Linear Nth Order Differential Equation Research Articles

Uniqueness intervals and two–point boundary value problems

Abstract Consider a linear nth order differential equation with continuous coefficients and continuous forcing term. The maximal uniqueness interval for a classical 2-point boundary value problem will be calculated by an algorithm that uses an auxiliary linear system of differential equations, called a Mikusinski system. This system always has higher order than n. The algorithm leads to a graphical representation of the uniqueness profile and to a new method for solving 2-point boundary value problems. The ideas are applied to construct a graphic for the conjugate function associated with the nth order linear homogeneous differential equation. Details are given about how to solve classical 2-point boundary value problems, using auxiliary Mikusinski systems and Green’s function.

On the General Solution of a Linear nth-Order Differential Equation with Constant Bounded Operator Coefficients in a Banach Space

On the General Solution of a Linear nth-Order Differential Equation with Constant Bounded Operator Coefficients in a Banach Space

Construction of asymptotic representations of solutions from the classL ? ? for systems of differential equations by the method of characteristic matrix functions

By the method of characteristic matrix functions, we construct asymptotic representations of solutions of a system ofq linearn th-order differential equations with a singularity of rankp/r, p, r∈ ℕ, in a sector of the complex plane whose central angle does not exceed πr/p.

Energetic consideration of the vibrational potential function in the effective nuclear charge model. VII

The homogeneity characteristics of the molecular energy with respect to the atomic number are discussed in detail on the basis of the local and nonlocal homogeneity hypothesis which has been proposed in a previous report [J. Chem. Phys. 85, 5882 (1986)]. It is first shown that the homogeneity hypothesis is convenient for describing the local and nonlocal properties of molecules. Inhomogeneous linear second-, third-, fourth-, and nth-order differential equations are constructed from the constrianed linear first-order differential equation and solved for understanding general potential forms of molecules. The results obtained give the theoretical basis to various models so far presented. The approximate potential function obtained by solving the constrained linear third-order differential equation is successfully applied to the calculations of higher-order potential constants as well as molecular energy components. Lastly, all the expressions of molecular energy components with the homogeneity constraint are given by making use of the general solution derived from the constrained linear nth-order differential equation.

Disconjugacy, disfocality, and differentiation with respect to boundary conditions

In particular, under certain assumptions concerning the disconjugacy or disfocality of (1 ), we will consider the differentiability of solutions of (1) with respect to boundary conditions (both boundary values and boundary points). In so doing, we will prove an analogue of a theorem of Peano’s which appears in Hartman’s text [ 1, Theorem 3.1, p. 951, with the consideration therein being with the differentiation of solutions of (1) with respect to initial conditions. In this setting, given a solution y(x) of (1) we are also interested in solutions of the linear nth order differential equation

Monotony Properties and Inequalities for Green’s Functions for Multipoint Boundary Value Problems

We derive monotony properties for $G( \cdot ,s)$ and deduce inequalities for G, where $G(t , s)$ is the Green’s function belonging to a nonsingular, disconjugate, linear nth order differential equation and to a multipoint boundary value problem.

On the extremal solutions of 𝑛th-order linear differential equations

Distribution of zeros of extremal solutions of linear nth-order differential equations is discussed. Existence and nonexistence of extremal solutions with certain zero distributions are established. For instance, it is proved that every extremal solution for [ α , η 1 ( α ) ] [\alpha , {\eta _1}(\alpha )] of the equation y ( n ) + p n − 1 y ( n − 1 ) + ⋯ + p 0 y = 0 {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y = 0 has a zero of order 2 at η 1 ( α ) {\eta _1}(\alpha ) and has no more than n − 2 n - 2 zeros on [ α , η 1 ( α ) ) if p i ≦ 0 , i = 0 , 1 , ⋯ , n − 2 [\alpha , {\eta _1}(\alpha ))\;{\text {if}}\;{p_i} \leqq 0,i = 0,1, \cdots , n - 2 .

The Use of Green's Functions to Solve Some General Initial-Value Problems

Ince [1 ] constructs a solution to an nth order comnpletely homogeneous linear differential system. His solution has continutous derivatives of order n -2 over its entire domain, with the (n t)st derivative discontinuotus at some point in the domain. In this paper, I use functions w^vhich solve initial value problemns whose differential equations are (in turn) nth order linear, nth order nonlinear and homogeneous in the dependent variable. If the equation satisfies a certain uniqueness condition, the solution is the only one. The Green's functions used here are not restricted to continuity of the first n-2 derivatives. One of the problems solved requires the solution to be a function with k cusp points (k is a positive integer greater than one). A nonsynmmetric function is obtained as the solution to a problem in which the linear nth order differential equation is complete, i.e., nonhomogeneous.

A note on the identification of linear systems

Abstract : The problem is considered of ascertaining when the conditions: b sub j = the integral from 0 to T with respect to + of u(t)phi sub j (t), where j = 1, 2,...,N, determine a unique solution of the linear Nth order differential equation in u with coefficients of the form a sub i (t), i = 1,2,...,N. This is a particular problem arising in the general study of the identification of linear systems.