Solutions Of Second-order Differential Equations Research Articles

Fixed sign solutions of second-order difference equations with Neumann boundary conditions

In this paper, we study the existence of solution of the nonlinear second-order difference problem with Neumann boundary conditions. Assuming the existence of a pair of ordered lower and upper solutions γ and β, we obtain optimal existence results for the case γ ≤ β and even for γ ≥ β. To this end, we do an exhaustive study of the values of the real parameters α and μ, for which the associated Green's function to the linear operator L[ α, μ] u κ ≡ u κ+2 − 2 α u κ+1 + μ u κ with Neumann boundary conditions has fixed sign.

Canonical vector polynomials for the computation of complex order Bessel functions with the tau method

A modification of the Lanczos Tau Method for the approximate solution of second-order differential equations with polynomial coefficients was proposed by the author earlier. This modification chooses a minimum perturbation, among all possible, on the right-hand side of the corresponding Volterra integral equation. A Tau Method computational scheme is applied to the approximate solution of a system of differential equations related to the differential equation of hypergeometric type. Various vector perturbations are discussed. Our choice of the perturbation term is a shifted Chebyshev polynomial with a special form of selected transition and normalization. The minimality conditions for the perturbation term are found for one equation. They are sufficiently simple for verification in a number of important cases. Ortiz's recursive approach to the Tau Method is extended to canonical vector-polynomial coefficients, and to the coefficients of the linear combination that generate an approximate solution. The advantages of the proposed approach are shown for modified Bessel functions computation. Seven digit tables of modified Bessel function K 1 2+iβ (x) have been computed and published. The computer programs for the calculations may be found in the State Fund of Algorithms and Programs and in the United Library of Numerical Analysis of the Moscow State University.

A novel technique for the solution of second-order difference equations

The class of second-order difference equations that arise in the diffraction of a plane wave at skew incidence on an impedance wedge is investigated. For a typical equation which was solved by the introduction of elliptic integrals, it is shown that the solution can be obtained in a very trivial manner from a convolution-type integral equation and the results are in complete agreement with those previously found.

Mild solutions of second-order differential equations on the line

A necessary condition for the existence and uniqueness of mild solutions of first- and second-order differential equations on the real line is given. It is known that the first-order problem is related to the operator equation AX − XD = −δ0. Here, it is shown how the second-order problem on the real line is connected to the operator equation AX − XD2 = −δ0 and also to some other operator equations.

On Unbounded Positive Solutions of Second-Order Difference Equations with a Singular Nonlinear Term

In this paper, several necessary and/or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the formΔrnΔxn+fn,xn=0,n∈Nn0,where N(n0)={n0,n0+1,…,} (n0 is a fixed nonnegative integer), rn>0 for n≥n0 such that ∑∞n=n01/rn=∞, and f: N(n0)×R→R is a real-valued continuous function such that f(n,x)≥0 for all (n,x)∈N(n0)×(0,∞).

Periodic Solutions for Second-Order Differential Equations involving Nonconvex Superpotentials

The paper establishes the existence of a nonconstant periodic solution of a general second order nonautonomous Hamiltonian system with discontinuous nonlinearities. The multiplicity of solutions is also studied.

Multivalued solutions of second-order differential equations

Multivalued (not set-valued as in the theory of differential inclusions!) solutions of ordinary differential equations (ODE) appear naturally in geometrical and physical problems in which the independent and dependent variablesx, y are geometric coordinates of a current point on the sought-for curve. This note contains some simple results concerning smooth multivalued solutions of real second-order ODE resolved with respect toy″; the special role of equations of the third degree with respect toy′ is underlined. The method of investigation is based on combining ODEs fory(x) andx(y).

Asymptotic Equivalence of Second-Order Difference Equations

The asymptotic behavior of solutions of second-order difference equations is discussed.

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations (*) Δ ( a n Δ x n ) = q n x n + 1 and (**) Δ ( a n Δ x n ) = q n f ( x n + 1 ) where a n > 0, q n > 0, and f: R→R is continuous with uf( u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.

Asymptotic behavior of the solutions of second-order difference equations

The asymptotic behavior of the solutions of the second-order difference equation Δ 2x(n) = ƒ(n,x(n)) is considered.

Some local properties of the solutions of second-order differential equations

AbstractWe investigate the asymptotics and zero distribution of solutions of ω″ + Aω = 0 where A is an entire function of very slow growth. The results parallel the classical case when A is assumed to be a polynomial.

Interval Oscillation Conditions for Difference Equations

Interval-type sufficient conditions for oscillation of solutions of second-order difference equations are established. These conditions are new even though their analogues for differential equations have been known for some time.

Some properties of orthogonal polynomials satisfying fourth order differential equations

In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

Asymptotic expansion of differential equations with an irregular singular point of finite rank

An asymptotic expansion is constructed for solutions of second-order differential equations with an irregular singular point of finite rank at infinity. Formal solutions are obtained. By reduction of the differential equation to a Volterra integral equation we prove that the formal solutions are asymptotic expansions of the solutions of the differential equation.

On conditions for oscillation and nonoscillation of the solutions of a semilinear second-order differential equation

We establish sufficient conditions for oscillation and nonoscillation of regular solutions of the secondorder differential equation Sign y=0, where λ<1 and p: [a,ω[→ ℝ, −∞< a < ω≤+∞ is a locally summable function.

The boundedness and convergence of solutions of second-order differential equations with applications

For multidimensional second-order differential equations we present new conditions for the boundedness and convergence of all solutions and their derivatives, and the existence of periodic solutions. In particular, for scalar case we give the necessary and sufficient conditions for the boundedness and convergence of all solutions and their derivatives. These results are used to improve a result of Levin and Nohel for the reactor dynamics and a result of Holmes for the evolution equations derived from a conveying fluid and to give an answer to the conjecture on the continuous stirred tank proposed by Uppal etc.

On a new method of solving a class of problems of applied mechanics usingR-functions, weighted discrepancies, and least squares

We propose a new approach to solving a class of problems of applied mechanics using R-functions and boundary integral equations. We give the results of numerical analysis of the solution of second-order differential equations with constant coefficients using the example of the torsion of orthotropic rods.

Ladder operators for central potential wave functions from the algebraic representation of orthogonal polynomials

A general algebraic procedure that yields to raising and lowering operators for the solutions of second-order differential equations is presented. The method is illustrated by applying it to the differential equations of Hermite and Laguerre polynomials. Taking advantage of the algebraic representation of these polynomials, the ladder operators for harmonic oscillator and hydrogen atom wavefunctions are straightforwardly deduced without resorting to specialized factorizations. The proposed algebraic approach can be extended to the determination of new sets of ladder operators that could be used in the calculation of matrix elements in specific applications.

Differential equations for total energy of atoms and atomic ions within thomas-fermi approximation

An approach to the calculation of the total energy of atoms and atomic ions as a function of atomic number Z and number of electrons N, based on the solution of secondorder differential equations together with auxiliary conditions, is presented. Some applications of the equations to the description of real atoms are also shown. Physical consequences of the approach are indicated. Attention is paid to the methodological aspects of the approach, which give the analytical form of the results and are very convenient for further treatment.

On the distribution of zeros of solutions of second-order differential equations

On the distribution of zeros of solutions of second-order differential equations