Periodic Solutions of Singular Second Order Equations at Resonance

In this paper, we study the existence of positive periodic solutions for singular second order equations x＂ ＋ n2/4＋h（x） = p（t）, where h has a singularity at the origin and n is a positive integer. We give an explicit condition to ensure the existence of positive periodic solutions when h is an unbounded perturbation at infinity by using qualitative analysis and topological dezree theory.

Picard Boundary Value Problems of Second Order p-Laplacian Differential Equations

Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.

An Oscillation Criterion for Second Order Superlinear Differential Equations

A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].

On the Asymptotic Behavior of Certain Second Order Differential Equations

Two criteria for asymptotic behavior of certain second order differential equations are obtained. The results extend and improve Feng's theorem.

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

SOLVING SECOND ORDER DIFFERENTIAL EQUATIONS IN QUANTUM MECHANICS BYORDER REDUCTION

Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.

SOLVABILITY IN HARDY SPACE FOR SECOND ORDER ELLIPTIC EQUATIONS

We obtain a priori estimates and solvability in Hardy type space in a bounded domain of Rn for second order elliptic equations with coefficients of limited smoothness. Such a result can be served as an endpoint case of the classical LP（1 〈 p 〈 ∞） theory for second order elliptic equations. Our approach is based on a standard technique of perturbation rather than that of integral representation formula.

AN ADI GALERKIN METHOD WITH MOVING FINITE ELEMENT SPACES FOR A CLASS OF SECOND-ORDER HYPERBOLIC EQUATIONS

An alternating direction implicit (ADI) Galerkin method with moving finite element spaces is formulated for a class of second order hyperbolic equations in two space variables. A priori H 1 error estimate is derived.

Oscillation Theorems for Second Order Damped Nonlinear Differential Equations

Several oscillation criteria are given for the second order nonlinear differential equation with damped term of the form [α(t)(y'(t))σ]' +p(t)(y'(t))σ+ q(t)f(y(t)) = 0, where α∈C(R, (0,∞)), p(t) and q(t) are allowed to change sign on [t0, ∞), and f∈C1 (R, R) such that xf(x) > 0 for x ≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.

ON LEVIN TYPE COMPARISON THEOREMS FOR CERTAIN SECOND ORDER DIFFERENTIAL EQUATIONS

In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison theorems.

Pseudo Almost Periodic Solutions for Second Order Abstract Differential Equations

On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract differential equation ( u″(t)+A 1u′(t)+A 0u(t)=0, t≥0 ) is strongly well posed, the necessary conditions for their solutions to be pseudo almost periodic are derived.

OSCILLATION THEOREMS FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DAMPING

Some new oscillation theorems are established for the second order nonlinear differential equations with damping of the form where p（t） and q（t） are allowed to change sign on [t0,∞).

ON WELL-CONDITIONED BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF SECOND ORDER DIFFERENCE EQUATIONS

In this paper well-conditioning of boundary value problems for systems of second order difference equa-tions is studied.First,a sufficient condition for the existence of a unique bounded solution (for large enough number of steps) of an associated homogeneous system is given.Finally,a sufficient condition for well-condi-tioning,intrinsically related to the problem data is proposed.

New Oscillation Results for Forced Second Order Differential Equations with Mixed Nonlinearities

Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend some known oscillation results in the literature.

Oscillation of Second Order Delay Difference Equations

In this paper we study the Oscillatory behaviour of the second order delay differenceequation.（1）△（rn△An）+PnAn-k=0,n=n0,n0+1……where{Pn}（?）is a nonnegative Sequenceof real number,（?）is a positive sequence of real number with sum from n=n0 to +∞（1/rn）=+∞,K is a positive integer and △An=An+1-An we prove that each one of following conditions.imples that al solutions of Eq（1）oscillate,where Rn=sum from i=n0 to n（1/ri

Subspaces for weak mild solutions of the second order abstract differential equation

The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.

Modelling of Energy Storage Photonic Medium by Wavelength-Based Multivariable Second-Order Differential Equation

Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium.The existence of this medium demands certain mathematical constraints,which have been derived in detail.Using reverse modelling,a medium satisfying the derived conditions is proven to store energy as the photon propagates from the entry to exit point.A single photon with a given intensity is considered in the analysis and hypothesized to possess a definite non-zero probability of maintaining its energy and velocity functions analytic inside the proposed optical medium,despite scattering,absorption,fluorescence,heat generation,and other nonlinear mechanisms.The energy and velocity functions are thus singly and doubly differentiable with respect to wavelength.The solution of the resulting second-order differential equation in two variables proves that energy storage or energy flotation occurs inside a medium with a refractive index satisfying the described mathematical constraints.The minimum-value-normalized refractive index profiles of the modelled optical medium for transformed wavelengths both inside the medium and for vacuum have been derived.Mathematical proofs,design equations,and detailed numerical analyses are presented in the paper.

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING

By the generalized Riccati transformation and the integral averaging technique, some sufficient conditions of oscillation of the solutions for second order nonlinear differential equations with damping were discussed.Some sufficient oscillation criteria for previous equations were built up.Some oscillation criteria have been expanded and strengthened in some other known results.

Particular Solutions of Generalized Linear Second Differential Equations by Fractional Calculus

In this parer, applications of the fractional calculus to the form (Az 2+Bz+C)ψ 2+(Dz+G)ψ 1+Eψ=f and the partial differential equation 2μz 2(Az 2+Bz+C)+(Dz+G)μz+δμ(z,t)=M 2μT 2+NμT, where ψ 1= d ψ d z and ψ 2= d 2ψ d z 2 are presented.

A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.