Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

Inverse problems for the heat-conduction equation with nonlocal boundary conditions

Conditions under which the time-dependent temperature conductivity coefficient is determined uniquely are established in the case where the boundary conditions and the overdetermination conditions are non local.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1066–1071, August, 1993.     

Stabilization of the solutions of boundary-value problems for the multidimensional heat-conduction equation

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 327–334, March, 1989.     

Matrix-differential methods of solving boundary-value problems for the nonlinear one-dimensional heat-conduction equation

Matrix numerical differentiation algorithms are applied to construct numerical-analytical methods for approximate solution of boundary-value problems for the nonlinear one-dimensional equation of heat conduction. The problems are reduced to a system of differential equations for the values of the sought approximate solution in the interior grid nodes and also to numerical formulas for the solution values at the boundary nodes. A numerical experiment is conducted. The error relative to grid spacing is established.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 37–43, 1986.     

Solvability of the initial-boundary-value problem for the equations of motion of a viscous compressible fluid

It is proved that the initial-boundary-value problem for the system of equations describing the motion of a compressible fluid with a constant viscosity is locally solvable with respect to time. The heat conductivity is not taken into account. The solution is found in the class W _q ^2.1 , q>3.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 128–142, 1976.     

Reduction of the fundamental initial-boundary-value problems for Stokes equations to initial-boundary-value problems for parabolic systems of pseudodifferential equations

It is shown that an initial-boundary-value problem for Stokes' system, in which on the boundary one prescribes the vector field of velocities , or the stress field, or the normal component of the velocity and the tangential stresses, reduces to an initial-boundary-value problem for a system of the form , where the operator A contains a nonlocal term (the so-called singular Green operator). For the solutions of these problems, coercive estimates in the spaces W₂ ^{l, l/2} and also estimates of the norm of the resolving operator in W₂ ^r are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 37–48, 1987.     

Solvability of certain inverse problems for the nonstationary kinetic transport equation

Linear and nonlinear inverse problems for the nonstationary multispeed anisotropic kinetic transport equation are studied. Sufficient conditions for the existence and uniqueness of weak solutions to these problems in various function spaces are found. The proofs of the corresponding theorems imply that solutions of the inverse problems under study can be obtained by applying the method of successive approximations.     

Solvability of some boundary-value problems for polyharmonic equation with Hadamard-Marchaud boundary operator

We investigate solvability conditions for certain non-classical boundary-value problems for the polyharmonic equation. As the boundary operators we consider fractional differential operators in the sense of Hadamard-Marchaud. The considered problems generalize well-known Dirichlet and von Neumann boundary-value problems for boundary operators of fractional type.     

Q-conditional symmetry of a nonlinear two-dimensional heat-conduction equation

We investigate theQ-conditional symmetry of a nonlinear two-dimensional heat-conduction equation. By using ansatzes, we obtain reduced equations. Poltava University, Poltava. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp.846–849, June, 2000.     

Stable second-order finite-difference methods for linear initial-boundary-value problems

Finite-difference methods of second order at the boundary points are presented for problems with one-dimensional second-order hyperbolic and parabolic equations with mixed boundary conditions. These methods do not require information at points outside the region of consideration. The linear stability of the algorithms developed is investigated. Numerical experiments are given for illustrating the accuracy and stability of the methods. Though the focus is on homogeneous boundary conditions, finite-difference methods with non-homogeneous mixed boundary conditions are also developed. To show the potential of the methods developed, in terms of CPU time, a comparison is made with the Crank–Nicolson method.     

On an approach to the approximation of a solution of problems for the heat-conduction equation without boundary-value or initial conditions

We propose an approach to the approximation of solutions of problems for the heat-conduction equation without initial or boundary-value conditions. The solution is given as a sum of odd and even functions, and this allows one to reconstruct conditions missed in the initial setting of the problem. The method is illustrated with test examples. Bibliography: 1 title. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 49–53.