Power series method and approximate linear differential equations of second order

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.

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".

59 - Exact Second Order Equations

An exact analytical solution of the Reynolds equation for the finite journal bearing lubrication

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.

In this paper we prove the Hyers-Ulam stability of the perfect linear differential equation f ( t ) y ″ ( t ) + f 1 ( t ) y ′ ( t ) + f 2 ( t ) y ( t ) = Q ( t ) , where f , y ∈ C 2 [ a , b ] , Q ∈ C [ a , b ] , f 2 ( t ) = f 1 ′ ( t ) − f ″ ( t ) and − ∞ < a < b < + ∞ . MSC:34K20, 26D10, 39B82, 34K06, 39B72.

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 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.

The thermodynamic theory for a fluid mixture in a gravitational field under a given pressure is developed. The local density and the local molfraction are varied to minimize the Gibbs free energy of the system. With the approximation of local thermodynamics, a set of coupled linear differential equations of first order is obtained. The non-local van der Waals-Landau contributions, leading to a set of linear differential equations of second order, are discussed for the two component lattice gas. Experimental methods allowing to determine concentration and density gradients are discussed also.

On inequalities of Lyapunov type

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.

<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>