Solution Of Mixed Problem Research Articles

Computing the variable coefficient telegraph equation using a discrete eigenfunctions method

This paper deals with the construction of discrete numerical solutions of mixed problems for the telegraph equation. After discretization, the two-variables partial difference mixed problem is solved by means of a discrete eigenfunctions method that mimics the advantages of the continuous eigenfunction method while eliminating its computational disadvantages. The solution is based on a closed form solution of the inhomogeneous second order difference equation without increasing the problem’s dimension.

Approximate solution of a mixed problem for a parabolic equation by means of a special basis of functions

Using the eigenfunctions of two Sturm-Liouville problems (with the same operator of the most general form but two different sets of boundary conditions), we propose a method for construction of specific basis functions such that the corresponding expansions of smooth and piecewise smooth functions lead to fast converging series. The last circumstance can be successfully employed for approximate solution of mixed problems for a parabolic equation when the sought function is approximated in the spatial variables by few basis functions. First, the case of one spatial coordinate is elaborated and the two-dimensional case is briefly discussed in Section 5. The method aims primarily at the case when the sought function is piecewise smooth in the spatial variable and the implementation of the method bases on the concept of generalized solution.

Asymptotic behaviour of supports of solutions of quasilinear many-dimensional parabolic equations of non-stationary diffusion-convection type

We study the phenomenon of the finiteness of the rate of propagation of the supports of generalized energy solutions of mixed problems for a broad class of doubly degenerate parabolic equations of high order; a model example here is the equation , , , . Bounds (that are sharp in a certain sense) for the early evolution of the supports of solutions (in particular, of the `right' and the `left' fronts of the solutions), which depend on local properties of the initial function and the parameters of the equation, are established. The behaviour of the supports for large times is also studied.

Generalized solutions of mixed problems for first-order partial functional differential equations

A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators.

A selfsimilar behavior of the urban structure in the spatially inhomogeneous model

At present there is a strong tendency to use new methods for the description of the regional and spatial economy. In increasing frequency we consider that any economic activity is spatially dependent. The problem of the evolution of internal urban formation can be described with the exact supposition. So that is why we use partial derivative equations set with the appropriate boundary and initial conditions for the solving the problem of the urban evolution. Here we describe the model of urban population's density modification taking into account a modification of the housing quality. A program has been created which realizes difference method of mixed problem solution for population's density. For the wide class of coefficients it has been shown that the problem's solution “quickly forgets” the parts of the initial conditions and comes out to the intermediate asymptotic form, which nature depends only on the problem's operator. Actually it means that the urban structure does not depend on external circumstances and is formed by the internal structure of the model.

Analytic solution of mixed problems for thegeneralized diffusion equation with delay

This paper deals with the construction of the exact series solution of mixed problems related to the generalized diffusion equation u t(t, ξ) = a 2u ξξ(t, ξ) + b 2u ξξ(t − τ, ξ), t > τ, 0 < ξ ≤ l . A separation of variables method is used to develop an exact theoretical series solution, which can be truncated to obtain a continuous numerical solution with prescribed accuracy in bounded domains.

The solution of mixed problems of the theory of elasticity for an anisotropic half-plane and coupled half-planes

The problem of the stretching of an elastic anisotropic plane with a bounded linear inclusion is solved. The problem is reduced to a Riemann boundary-value problem on two sheets of the complex plane glued together along the real axis. Computations are carried out for an inclusion situated at an angle to the anisotropy axis.

Constructive solution of strongly coupled continuous hyperbolic mixed problems

This paper deals with the construction of exact series solutions of mixed problems for strongly coupled hyperbolic partial differential systems using a matrix separation of variables method. Algebraic methods are used to study the underlying Sturm–Liouville type vector problem with matrix coefficients appearing in the boundary value conditions.

Monotone Difference Schemes for Nonlinear Parabolic Equations

The maximum principle [1] is used for the study of the uniqueness and continuous dependence of a classical solution of mixed problems for parabolic and elliptic equations on the input data. For example, by using this principle, one can show that the corresponding solution of the homogeneous equation attains its maximum or minimum on the boundary of a domain. In a more general form, it permits one to derive a priori estimates for the maximum absolute value of the solution. Great attention is paid to the maximum principle in the theory of finite-difference approximations [2–7]. In particular, it is used in convergence analysis of difference schemes in the uniform norm. Difference schemes satisfying the maximum principle are said to be monotone. The monotonicity condition plays an important role in computational practice, since systems of algebraic equations obtained on the basis of such methods are well-conditioned. From the viewpoint of stability, the most rigorous results were obtained for linear problems in the investigation of difference schemes in grid Hilbert spaces as well as in the uniform norm. The maximum principle differs from the method of energy inequalities by the fact that it permits one to derive a priori estimates in the norm of C for problems of arbitrary dimension. The investigation of the well-posedness of a scheme in the uniform norm (in the grid Banach space L∞) is a key point in many problems. As an example, we note problems with unbounded nonlinearity. These problems are characterized by the fact that specific conditions imposed on the coefficients depending on the solution are valid either in the range of the exact solution or in a small neighborhood of it. To provide the validity of these properties for the coefficients of a difference scheme, one has to show that the approximate solution belongs to the range of the exact solution. This necessitates studying the properties of the solution of the difference scheme in the norm of C [8]. In the present paper, we state the maximum principle for general grid equations written out in a common canonical form for both interior and boundary nodes of the grid. On the basis of its corollaries, we prove the monotonicity of difference schemes (including vector-additive schemes [9, 10]) approximating a multidimensional quasilinear parabolic equation. We obtain the corresponding a priori estimates. In monotone schemes, it is very important to preserve the second-order accuracy of the approximation to a quasilinear parabolic equation with boundary conditions of the third kind. One usually increases the order of approximation of the boundary conditions by using the main differential equation on the boundary of the domain. However, it is difficult to prove the second-order convergence in the norm of C in the framework of this classical approach. Matus [11] suggested an approach to the construction of monotone difference schemes preserving the second-order approximation and accuracy for linear differential problems with boundary conditions of the second and third kind without using the main differential equation on the boundary of the domain. The main idea is based on the assumption of the existence and uniqueness of a smooth solution in some sufficiently small neighborhood of the domain of the problem and the use of only half-integer grid points. In this case, the boundary conditions are approximated with second order on the two-point stencil. If the equation is assumed to be well-posed at the boundary points as well, then one can also construct fourth-order schemes in this case [11]. In the present paper, we apply this approach to nonlinear equations. In particular, we prove the monotonicity and obtain a priori estimates for a difference scheme approximating the third boundary value problem for a nonlinear parabolic equation with nonlinearities of unbounded growth. Some results in this direction were announced in [12, 13].

Stabilization of the solution of a two-dimensional system of Navier-Stokes equations in an unbounded domain with several exits to infinity

The behaviour as of the solution of the mixed problem for the system of Navier-Stokes equations with a Dirichlet condition at the boundary is studied in an unbounded two-dimensional domain with several exits to infinity. A class of domains is distinguished in which an estimate characterizing the decay of solutions in terms of the geometry of the domain is proved for exponentially decreasing initial velocities. A similar estimate of the solution of the first mixed problem for the heat equation is sharp in a broad class of domains with several exits to infinity.

Mixed problems for separated variable coefficient wave equations: analytic–numerical solutions with a priori error bounds

This paper deals with the construction of accurate analytic-numerical solutions of mixed problems related to the separated variable dependent wave equation u tt=(b(t)/a(x))u xx, 0<x<L, t>0 . Based on the study of the growth of eigenfunctions of the underlying Sturm–Liouville problems, an exact theoretical series solution is firstly obtained. Explicit bounds allow truncation of the series solution so that the error of the truncated approximation is less than ε 1 in a bounded domain Ω(d)={(x,t); 0⩽x⩽L, 0⩽t⩽d } . Since the approximation involves only a finite number of exact eigenvalues λ i 2, 1⩽i⩽n 0 , the admissible error for the approximated eigenvalues λ ̃ i 2, 1⩽i⩽n 0 , is determined in order to construct an analytical numerical solution of the mixed problem, involving only approximated eigenvalues λ ̃ i 2 , so that the total error is less than ε uniformly in Ω(d). Uniqueness of solutions is also treated.

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, I

In this paper, we consider mixed problems with a spacelike boundary derivative condition for semilinear wave equations with exponential nonlinearities in a quarter plane. Results similar to those obtained earlier by Caffarelli-Friedman for Cauchy problems and power nonlinearities are proved in the present situation, namely we show that solutions either are global or blow up on a $C^1$ spacelike curve. Weaker results are also obtained if the boundary vector field is tangent to the characteristic which leaves the domain in the future.

Integral solutions of mixed problems in a theory of plane strain elasticity with microstructure

The real boundary integral equation method is used to solve mixed problems in a theory of plane elasticity which includes the effects of material microstructure. In the exterior case, a specific far-field pattern for the displacements and microrotation is introduced without which the classical scheme fails to work.

Exact solution of mixed problems for variable coefficient one-dimensional diffusion equation

This paper deals with diffusion problems modeled by the equation a( t) u xx = u t , x > 0, t > 0, u( x, 0) = c( x) together with the boundary condition u(0, t) = b( t) or u x (0, t) = b( t). By using Fourier transforms, existence conditions and exact solutions of the above mixed problems are given.

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, II

In this paper, we consider mixed problems with a timelike boundary derivative (or a Dirichlet) condition for semilinear wave equations with exponential nonlinearities in a quarter plane. The case when the boundary vector field is tangent to the characteristic which leaves the domain in the future is also considered. We show that solutions either are global or blow up on a C1 curve which is spacelike except at the point where it meets the boundary; at that point, it is tangent to the characteristic which leaves the domain in the future.

An analytical method of solution of mixed problems of the theory of elasticity

An anllytical method of solution of integral, integrodifferential equations and their systems of special forms, is described here. The method consists of reducing the mentioned equations to Riemann's problem of the theory of analytic functions for a vector of the second or more order and its effective solution by reducing them to infinite algebraic systems with exponential rate of convergence of the approximate solution to the coexact one. Solutions of some problems of the theory of cracks in composite bodies carried out with the help of this method are also given.

A Fast Spectral Solver for a 3D Helmholtz Equation

We present a fast solver for the Helmholtz equation $$\Delta u \pm \lambda^2 u = f,$$ in a 3D rectangular box. The method is based on the application of the discrete Fourier transform accompanied by a subtraction technique which allows us to reduce the errors associated with the Gibbs phenomenon and achieve any prescribed rate of convergence. The algorithm requires O(N3 log N) operations, where N is the number of grid points in each direction. We solve a Dirichlet boundary problem for the Helmholtz equation. We also extend the method to the solution of mixed problems, where Dirichlet boundary conditions are specified on some faces and Neumann boundary conditions are specified on other faces. High-order accuracy is achieved by a comparatively small number of points. For example, for the accuracy of 10-8 the resolution of only 16--32 points in each direction is necessary.

Dynamics of the supports of energy solutions of mixed problems for quasi-linear parabolic equations of arbitrary order

We study the geometry of the supports of solutions of the Cauchy-Dirichlet problem for a wide class of quasi-linear degenerate parabolic equations of any order, whose model representative is the equation of non-stationary filtration with non-linear absorption: In the cases when and , which correspond to “fast” and “slow” diffusion, we find conditions on the behaviour of the initial function in a neighbourhood of the boundary of its support that ensure the effect of finite and infinite inertia of the support of an arbitrary energy solution; these conditions are, in a certain sense, exact. We establish a condition for the reverse motion of the front of the support boundary.

Leningradskogo universiteta

The present paper deals with qualitative properties of the solutions of the mixed Zaremba's problem. In part I two aspect of the Zaremba's problem are studied. First the existence of Wiener's generalized solutions in limited domains is proven. The part of the boundary supporting Neumann's conditions is assumed to be locally smooth, whereas the other part of the alternating Schwarz's method, reducing the mixed problem in an arbitrary domain to a sequence of mixed problems in standard domains and corresponding Dirichlet's problem in domains adjoining adjoining the supports of Dirichlet's boundary conditions. Second, in the domains, satifying an isoperimertic inequality, the so called Growth Lemma, giving a quantitative estimate of the degree of “saddle” of the mixed problem solution, is proven. Being important in itself, this lemma be widely used in part 2 of the present paper.

Mixed problems for the time-dependent telegraph equation: Continuous numerical solutions with a priori error bounds

This paper deals with the construction of continuous numerical solutions of mixed problems described by the time-dependent telegraph equation u tt + c( t) u t + b( t) u = a( t) u xx , 0 < x < d, t > 0. Here a( t), b( t), and c( t) are positive functions with appropiate additional alternative hypotheses. First, using the separation of variables technique a theoretical series solution is obtained. Then, after truncation using one-step matrix methods and interpolating functions, a continuous numerical solution with a prefixed accuracy in a bounded subdomain is constructed.