Third-order Linear Differential Equation Research Articles

On a Two-point Boundary Value Problem for Third-order Linear Functional Differential Equations. Part I.

Ecient conditions sucient for the solvability of the problem u 000 (t) = g(t)u( (t)) p(t)u( (t)) +q(t); u(a) = c1; u 0 (a) = c2; u(b) = c3 are derived using the general results obtained in our recent paper (1). Here, p;g2 L (a;b);R + , q 2 L (a;b);R , ; : (a;b) ! (a;b) are measurable functions, and ci2 R (i = 1; 2; 3). Sign-constant solutions are discussed as well. '(v)(t) = p(t)v( (t)) for a.e. t2 (a;b); v2 C (a;b);R ; (1.3) '(v)(t) = g(t)v( (t)) for a.e. t2 (a;b); v2 C (a;b);R ; (1.4)

Analytic solutions of optimal angularly constrained guidance for first-order lag system

Closed-form solutions of an optimal impact-angle-control guidance for a first-order lag system are derived. Under the assumptions of a stationary target and a first-order missile with constant speed and small flight path angle, they are obtained by solving a third-order linear time-varying ordinary differential equation. It is shown that the solutions are expressed by combinations of polynomial, logarithmic, and hypergeometric functions. Moreover, terminal misses due to the system lag and homing geometries for zero miss-distance have been studied. Linear and non-linear simulations are performed to verify the proposed results.

On Oscillations of Solutions of Third‐Order Dynamic Equation

We are proving the new oscillation theorems for the solutions of third‐order linear nonautonomous differential equation with complex coefficients. In the case of real coefficients we derive the oscillation criterion that is invariant with respect to the adjoint transformation. Our main tool is a new version of Levinson′s asymptotic theorem.

Geometric Properties Solutions of a Class of Third-Order Linear Differential Equations

We aim at investigating the geometric properties of the solutions of the initial-value problem which involves the following third-order linear differential equations: ω′′′(z)+Q(z)ω′(z)=0, ω(0)=0, ω′(0)=1, ω′′(0)=0, where Q(z) is analytic in the open unit disk U.

Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

We consider the double scaling limit for a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t=0$ at the same positive value $x=a$, remain positive, and are conditioned to end at time $t=1$ at $x=0$. After appropriate rescaling, the paths fill a region in the $tx$--plane as $n\to \infty$ that intersects the hard edge at $x=0$ at a critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling limits for the positions of the paths at time $t\neq t^{*}$ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a $3\times 3$ matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix.

Third-order linear differential equation with three additional conditions and formula for Green’s function

In this paper, we investigate a third-order linear differential equation with three additional conditions. We find a solution to this problem and give a formula and an existence condition for Green’s function. We compare two Green’s functions for two such problems with different additional conditions: nonlocal and classical boundary conditions. Formula applications are shown by examples.

New exact solutions for a generalization of the Korteweg–de Vries equation (KdV6)

The generalized tanh–coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg–de Vries equation with a source and the second equation is a third-order linear differential equation.

Asymptotic Dichotomy in a Class of Fourth-Order Nonlinear Delay Differential Equations with Damping

All solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate. Novel Riccati type techniques involving third-order linear differential equations are employed. Implications in the deflection of elastic horizontal beams are also indicated.

A second Wronskian formulation of the Boussinesq equation

A Wronskian formulation leading to rational solutions is presented for the Boussinesq equation. It involves third-order linear partial differential equations, whose representative systems are systematically solved. The resulting solutions formulas provide a direct but powerful approach for constructing rational solutions, positon solutions and complexiton solutions to the Boussinesq equation. Various examples of exact solutions of those three kinds are computed. The newly presented Wronskian formulation is different from the one previously presented by Li et al., which does not yield rational solutions.

Asymptotic form at largerof a third-order linear homogeneous differential equation for the ground-state electron density of the He atom

In earlier work a linear differential equation satified by the Schwartz ground-state electron density $\ensuremath{\varrho}(r)$ for (non-relativistic) He-like atomic ions with large atomic number $Z$ has been derived. Here, we utilize the asymptotic expansion at large $r$ given by Amovilli and March for the neutral He atom. We thereby show that a linear differential equation of the same general shape as that satisfied by the Schwartz $\ensuremath{\varrho}(r)$ again emerges for the neutral He atom itself, in the asymptotic limit of large $r$. We argue that essential input into the final differential equation for the He ground-state electron density will be the ionization potential plus the atomic polarizability.

Periodic boundary-value problem for third-order linear functional differential equations

For a linear functional differential equation of the third order {fx481-01}, we establish the theorems on existence and uniqueness of a solution satisfying the conditions {fx481-02}. Here, ℓ is a linear continuous operator transforming the space C([0, ω]; R) into the space L([0, ω]; R) and q ∈ L([0, ω]; R). The problem of nonnegativity of the solution of the analyzed boundary-value problem is also studied.

The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations

We use a spectral transform method to study general boundary-value problems for third-order, linear, evolution partial differential equations with constant coefficients, posed on a finite space domain. We show how this method yields a simple characterization of the discrete spectrum of the associated spatial differential operator, and discuss the obstructions that arise when trying to represent the solution of such a problem as a series of exponential functions. We first review the theory for second-order two-point boundary-value problems, and present an alternative way to derive the classical series representation, as well as an equivalent integral representation, which generally involves complex contours. We illustrate the advantages of the integral representation by studying in some detail the case where Robin-type boundary conditions are prescribed. We then consider the third-order case and show that the integral representation is in general not equivalent to a discrete series representation, justifying a posteriori the failure of some of the classical approaches. We illustrate the third-order case in detail, using the example of the equation q t + q xxx =0 for various types of boundary conditions. In contrast with the second-order case, the qualitative properties of the spectrum of the associated spatial differential operator depend in this case not only on the equation but also on the type of boundary conditions. In particular, the solution appears to admit a series representation only when the prescribed boundary conditions couple the two endpoints of the interval.

Stability criteria for certain third-order delay differential equations

In this paper, we study the asymptotic stability of the zero solution of third-order linear delay differential equations of the form y ‴ ( t ) = p 1 y ″ ( t ) + p 2 y ″ ( t - τ ) + q 1 y ′ ( t ) + q 2 y ′ ( t - τ ) + v 1 y ( t ) + v 2 y ( t - τ ) , where p 1 , p 2 , q 1 , q 2 , v 1 and v 2 are certain constants. Here τ > 0 is a constant delay. In proving our results we make use of Pontryagin's theory for quasi-polynomials.

Towards a differential equation for the nonrelativistic ground-state electron density of the He-like sequence of atomic ions

The early study of Schwartz [Ann. Phys. (N.Y.) 6, 156 (1959)] led to an explicit expression for the ground-state electron density {rho}(r) of He-like atomic ions with nuclear charge Ze in the limit of large Z. Much later, Gal, March, and Nagy [Chem. Phys. Lett. 305, 429 (1999)] derived a third-order linear homogeneous differential equation satisfied by the Schwartz limiting density. Our aim here has been to solve a still correlated model more closely related to the He atom itself. Motivated by semiclassical studies (as by Handke [Phys. Rev. A 50, R3561 (1994)]), we have solved a fully quantal model in which the Coulomb repulsion energy e{sup 2}/r{sub 12} of the two electrons at separation r{sub 12} is expanded in a one-center form about the atomic nucleus and only the s-wave term is retained. From the resulting analytical ground-state density, a differential equation has been derived, which is contrasted with that satisfied by the large-Z limiting form of Schwartz. Suggestions are finally made as to the way the density of the He atomic ion series may be characterized, the ionization potential I(Z) being proposed as crucial input data.

Effective one-body potential of DFT plus correlated kinetic energy density for two-electron spherical model atoms

For three model problems concerning two-electron spin-compensated ground states with spherical density, the third-order linear homogeneous differential equation constructed for the determination of ρ ( r ) is used here in conjunction with the von Weizsäcker functional to characterize the one-body potential of density functional theory (DFT). Correlated von Weizsäcker-type terms are compared to the exact DFT functional.

Third-order equation for harmonic generation: complex canonical transformation and JWKB solution

The quantum description of third harmonic generation can be formulated as an eigenvalue problem for a third-order linear differential equation. We perform a semiclassical study of this third-order equation, generalizing the familiar JWKB theory for the second-order Schrodinger equation, and deriving explicit (albeit approximate) formulas for the eigenvalues within this semiclassical context. A central role in this analysis is played by a nonlinear complex canonical transformation which permits a complete description of the classical motion (generated by a complex polynomial Hamiltonian function) in the complexified position and momentum planes.

Three-level models solvable in terms of the Clausen function

The problem of analytical integrability of the three-level problem by reduction of the time-dependent Schrodinger equations to the third-order linear differential equation satisfied by the generalized hypergeometric functions 3F2 is considered. A total of 12 infinite classes of models solvable in terms of these functions is found, most of which are new and others are generalizations of the previously known families.

Local density of statesN(r,E) for central fields, with energyEin the continuum: especially the Coulomb potential

A third-order linear homogeneous differential equation is first derived for the local density of s-states only, Ns(r, E), for a general central potential V(r). The major example presented is then for the bare Coulomb potential −Ze2/r, for which analytical forms are obtained for both Ns(r, E) and for the total density of states N(r, E). In the appendix, the repulsive linear potential case is also solved analytically.

MOS Realization of the Conjectured Simplest Chaotic Equation

This paper presents a general block diagram of a third-order linear differential equation using current mode techniques. The realization of the conjectured simplest chaotic equation of Elwakil and Kennedy based on Gm − C technology is given. The metal oxide semiconductor circuit is composed of 20 transistors and three grounded capacitors, can operate from a supply voltage as low as ±1.5 V, and covers a very wide range of frequencies. PSpice simulation results using 0.5 µm Mietec technology are given. A numerical solution is also included to verify the circuit operation.

Explicit solutions of some linear ordinary and partial fractional differintegral equations

In recent years, many authors demonstrated the usefulness of some fractional calculus operators in the derivation of (explicit) particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how readily several recent contributions on this subject, involving second- as well as third-order linear ordinary and partial differential equations, can be obtained (in a unified manner) by appropriately applying some general theorems on (explicit) particular solutions of a certain family of linear ordinary and partial fractional differintegral equations.