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.     

Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems

Finite-difference methods of orders six and eight are presented for second-order, non-linear, boundary-value problems. Both methods are economical in the sense that they use few function evaluations at interior grid points. The implementation of the methods is straightforward. The convergence of the methods is discussed. Numerical examples are considered to demonstrate computationally their order of convergence.     

Parallel algorithms for solving nonlinear two-point boundary-value problems which arise in optimal control

This paper describes a collection of parallel optimal control algorithms which are suitable for implementation on an advanced computer with the facility for large-scale parallel processing. Specifically, a parallel nongradient algorithm and a parallel variablemetric algorithm are used to search for the initial costate vector that defines the solution to the optimal control problem. To avoid the computational problems sometimes associated with simultaneous forward integration of both the state and costate equations, a parallel shooting procedure based upon partitioning of the integration interval is considered. To further speed computations, parallel integration methods are proposed. Application of this all-parallel procedure to a forced Van der Pol system indicates that convergence time is significantly less than that required by highly efficient serial procedures.This research was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-77-3418.     

Third-order variable-mesh cubic spline methods for nonlinear two-point singularly perturbed boundary-value problems

A family of third-order variable-mesh methods for singularly perturbed two-point boundary-value problems of the form y=f(x,y,y),y(a)=A, y(b)=B is derived. The convergence analysis is given, and the method is shown to have third-order convergence properties. Several test examples are solved to demonstrate the efficiency of the method.     

Variational iteration technique for solving nonlinear equations

In this paper, we use the variational iteration technique to suggest some new iterative methods for solving nonlinear equations f(x)=0. We also discuss the convergence criteria of these new iterative methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. We also give several examples to illustrate the efficiency of these methods. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations.     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

Cubic spline method for solving two-point boundary-value problems

In this paper, we use uniform cubic spline polynomials to derive some new consistency relations. These relations are then used to develop a numerical method for computing smooth approximations to the solution and its first, second as well as third derivatives for a second order boundary value problem. The present method outperforms other collocations, finite-difference and splines methods of the same order. Numerical illustrations are provided to demonstrate the practical use of our method.     

General technique for solving nonlinear,two-point boundary-value problems via the method of particular solutions

In this paper, a general technique for solving nonlinear, two-point boundary-value problems is presented; it is assumed that the differential system has ordern and is subject top initial conditions andq final conditions, wherep+q=n. First, the differential equations and the boundary conditions are linearized about a nominal functionx(t) satisfying thep initial conditions. Next, the linearized system is imbedded into a more general system by means of a scaling factor , 01, applied to each forcing term. Then, themethod of particular solutions is employed in order to obtain the perturbation x(t)=A(t) leading from the nominal functionx(t) to the varied function (t); this method differs from the adjoint method and the complementary function method in that it employs only one differential system, namely, the nonhomogeneous, linearized system.The scaling factor (or stepsize) is determined by a one-dimensional search starting from =1 so as to ensure the decrease of the performance indexP (the cumulative error in the differential equations and the boundary conditions). It is shown that the performance index has a descent property; therefore, if is sufficiently small, it is guaranteed that <P. Convergence to the desired solution is achieved when the inequalityP is met, where is a small, preselected number.In the present technique, the entire functionx(t) is updated according to (t)=x(t)+A(t). This updating procedure is called Scheme (a). For comparison purposes, an alternate procedure, called Scheme (b), is considered: the initial pointx(0) is updated according to (0)=x(0)+A(0), and the new nominal function (t) is obtained by forward integration of the nonlinear differential system. In this connection, five numerical examples are presented; they illustrate (i) the simplicity as well as the rapidity of convergence of the algorithm, (ii) the importance of stepsize control, and (iii) the desirability of updating the functionx(t) according to Scheme (a) rather than Scheme (b).This research, supported by the National Science Foundation, Grant No. GP-18522, is based on Ref. 1. The authors are indebted to Mr. A. V. Levy for computational assistance.     

Iterative stochastic methods for solving variational problems of mathematical physics and operations research

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 3, Analiz—2, 1993.     

Iterative methods for solving inverse problems for nonlinear partial differential equations

Inverse coefficient problems are considered for the mathematical models of sorption dynamics and heat conduction. Iterative methods proposed for solving these inverse problems transform a supplementary condition into an integral relationship containing the unknown coefficient. Combined with the original boundary-value problem, this integral relationship makes it possible to construct an iterative process. A priori representation of the unknown nonlinear coefficients in parametric form is not required. Results of computational experiments are reported.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 142–149.     

Stationary boundary-value problems for Boltzmann's nonlinear equation

For Boltzmann's nonlinear equation one investigates the solvability of the stationary flow problem. One describes the asymptotic behavior of its solution for ¦x¦.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 100–104, 1981.     

Variational treatment of exterior boundary-value problems for thin elastic plates

Existence, uniqueness and continuous dependence on the dataare investigated for weak solutions of exterior problems inthe theory of bending of plates with transverse shear deformation.     

A finite element method for solving singular boundary-value problems

It is proved that under certain assumptions on the functions q(t) and f(t), there is one and only one function u₀(t) ∈ at which the functional

Interval length continuation method for solving two-point boundary-value problems

We consider the solution of the following two-point boundary-value problem: $$\begin{gathered} \dot x(t) = f(t,x(t),p(t)), \dot p(t) = g(t,x(t),p(t)), t \in [0,T], \hfill \\ h(x(0),p(0)) = 0, p(T) = q. \hfill \\ \end{gathered} $$ We propose a combination technique consisting of the interval length continuation method and the back-and-forth shooting method. Certain alternative ways of employing continuation are discussed, and some of them are well suited for the problem under consideration. As a test for the method, a numerical example of a problem originating in optimal control is given.     

Incremental unknowns for solving nonlinear eigenvalue problems: New multiresolution methods

In this article, we present several numerical multilevel schemes for the solution of nonlinear eigenvalue problems. Using the Incremental Unknowns, we construct some generalization of the Marder and Weitzner method, which is well suited for the calculation of unstable solutions. The new methods that we present are based on a different treatment of the several structures appearing with the utilization of the hierarchical preconditioner. We illustrate the efficiency of the new methods with the calculation of unstable solutions of a reaction diffusion problem. © 1995 John Wiley & Sons, Inc.     

Variational methods for nonlinear Steklov eigenvalue problems with an indefinite weight function

We consider the problem of finding a harmonic function u in a bounded domain ${\Omega\subset\mathbb{R}^N}$ , N ≥ 2, satisfying a nonlinear boundary condition of the form ${\partial_{\nu}u(x)=\lambda\,\mu(x)h(u(x)), x\in \partial\Omega}$ where μ changes sign and h is an increasing function with superlinear, subcritical growth at infinity. We study the solvability of the problem depending on the parameter λ by using min-max methods.