Convergence with orderh 2p-1 of a 2p+1-point scheme of the method of lines for one boundary-value problem

A multipoint scheme of the method of lines is applied to the boundary-value problem The second derivative replaced by a-point ( is any positive integer) central-difference approximation with an error of order where is the step of the net of lines. An approximating system of ordinary differential equations, associated with problem (1), (2), is transformed into a reducing one. The uniform convergence of the approximate solution of the method of lines to the solution of the original boundary-value problem with order is established. For this the solution of the reducing system with zero boundary conditions is examined for the difference between the exact solution of problem (1), (2) and the approximate solution obtained by the method of lines. The behavior of this solution as is studied at point and, next, at any point by transforming the independent variable that transfers point z to the origin.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 39–45, 1979.     

Rate of convergence bound of difference schemes for the first boundary-value problem of elasticity theory in arbitrary domains

The fictitious domain method and exact difference scheme operators are used to construct a difference scheme for solving the first boundary-value problem of elasticity theory in arbitrary domains. The rate of convergence bound is established for this theorem, where (x) is the polylinear completion of the grid function is the solution of the original problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 49–58, 1988.     

Vibrations of Elastic Strings: Mathematical Aspects, Part Two

Dedicated to Professor Jacque-Louis Lions on the occasion of his 70th birthday We consider a mixed problem for the operator

ℓ1-Minimization with Magnitude Constraints in the Frequency Domain

In this paper, we study the -optimal control problem with additional constraints on the magnitude of the closed-loop frequency response. In particular, we study the case of magnitude constraints at fixed frequency points (a finite number of such constraints can be used to approximate an -norm constraint). In previous work, we have shown that the primal-dual formulation for this problem has no duality gap and both primal and dual problems are equivalent to convex, possibly infinite-dimensional, optimization problems with LMI constraints. Here, we study the effect of approximating the convex magnitude constraints with a finite number of linear constraints and provide a bound on the accuracy of the approximation. The resulting problems are linear programs. In the one-block case, both primal and dual programs are semi-infinite dimensional. The optimal cost can be approximated, arbitrarily well from above and within any predefined accuracy from below, by the solutions of finite-dimensional linear programs. In the multiblock case, the approximate LP problem (as well as the exact LMI problem) is infinite-dimensional in both the variables and the constraints. We show that the standard finite-dimensional approximation method, based on approximating the dual linear programming problem by sequences of finite-support problems, may fail to converge to the optimal cost of the infinite-dimensional problem.     

Initial value problems for a system of nonlinear equations on a half-line

The paper deals with initial value problems for the system of nonlinear equations on a half-line where ₁, ₁ are real constant connections.The first problem for the nonlinear Schrödinger equation (NLS) is obtained from the system (A) when . In this case, the boundary problem associated with this NLS does not have a discrete spectrum, and the solution of the NLS is uniquely found from the given initial condition. Further, we show two remarkable classes of matrix potentials, in which the inverse scattering problem associated with system (A) can be solved exactly. In the case where the given scattering data for (A) consist of only two simple poles, the exact soliton solution of system (A) is presented. This happens if and only if the time evolution of the scattering data for (A) obeys some time evolution equations. In this case, the initial value problem for the NLS which is obtained from (A) when is solved exactly.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

Quasilinearization and Rate of Convergence for Higher-Order Nonlinear Periodic Boundary-Value Problems

We study the convergence of a sequence of approximate solutions for thefollowing higher-order nonlinear periodic boundary–value problem:

Critical eigenvalues for a nonlinear problem

We study the application, , where is the supremum of positive s such that the problem admits a solution. Where B ₁ is the unit ball in We show that is a decreasing function, with where is the unique solution of the problem . We also give the explicit solutions of the problem , when and show that . We show that the problem doesnt admit a solution. In the end, we give a numerical approximation of , when .     

Quintic Spline Approach to the Solution of a Singularly-Perturbed Boundary-Value Problem

A fourth-order uniform mesh difference scheme using quintic splines for solving a singularly-perturbed boundary-value problem of the form

An estimate of the maximum modulus of a solution of the linear,time-dependent system of Navier-Stokes equations

It is proved that the maximum modulus of a solution of the initial boundary-value problem for the time-dependent Stokes system with zero boundary data is bounded in terms of the maximum modulus of the initial data with a constant which depends only on the domain.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 34–44, 1977.     

Convergence bounds of difference schemes for secondorder elliptical equations in domains of arbitrary shape

The fictitious domain method is applied to construct a difference scheme for the first boundary-value problem for elliptical equations of second order in domains of arbitrary shape. The rate of convergence bound

Behavior of the Effective QCD Coupling in the Infrared Region

The summary of nonperturbative results for the QCD invariant coupling obtained by numerical lattice simulations for the path integral and by solution of the approximate Dyson–Schwinger equations reveals a remarkable variety of IR behaviors of even at the qualitative level. In turn, this raises the question of the correspondence between the results obtained by different groups. We analyze this issue in terms of mass-dependent coupling-constant transformations and conclude that the problem of the IR behavior of the effective QCD coupling and of propagators is not well defined and requires a further clarification.     

Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations

Under a nondegeneracy condition on the boundary, we prove a comparison principle for discontinuous viscosity sub- and supersolutions of the generalized Dirichlet boundary-value problem for a first-order Hamilton-Jacobi equation

Functional a posteriori error estimates for incremental models in elasto-plasticity

We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants and are valid for any conforming approximations regardless of the method used for their derivation. It is shown that the structure of error majorant reflects properties of the exact solution so that the majorant vanishes only if an approximate solution coincides with the exact one. Moreover, it possesses necessary continuity properties, so that any sequence of approximations converging to the exact solution in the energy space generates a sequence of positive numbers (explicitly computable by the majorant functional) that tends to zero.      

Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Variable Coefficients

We establish conditions for the unique solvability of a boundary-value problem for a weakly nonlinear hyperbolic equation of order 2n, n > (3p + 1)/2, with coefficients dependent on the space coordinates and data given on the entire boundary of a cylindric domain . The investigation of this problem is connected with the problem of small denominators.     

A Posteriori Estimates in Local Norms

We derive a posteriori estimates for the difference between exact solutions and approximate solutions to boundary-value problems in terms of local norms. The diffusion problem, linear elasticity and generalizations to other boundary-value elliptic problems are considered. Computable estimates for the deviation from the exact solution are also obtained in terms of linear functionals. Unlike published works of other authors, the construction of such estimates is not connected with any analysis of the adjoint boundary-value problem. On the basis of multiplicative inequalities, local estimates in certain norms subject to the energy norm are derived. Bibliography: 10 titles.     

Axially symmetric flow with free boundary

We prove the solvability of a boundary-value problem with the Bernoulli condition in the form of an inequality on a free boundary. By using the Rietz method, we construct an approximate solution that converges to an exact solution in the integral metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 477–487, April, 1995.