Functional Differential Equations of Second Order

The idea of fuzzy systems as models for the complex or incomplete processes is not a new idea. The determination of a satisfactory model for an imprecise, complex or incomplete process is basically a matter of approximation. Namely, determining an admissible model best deserving the particular physical phenomena. We suggest that one of the ways to represent imprecise or complex dynamical processes in a fuzzy environment is via fuzzy differential equations and some analysis of their typical solution. In particular, total differential equations of first and second order and partial differential equations of second order are considered. They are unusual in that the coefficients appearing in the equation are "fuzzy".

<p><em> Розглянуто методику застосування тригонометричних сплайнів для задач побудови наближених розв’язків першої крайової задачі для звичайних диференціальних рівнянь другого порядку зі змінними коефіцієнтами. Невизначені параметри зна</em><em>йдено</em><em> методом колокацій</em>.</p>

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.

In Chap. 16, boundary value problems for system of linear functional differential equations are studied, and results on positivity of Green’s matrices are obtained. A necessary and sufficient condition for nonnegativity of all the entries of the Cauchy matrix for linear systems of ordinary differential equations of the first order (obtained by Wazewski) is nonpositivity of off-diagonal coefficients. In the first part of this chapter, extension of this result to various boundary value problems for functional differential equations is obtained, and all the results include an analogue of Wazewski’s sign condition.

Stochastic differential delay equations and their asymptotic behavior have received much attention in recent years (see [1, 2, 5, 7–10, 12, 13] and the references therein), since these retarded problems often appear in physics, biology, engineering, etc. The delays can enter into the formulations in very different ways, e.g., as a constant or variable delay or as a distributed one; some of them can even appear in the model at the same time. However, it is possible to consider all of them under a unified formulation by using appropriate differential functional equations. In this sense, we will carry out our analysis in a functional framework that will cover a wide variety of situations containing finite delays. There exists a vast literature concerning pathwise exponential stability of parabolic stochastic evolution equations (with and without delays). We mention here, among many others, [4, 11–13] and the references therein. In the case without hereditary characteristics, the problem of the asymptotic stability of delay stochastic partial differential equations of second order in time has been considered by Curtain in [6], where one can find sufficient conditions for the exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths, when the main operator generates a strongly continuous contraction semigroup. In this paper, we shall develop the theory in a variational framework for nonlinear operators in general. We are going to consider the following model:

We show that under suitable conditions the covariance of a Gaussian reciprocal process satisfies a self-adjoint linear differential equation of second order. We also give a revised definition of a linear stochastic differential equation of second order and necessary and sufficient conditions for the existence of solutions of such equations with Dirichlet boundary conditions. We close with a series of examples of the theory applied to the scalar stationary Gaussian Reciprocal processes which have been completely classified.

59 - Exact Second Order Equations

Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces

On the interval [0,ω], consider the periodic boundary value problem where 𝑛 ≥ 2, 𝑙𝑖 : 𝐶([0,ω];𝑅) → 𝐿([0,ω];𝑅) (𝑖 = 0,…,𝑛 – 1) are linear bounded operators, 𝑞 ∈ 𝐿([0,ω];𝑅), 𝑐𝑗 ∈ 𝑅 (𝑗 = 0,…,𝑛 – 1). The effective sufficient conditions guaranteeing the unique solvability of the considered problem are established.

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

Article On the Landweber iteration for the solution of a parameter identification problem in a hyperbolic partial differential equation of second order was published on January 1, 1998 in the journal Journal of Inverse and Ill-posed Problems (volume 6, issue 5).

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

This paper is devoted to the mathematical analysis of the stability, boundedness and uniform boundedness of solutions of a class of nonlinear differential equations of second order with multiple constant deviating arguments. We use Lyapunov functionals to verify the stability and boundedness of the solutions and some examples are given to illustrate the theoretical analysis in this work.

In this paper, we study the existence of quasi-periodic solutions and the boundedness of solutions for a wide class nonlinear differential equations of second order. Using the KAM theorem of reversible systems and the theory of transformations we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.