Value Problem For System Research Articles

Boundary value problems for systems of second order differential equations

SynopsisWe consider boundary value problems for second order differential systems of the form (1)x” = A(t)x′ + f(t, x) and (2) x” = A(t)x′ + f(t, x) + q(t, x). By assuming the existence of a solution to (1) with a given region in (t, x) space, we derive conditions under which there exists a solution to (2) which stays in a certain neighbourhood of and satisfies given boundary conditions.

On the Boundary Value Problem for Systems of Ordinary Second Order Differential Equations with a Singularity of the First Kind

Analytical properties like existence, uniqueness and smoothness of continuous solutions of nonlinear boundary value problems are considered. Fredholm theory for linear boundary value problems is established. The results are applied to two practical examples from the theory of spherical shells.

Factored Product Expansions of Solutions of Nonlinear Differential Equations

Lie transformations are used to represent solutions of initial value problems for systems of nonlinear ordinary differential equations as exponentials of first order linear partial differential operators. These exponentials are then expanded using an analog of the usual exponential identities. This expansion is called the factored product expansion. Such expansions have been found useful in the study of magnetic and optical lenses.

Determination of the solutions of the first boundary value problem for a system of K�rm�n equations having an unbounded energy integral

For the nonzero solutions of a homogeneous Karman system, satisfying homogeneous boundary conditions of the first kind at the boundary of an infinite strip, one proves that the energy integral, taken over a piece of the strip of length t, increases not slower than t2 when t→∞. Then, one poses and one solves a boundary value problem for the nonhomogeneous system, where the applied actions need not decrease at infinity.

First boundary value problem for systems of quasilinear elliptic equations

One proves the solvability of the Dirichlet problem for a certain classes of degenerate, quasilinear, second-order elliptic system. This is done with the aid of operator multiplicative functionals of Markov process.

Differential Inequalities of Higher Order and the Asymptotic Solution of Nonlinear Boundary Value Problems

Two differential inequality results of Nagumo on initial and boundary value problems for systems [Proc. Phys. Math. Soc. Japan, 19 (1937), pp. 861–866; 21 (1939), pp. 529–534] are combined to yield existence and comparison results on certain boundary value problems for nth order scalar nonlinear differential equations and their system analogues. This theory is then applied to several classes of singularly perturbed boundary value problems of higher order. Many examples are discussed in order to motivate the theory and indicate avenues of further study.

On the spectrum of a family of non-self-adjoint boundary value problems for systems of helmholtz equations

On the spectrum of a family of non-self-adjoint boundary value problems for systems of helmholtz equations

On the Boundary Value Problem for Systems of Ordinary Differential Equations with a Singularity of the Second Kind

A Fredholm theory for linear boundary value problems $t^\alpha y' = G(t)y + g(t),\,0 < t \leqq 1,\,y \in C[0,1] \cap C^1 (0,1],\,\alpha \geqq 1$; $B_0 y(0) + B_1y(1) = \gamma $ is established, together with existence and regularity results for continuous solutions of nonlinear systems of ordinary differential equations $t^\alpha y' = f(t,y)$. This theory is applied to boundary value problems on infinite intervals and is illustrated by two examples. Finally, some fundamental properties of the generalized linear eigenvalue problem $t^\alpha y' - (G(t) - \lambda H(t))y = 0,\,0 < t \leqq 1,\,y \in C[0,1] \cap C^1 (0,1]$; $B_0 y(0) + B_1 y(1) = 0$, are derived.

Systems of second order equations with nonnegative characteristic form

The object of this paper is to study systems of second order equations with nonnegative characteristic form. The systemsconsidered will have the form and a bounded domain. Such equatiom are said to be of nonnegative characteristic form if for a , and . The primary results are existence and uniqueness theorems for weak solutions to boundary value problems for systems such a s (1.1).

Stability on the imaginary axis andA-stability of linear multistep methods

In order to solve initial value problems of systems of first order differential equations where the Jacobian of the right hand side has purely imaginary eigenvalues, one requests that the method be stable along the imaginary axis. It is then shown that such methods have to beA-stable and hence their error order is at most 2 and the trapezoidal rule has the smallest error constant. This result is then related to a similar result by Dahlquist on second order differential equations.

Stability of boundary value problems for systems of difference equations with various structures

Stability of boundary value problems for systems of difference equations with various structures

Non-linear boundary value problems for systems of differential equations†

SynopsisThe paper deals with boundary value problems for second-order vector differential equations x″ = f (t, x, x′). Given a region Ω in (t, x)-space we ask whether there exists a solution x(t) of the problem satisfying (t, x(t)) ∊Ω. We arrive at a rather general type of conditions which are sufficient in order that Ω has the desired property. One of these conditions is geometric in nature and depends upon the boundary data only. The second condition can be expressed in terms of inequalities and depends upon the values of f on ∂Ω. These inequalities turn out to be the common background of a variety of conditions which can be found in the literature on boundary value problems and which in the case of a scalar equation reduce to the well-known properties of upper and lower solutions.

Convergence and Error Estimates for the Method of Lines for Certain Nonlinear Elliptic and Elliptic-Parabolic Equations

Differential inequality theory for systems of second order ordinary differential equations is applied to obtain convergence and error estimates for the method of lines (MOL). Comparison theorems for boundary value problems for systems of ordinary differential equations are developed which are then used to compare solutions to elliptic and elliptic-parabolic boundary value problems with the solutions to their MOL analogs. Included are some results on equations defined on unbounded domains. These theorems extend some results of W. Walter on parabolic equations to the elliptic and elliptic-parabolic cases.

Upper and lower bounds for solutions of generalized two-point boundary value problems

This paper is concerned with two-point boundary value problems for systems of differential equations and integro-differential equations. If ?, ? and ?, ? are functions which satisfy certain differential (integro-differential) inequalities, then the given problem has a solutionu * such that ??u *?? and ??u *???.

Fehlerabsch�tzung bei Anfangswertaufgaben f�r Systeme von gew�hnlichen Differentialgleichungen mit Anwendung auf das REENTRY-Problem

An inclusion theorem for the solution of initial value problems for system of ordinary differential equations is formulated. This allows an error estimation of an arbitrary approximationu 0. The method is based on the estimation of a solution of a system of nonlinear differential inequalities, which depends essentially on the defect ofu 0. A method of numerical realization also developed herein is applied to a difficult practical problem.

The application of the method of averaging to a two-point boundary value problem for systems of integrodifferential equations of Volterra type

The application of the method of averaging to a two-point boundary value problem for systems of integrodifferential equations of Volterra type

On Two-Point Boundary Value Problems for Nonlinear Second Order Systems

This note deals with a priori bounds for solutions (and their derivatives) and applications to existence theorems for two-point boundary value problems for systems of nonlinear, second order, ordinary differential equations. The methods are based on those of the author and McLeod employed in the particular case of the swirling flow problem in fluid mechanics. It depends on the construction of general “Lyapunov-type” functions and comparison theorems for (scalar) second order equations.

Two-Point Boundary Value Problems for Systems of Second Order Ordinary Differential Equations

Two-Point Boundary Value Problems for Systems of Second Order Ordinary Differential Equations

Extension of the Goodman-Lance method of adjoints

The Goodman-Lance method of adjoints for solving two point boundary value problems of systems of non-linear ordinary differential equations is extended to handle the most general boundary conditions and specialized to several important subcases.

A one sweep method for linear difference equations with two point boundary conditions

A new method is proposed for reducing two point boundary value problems for systems of linear difference equations to initial value problems. The method has the advautage that only one sweep is required. Comparisons against classical reduction methods are made.