Second-Order Differential Equations

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".

Fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second-order differential equations. There are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second-order differential equations. These are (i) fractional differential operators are nonlocal operators, and (ii) the adjoint of a fractional differential operator is not the negative of itself. The wellposedness of a Galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously. Many subsequent works were carried out to extend the analysis to other numerical methods. A constant diffusivity coefficient has been assumed in all these works. In this paper we present a counterexample which shows that the Galerkin weak formulation loses coercivity in the context of variable-coefficient conservative fractional elliptic differential equations. Hence, the previous results cannot be extended to variable-coefficient conservative fractional elliptic differential equations. We adopt an alternative approach to prove the existence and uniqueness of the classical solution to the variable-coefficient conservative fractional elliptic differential equation and characterize the solution in terms of the classical solutions to second-order elliptic differential equations. Furthermore, we derive a Petrov--Galerkin weak formulation to the fractional elliptic differential equation. We prove that the bilinear form of the Petrov--Galerkin weak formulation is weakly coercive and so the weak formulation has a unique weak solution and is well posed. Finally, we outline potential application of these results in the development of numerical methods for variable-coefficient conservative fractional elliptic differential equations.

59 - Exact Second Order Equations

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.

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + p y u c ϑ y = 0 , for y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a = ∞ . Two cases are considered for a < c and a > c , where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.

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:

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.

<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>

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 study a boundary value problem for functional differential equations of second order. Applying a quasilinearization technique we obtain two monotone sequences showing that they converge to the unique solution and this convergence is superlinear.

We study the asymptotic behavior of solutions to a nonlinear differential equation of the second order whose coefficient of nonlinearity is a bounded function for arbitrarily large values of \(x\) in \(R\). We obtain certain sufficient conditions which guarantee boundedness of solutions, their convergence to zero as \(x\rightarrow \infty\) and their unboundedness.

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.

In this paper, we study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the differential equation of second order of the form u”(t) = h(t,u(t)) by using fixed point method.