On the Landweber iteration for the solution of a parameter identification problem in a hyperbolic partial differential equation 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".

Parallel Sturmian comparison theorems are proved for hyperbolic and elliptic linear differential equations of second order, with special emphasis on cylindrical domains with the time axis parallel to the axis of the cylinder. These theorems have the following form: If a boundary problem I for a partial differential equation in a bounded domain has a nontrivial solution, then every solution of a second boundary problem II of the same type has a zero in the domain if the coefficient functions and boundary functions of II majorize those of I. The results extend and unify earlier results given for either hyperbolic equations or elliptic equations. In particular, the majorization hypotheses in Theorem 3, in the form of inequalities between the data functions in the differential equations and in the boundary conditions, indicate parallel sets of sufficient conditions for the Sturmian conclusion in the elliptic and hyperbolic cases. Counterexamples are given to show that two known theorems in the elliptic case do not extend to the hyperbolic case. Extensions to singular boundaries and mixed problems are included. The dichotomy referred to in the title pertains to the different but parallel Sturmian theories for elliptic and hyperbolic boundary problems: Results of similar type are true for both types of problems and yet other results true in the elliptic case are false in the hyperbolic case.

This paper deals with an initial value problem that can be set for any analytic hyperbolic equation (rank 2 or 3) linear or non-linear in three independent variables with any inclination of the initial surface. This initial value problem is a Cauchy problem in which we require that the initial data be analytic in one of their arguments but not in the other. Special instances of this problem have been considered by Volterra2 and Hadamard on the one hand and Hamel3 on the other. The (n 1) dimensional characteristic strips employed in this paper are an extension on the one hand of a concept well known in the theory of the second order equation in two independent variables4 and on the other hand of a concept which occurs in the theory of the first order equation in more than two independent variables and which has not been clearly recognized.5 The method used in handling our initial value problem in a great many places bears a resemblance to the method used by Lewy6 in showing that Cauchy's problem for a non-linear hyperbolic equation in two independent variables has a unique solution. In the proof of our existence theorems we

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

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.

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.

The oscillations of a viscous compressible fluid in an arbitrarily-shaped pore

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.

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.