Odd-order Differential Equations Research Articles

New Monotonic Properties for Solutions of Odd-Order Advanced Nonlinear Differential Equations

The present paper studies the asymptotic and oscillatory properties of solutions of odd-order differential equations with advanced arguments and in a noncanonical case. By providing new and effective relationships between the corresponding function and the solution, we present strict and new criteria for testing whether the studied equation exhibits oscillatory behavior or converges to zero. Our results contribute uniquely to oscillation theory by presenting some theorems that improve and expand upon the results found in the existing literature. We also provide an example to corroborate the validity of our proposed criteria.

Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms

ABSTRACT This paper is concerned with the oscillation criteria of odd-order non-linear differential equations with mixed non-linear neutral terms. We provide new oscillation criteria that improve, expand, and simplify existing ones. Moreover, some examples are provided to demonstrate the theoretical findings.

On the Asymptotic Behavior of Solutions of Odd-Order Differential Equations with Oscillating Coefficients

On the Asymptotic Behavior of Solutions of Odd-Order Differential Equations with Oscillating Coefficients

Oscillation of odd-order differential equations with a nonpositive sublinear neutral term and distributed deviating arguments

New sufficient conditions for the oscillation of all solutions to a class of oddorder differential equations with a nonpositive sublinear neutral term and distributed deviating arguments are established. Example are included to illustrate the results.

Lidstone\u2013Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

General nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

Efficient and Accurate Chebyshev Dual-Petrov-Galerkin Methods for Odd-Order Differential Equations

Efficient and Accurate Chebyshev Dual-Petrov-Galerkin Methods for Odd-Order Differential Equations

On a Method for Studying the Asymptotics of Solutions of Odd-Order Differential Equations with Oscillating Coefficients

On a Method for Studying the Asymptotics of Solutions of Odd-Order Differential Equations with Oscillating Coefficients

Odd-order differential equations with deviating arguments: asymptomatic behavior and oscillation.

Despite the growing interest in studying the oscillatory behavior of delay differential equations of even-order, odd-order equations have received less attention. In this work, we are interested in studying the oscillatory behavior of two classes of odd-order equations with deviating arguments. We get more than one criterion to check the oscillation in different methods. Our results are an extension and complement to some results published in the literature.

Efficient Laguerre and Hermite Spectral Methods for Odd-order Differential Equations in Unbounded Domains

Efficient Laguerre and Hermite Spectral Methods for Odd-order Differential Equations in Unbounded Domains

Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations

Efficient Legendre dual-Petrov-Galerkin methods for solving odd-order differential equations are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like series. Numerical results indicate that the suggested methods are extremely accurate and efficient, and suitable for the odd-order equations.

Transmission problem for odd-order differential equations with two time variables and a varying direction of evolution

The solvability of a boundary value problem for the differential equation \(h\left( x \right){u_t} + {\left( { - 1} \right)^m}\frac{{{\partial ^{2m + 1}}u}}{{\partial {a^{2m + 1}}}} - {u_{xx}} = f\left( {x,t,a} \right)\) is studied in the case where h(x) has a jump discontinuity and reverses its sign on passing through the discontinuity point. Existence and uniqueness theorems are proved in the case of solutions having all Sobolev generalized derivatives involved in the equation.

Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations

We study asymptotic behavior of solutions to a class of higher-order quasilinear neutral differential equations under the assumptions that allow applications to even- and odd-order differential equations with delayed and advanced arguments, as well as to functional differential equations with more complex arguments that may, for instance, alternate indefinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

Complete asymptotic analysis of positive solutions of odd-order nonlinear differential equation

We study the asymptotic behavior of solutions of the odd-order differential equation of Emden–Fowler type $$ {x^{{\left( {2n+1} \right)}}}(t)=q(t){{\left| {x(t)} \right|}^{\gamma }}\operatorname{sgn}x(t) $$ in the framework of regular variation under the assumptions that 0 < γ < 1 and q(t) : [a, ∞) → (0, ∞) is regularly varying function. We show that complete and accurate information can be acquired about the existence of all possible positive solutions and their asymptotic behavior at infinity.

Coercive solvability of odd-order differential equations and its applications

Coercive solvability of odd-order differential equations and its applications

Lyapunov-type inequality for a class of odd-order differential equations

In this paper, we give a generalization of the well-known Lyapunov-type inequality for a class of odd-order differential equations, the result of this paper is new and generalizes some early results on this topic.

A singular boundary value problem for odd-order differential equations

The odd-order differential equation (−1) n x (2 n+1) = f( t, x,…, x (2 n) ) together with the Lidstone boundary conditions x (2 j) (0)= x (2 j) ( T)=0, 0⩽ j⩽ n−1, and the next condition x (2 n) (0)=0 is discussed. Here f satisfying the local Carathéodory conditions can have singularities at the value zero of all its phase variables. Existence result for the above problem is proved by the general existence principle for singular boundary value problems.

A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Differential Equations: Application to the KDV Equation

A new dual-Petrov-Galerkin method is proposed, analyzed, and implemented for third and higher odd-order equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the "dual" boundary conditions. The method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients. Our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd-order equations.

Symmetry Factors for Differential Equations

(3) a2(x)f'(X) + (a2(x) a,(x))f(x) = 0. Likewise, it is a familiar fact ([2, p. 45], [4]) that an odd order differential equation with real coefficients cannot be formally self adjoint. However, the answer to the following is less well known: can a differential equation of order 2n (n >? 2), with real coefficients, be made formally self adjoint by multiplying it by a suitable function f(x)? If so, how does one find this f(x)? It is the purpose of this article to answer these questions.