An application of a multipoint differential-difference scheme to a boundary-value problem

For the boundary-value problem (2) we construct a scheme of the method of lines with a central-difference approximation of the derivative for any odd pattern. In particular cases we investigate the behavior at the net refinement of the direct solution of the boundary-value problem for the determination of the difference between the approximate solution obtained by the method of lines and the exact solution of the problem (1), (2). We also consider some modifications of the method of lines: the number of the lines of the net is taken to be equal to that of the pattern. We give an estimate for the norm of the difference between the approximate solution obtained by this method and the exact solution of the problem (1), (2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 76–88, 1977.     

A biplicit spectral-collocation-type ansatz for the numerical integration of partial differential equations with the transversal method of lines

In the implicit formulation of the transversal method of lines, numerical instabilities (singular perturbation) do occur whenever “small” step sizes of the discretized variable have to be used for some reason. This problem can effectively be avoided if the derivatives with respect to the discretized variable are chosen using a combination of implicit and explicit methods (the biplicit method). This combination method uses piecewise defined trial functions involving a certain number of free parameters. The values of these parameters are found by the requirement that the trial functions approximate the solution of the implicit formulation of the method of lines. From a mathematical point of view, this spectral-collocation-type ansatz results in a multipoint boundary-value problem with added parameters to be determined. Two numerical examples are presented in order to illustrate the performance of the method. © 1995 John Wiley & Sons, Inc.     

关于耦合型对流-扩散方程组的奇异摄动边值问题

本文考虑下述耦合型对流-扩散方程组的奇异摄动边值问题:本文提出两种方法:一种是初值化解法,用这种方法,原始问题转化成一系列没有扰动的一阶微分方程或方程组的初值问题,从而得到一个渐近展开式;第二种是边值化解法,用这种方法,原始问题转化成一组没有边界层现象的边值问题,从而可以得到精确解和使用经典的数值方法去得到具有关于摄动参数ε一致的高精度数值解.     

Monte Carlo path-integral calculations for two-point boundary-value problems

A computational technique based on the method of path integral is studied with a view to finding approximate solutions of a class of two-point boundary-value problems. These solutions are rough solutions by Monte Carlo sampling. From the computational point of view, however, once these rough solutions are obtained for any nonlinear cases, they serve as good starting approximations for improving the solutions to higher accuracy. Numerical results of a few examples are also shown.     

Estimates of the solutions of two-point boundary-value problems for systems of first-order integrodifferential equations and the method of lines

One proves theorems on the estimates of the solutions of the systems of first-order integrodifferential equations with the boundary conditions On the basis of these theorems, one suggests a method for estimating the norms of integrodifferential equations by the method of the lines for the solutions of the periodic boundary-value problems for second-order integrodifferential equations of parabolic type. On the basis of the established theorem, on the solvability and on the estimate of the solution of the nonlinear equation $$Tx + F\left( x \right) = 0$$ in a Banach space X, where T is a linear unbounded operator, one investigates the convergence of the method of lines for solving the periodic boundary-value problem for a second-order nonlinear integrodifferential equation of parabolic type.     

Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variations

In the parameter variation method, a scalar parameterk, k[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method.The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables).     

Sinc-Collection Methods for Two-Point Boundary Value Problems

A collocation method is developed for the approximate solutionof two-point boundary-value problems with mixed boundary conditions.The method is based on replacing the exact solution by a linearcombination of Sinc functions. No integrals need to be evaluatedapproximately when setting up the resulting system of linearequations. The error of the method converges to zero like O(exp(-cN²)),as N, where N is the number of collocation points used, andwhere c is a positive constant independent of N. It is claimedthat the method is superior to the Sinc-Galerkin method dueto its simple implementation and possible extensions to moregeneral boundary-value problems.     

On one direct method for the approximate solution of a periodic boundary-value problem

We propose a direct method for the approximate solution of integral equations that arise in the course of approximate solution of a periodic boundary-value problem for linear differential equations by the method of boundary conditions. We show that the proposed direct method is optimal in order.     

$$S_{1/2}$$ regularization methods and fixed point algorithms for affine rank minimization problems

The affine rank minimization problem is to minimize the rank of a matrix under linear constraints. It has many applications in various areas such as statistics, control, system identification and machine learning. Unlike the literatures which use the nuclear norm or the general Schatten \(q~ (0<q<1)\) quasi-norm to approximate the rank of a matrix, in this paper we use the Schatten 1 / 2 quasi-norm approximation which is a better approximation than the nuclear norm but leads to a nonconvex, nonsmooth and non-Lipschitz optimization problem. It is important that we give a global necessary optimality condition for the \(S_{1/2}\) regularization problem by virtue of the special objective function. This is very different from the local optimality conditions usually used for the general \(S_q\) regularization problems. Explicitly, the global necessary optimality condition for the \(S_{1/2}\) regularization problem is a fixed point inclusion associated with the singular value half thresholding operator. Naturally, we propose a fixed point iterative scheme for the problem. We also provide the convergence analysis of this iteration. By discussing the location and setting of the optimal regularization parameter as well as using an approximate singular value decomposition procedure, we get a very efficient algorithm, half norm fixed point algorithm with an approximate SVD (HFPA algorithm), for the \(S_{1/2}\) regularization problem. Numerical experiments on randomly generated and real matrix completion problems are presented to demonstrate the effectiveness of the proposed algorithm.     

Rate of convergence bound of difference schemes for quasilinear elliptical equations with solutions inW 2 1

We consider a mixed boundary-value problem for a quasilinear elliptical equation of second order in the rectangle with a generalized solution in W ₁ ² (). Exact difference scheme operators are applied to construct a first-order accurate difference scheme in the L₂ grid norm.Kiev Technological Institute of Light Industry. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 50–55, 1991.     

Inverse problems of reconstructing parameters of the Navier-Stokes system

An inverse problem of reconstructing parameters not known a priori of the dynamical system described by the boundary-value problem for the Navier-Stokes system is considered. The reconstruction is based on one piece of admissible information or another about the motion of the dynamical system (solution of the corresponding boundary-value problem). In particular, one of the problems considered is the inverse problem consisting of reconstruction of the a priori unknown right-hand side of the Navier-Stokes system. The right-hand side characterizes the density of exterior mass forces acting on the system. This problem, as well as many other similar problems, is ill-posed. Two methods are proposed for its solution: the statistical method and the dynamical method. These methods use different initial information. In solving the problem by using the statistical method, initial information for the solution is the results of approximate measurement (in one sense or another) of the motion of the dynamical system on a given interval of time. Here, the reconstruction is performed after the corresponding interval of time. For solution of the problem by this method, the concepts and constructions of open-loop control theory are used. In solving the problem by using the dynamical method, initial information for its solution is the results of approximate (in one sense or another) measurements of the current states of the system, which are dynamically obtained by the observer. Here, the reconstruction is dynamically performed during the process. For solution of the problem by the dynamical method, the concepts and constructions of the dynamical regularization method based on positional control theory are used. Also, the author considers various modifications and regularizations of the methods for solution of problems proposed that are based on one piece of a priori information or another about the desired solution and solvability conditions of the problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

An <Emphasis Type="BoldItalic">O</Emphasis>(<Emphasis Type="BoldItalic">h</Emphasis><Superscript><Emphasis Type="Bold">6</Emphasis></Superscript>) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h ²) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h ⁶) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.     

Numerical solution of eighth order boundary value problems in reproducing Kernel space

In this paper, the approximate solutions to the eighth-order boundary-value problems are presented using the reproducing kernel space method. The procedure is applied on both linear and nonlinear problems. Searching least value (SLV) method is investigated for nonlinear boundary value problems. The argument is based on the reproducing kernel space $W_{2}^{9}[a,b]$ . The approach provides the solution in the form of a convergent series with easily computable components. Analytical results are given for several examples to illustrate the implementation and efficiency of the method. A comparison of the results obtained by the present method with results obtained by other methods reveals that the present method is more effective and convenient.     

Uniform convergence with orderh 4 of a scheme of the method of lines for quasilinear parabolic and hyperbolic equations

The order of uniform convergence of the method of lines is considered in connection with the solution of the first boundary-value problem for quasilinear second-order parabolic and hyperbolic equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 48, pp. 205–211, 1974.     

Inverse Problem for an Integro-Differential Equation of Acoustics

We consider the hyperbolic integro-differential equation of acoustics. The direct problem is to determine the acoustic pressure created by a concentrated excitation source located at the boundary of a spatial domain from the initial boundary-value problem for this equation. For this direct problem, we study the inverse problem, which consists in determining the onedimensional kernel of the integral term from the known solution of the direct problem at the point x = 0 for t &gt; 0. This problem reduces to solving a system of integral equations in unknown functions. The latter is solved by using the principle of contraction mapping in the space of continuous functions. The local unique solvability of the posed problem is proved.     

Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition

An initial boundary-value problem for a quasilinear system of partial differential equations with a nonlocal boundary condition involving a delayed argument is considered. The existence of a unique solution to this problem is proved by reducing it to a system of nonlinear integral-functional equations. The inverse problem of finding a solution-dependent coefficient of the system from additional information on a solution component specified at a fixed point of space as a function of time is formulated. The uniqueness of the solution of the inverse problem is proved. The proof is based on the derivation and analysis of an integral-functional equation for the difference between two solutions of the inverse problem.     

Coefficients in the asymptotics of the solutions of elliptic boundary-value problems in a cone

The asymptotics near a conical point of the solution of an elliptic boundary-value problem contains linear combinations of the special solutions of the model homogeneous problem in the cone. One gives formulas for the coefficients of these linear combinations under the assumption that the domain is a cone.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 110–127, 1975.     

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C ^∞ (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.     

An efficient numerical method for singular perturbation problems

In this paper, we propose a method for the numerical solution of singularly perturbed two-point boundary-value problems (BVPs). First, we develop two schemes to integrate initial–value problem (IVP) for system of two first-order differential equations, and then by using these schemes we solve the BVP. Precisely, we convert the second-order BVP into a system of first-order differential equations, and then apply the numerical schemes to obtain the solution. In order to get an initial condition for the system, we use the asymptotic approximate solution. Error estimates are derived and numerical examples are provided to illustrate the present method.