms of the spaces W ₂ ⁿ [0, 1] and $C\left( {\bar D} \right)$ .     

Boundary value problem for a mixed type equation with an advanced-retarded argument

We consider the Tricomi problem for the Lavrent??ev-Bitsadze equation with a mixed deviation of the argument. The uniqueness theorem for the problem is proved under constraints on the deviation of the argument. The existence of a solution is related to the solvability of a difference equation. We obtain integral representations of solutions in closed form.     

Dirichlet problem for a third-order equation of mixed type in a rectangular domain

For a third-order differential equation of parabolic-hyperbolic type, we suggest a method for studying the first boundary value problem by solving an inverse problem for a second-order equation of mixed type with unknown right-hand side. We obtain a uniqueness criterion for the solution of the inverse problem. The solution of the inverse problem and the Dirichlet problem for the original equation is constructed in the form of the sum of a Fourier series.     

Boundary value problem for an advanced-retarded equation of mixed type with a nonsmooth degeneration line

We consider the Tricomi problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator in the leading part, with advanced-retarded arguments, and with parallel degeneration lines. We prove the uniqueness theorem under restrictions on the values of the argument deviations. The problem is uniquely solvable. We find integral representations of solutions in closed form.     

Gellerstedt problem for a differential-difference equation of mixed type with advanced-retarded multiple deviations of the argument

We consider the Gellerstedt problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator in the leading part and with advanced-retarded multiple deviations of the argument in the derivatives and the function. We prove the uniqueness theorem for the problem without restrictions on the deviation value. The problem is uniquely solvable. We derive closed-form integral representations of the solutions.     

Boundary value problems for a third-order hyperbolic equation on the plane

For a factorized third-order hyperbolic equation on the plane, we obtain sufficient conditions for the solvability of some boundary value problems with conditions that have not been considered for this equation earlier.     

Initial-boundary value problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain

The spectral analysis method is used to establish a uniqueness criterion and prove the existence of a solution of the first initial-boundary value problem for a special equation of mixed type.     

Boundary value problem for one evolution equation

The aim of the paper is to investigate the boundary value problem of the evolution equation Lu = K (x,t) _ut - Δu + a (x,t) u = f (x,t). The characteristic property of this type of equations is the failure of the Petrovski’s “A” condition when coefficients are constant [1]. In this case, Cauchy problem is incorrect in the sense of Hadamard. Hence in this paper, the space, guaranteeing the correctness of the boundary value problem in the sense of Hadamard, is selected by adding some additional conditions to the coefficients of the equation.     

Finite-difference method for a nonlocal boundary value problem for a third-order pseudoparabolic equation

We consider a nonlocal boundary value problem for a third-order pseudoparabolic equation with variable coefficients. For its solution, in the differential and finite-difference settings, we derive a priori estimates that imply the stability of the solution with respect to the initial data and the right-hand side on a layer as well as the convergence of the solution of the difference problem to that of the differential problem.     

Boundary value problem for a multidimensional fractional partial differential equation

We construct a fundamental solution of a linear fractional partial differential equation. For an equation with Dzhrbashyan-Nersesyan fractional differentiation operators, we solve a boundary value problem and find a closed-form representation for its solution. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are special cases of the assertions proved here.     

Boundary value problem for the KDV equation on a half-line

The L-A pair corresponding to the boundary value problem with the conditionu| _x=0=a for the KdV equation is presented. A broad class of exact solutions to this equation is constructed and the conservation laws are discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 98–113, January, 1997.     

BOUNDARY VALUE PROBLEM FOR A GENERALIZED LIENARD EQUATION

By coincidence degree, the existence of solution to the boundary value problem of a generalized Liénard equation

Characteristic boundary value problem for a third-order functional-differential equation with the Bianchi operator

We consider a version of the Goursat problem for a previously unstudied third-order equation and prove the existence and uniqueness of the solution of this problem.     

Boundary value problem for a mixed-type equation with a difference-differential operator

The Tricomi problem for a mixed-type equation with retarded argument in an unbounded domain is considered. The unique solvability of the problem is proved without restrictions on the delay magnitude. The existence of a solution follows from the solvability of a difference equation. Closed-form integral representations for the solutions are derived.