Nth-order Differential Equations Research Articles

Existence of solutions or nonlinear nth-order differential equations and inclusions with nonlocal and integral boundary conditions via fixed point theory

In this paper, a class of boundary value problems of nonlinear nth-order differential equations and inclusions with nonlocal and integral boundary conditions is studied. New existence results are obtained by means of some fixed point theorems. Examples are given for the illustration of the results.

The generalized continuous wavelet transform associated with the fractional Fourier transform

The main objective of this paper is to study the fractional Fourier transform (FrFT) and the generalized continuous wavelet transform and some of their basic properties. Applications of the FrFT in solving generalized nth-order linear nonhomogeneous ordinary differential equations and a generalized wave equation are given. The generalized continuous wavelet transform, and its inversion formula, and the Parseval relation are also studied using the fractional Fourier transform.

Computation of Green’s functions for boundary value problems with Mathematica

This paper is devoted to construct an algorithm that allows us to calculate the explicit expression of the Green’s function related to a nth-order linear ordinary differential equation, with constant coefficients, coupled with two-point linear boundary conditions. We develop this algorithm by making a Mathematica package.

A new method to solve time-dependent intuitionistic fuzzy differential equations and its application to analyze the intuitionistic fuzzy reliability of industrial systems

To the best of our knowledge, there is no method in the literature for solving such differential equations in which all the parameters except independent variables are represented by intuitionistic fuzzy numbers. In this article, a new method is proposed for solving such nth-order time-dependent intuitionistic fuzzy linear differential equations. To show the application of the proposed method in real-life problems, the time-dependent intuitionistic fuzzy Kolmogorov’s differential equations, obtained by Markov model of condensate system, are solved by the proposed method, and the obtained solution is used to analyze the intuitionistic fuzzy reliability of condensate system.

A Numerical Method For Solving Nth-Order Fuzzy Differential Equation by using Characterization Theorem

In this paper, we study a numerical method for solving Nth-Order fuzzy differential equation (NOFDE). A Characterization Theorem that is presented shows that the NOFDE is equivalent to system of ODEs. So the numerical method for ODE can be used for NOFDE. Two examples are provided.

The variational iteration method for solving n-th order fuzzy differential equations

Abstract The variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

A study of higher-order nonlinear ordinary differential equations with four-point nonlocal integral boundary conditions

This paper investigates some new existence results for an nth-order nonlinear differential equation with four-point nonlocal integral boundary conditions (strip/slit like conditions). Our results are based on some standard fixed point theorems and Leray-Schauder degree theory.

On nth-order nonlinear ordinary random differential equations

An existence result for a nonlinear nth-order ordinary random differential equation is proved under the Caratheodory condition. Two existence results for extremal random solutions are also proved in the Caratheodory case and the discontinuous case of the nonlinearity involved in the equations. Our investigations are carried out in the Banach space of continuous real-valued functions on closed bounded intervals of the real line together with the application of a random version of the Leray–Schauder principle.

Nth-order Fuzzy Differential Equations Under Generalized Differentiability

In this paper, the multiple solutions of Nth-order fuzzy differential equations by the equivalent integral forms are considered. Also, an Existence and uniqueness theorem of solution of Nth-order fuzzy differential equations is proved under Nth-order generalized differentiability in Banach space.

On the lower and upper solution method for higher order functional boundary value problems

The authors consider the nth-order differential equation ?(?(u(n?1)(x)))?= f(x, u(x), ..., u(n?1)(x)), for 2?(0, 1), where ?: R? R is an increasing homeomorphism such that ?(0) = 0, n?2, I:= [0,1], and f : I ?Rn ? R is a L1-Carath?odory function, together with the boundary conditions gi(u, u?, ..., u(n?2), u(i)(1)) = 0, i = 0, ..., n? 3, gn?2 (u, u?, ..., u(n?2), u(n?2)(0), u(n?1)(0)) = 0, gn?1 (u, u?, ..., u(n?2), u(n?2)(1), u(n?1)(1)) = 0, where gi : (C(I))n?1?R ? R, i = 0, ..., n?3, and gn?2, gn?1 : (C(I))n?1?R2 ? R are continuous functions satisfying certain monotonicity assumptions. The main result establishes sufficient conditions for the existence of solutions and some location sets for the solution and its derivatives up to order (n?1). Moreover, it is shown how the monotone properties of the nonlinearity and the boundary functions depend on n and upon the relation between lower and upper solutions and their derivatives.

A new technique of using homotopy analysis method for solving high-order nonlinear differential equations

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high-order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth-order nonlinear differential equation to a system of n first-order equations. Second- and third- order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

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.

Weighted Yosida functions

We study meromorphic functions on the complex plane with given growth of the spherical derivative. We find a criterion of Lohwater–Pommerenke type for non-weighted Yosida functions, and apply this result to obtain a version of Lappan's 5-point theorem. Further, a necessary condition for all solutions of algebraic differential equation of nth order to be weighted Yosida functions is found.

Classification and Reduction of Generalized Thin Film Equations

Classification and reduction of the generalized fourth-order nonlinear differential equations arising from the liquid films are considered. It is shown that these equations have solutions on subspaces of the polynomial, exponential or trigonometric form defined by linear nth-order ordinary differential equations with constant coefficients for n = 4, …, 9. Several examples of exact solutions are presented.

On special solutions of second and fourth Painlevé hierarchies

In this Letter, we give special solutions of second and fourth Painlevé hierarchies derived by Gordoa, Joshi, and Pickering. We show that for certain choice of the parameters each nth member of these hierarchies has a special solution in terms of an nth order differential equation. Furthermore we derive a relation between these two hierarchies.

On linear homogeneous differential equation of Chebyshev type

Let L[y] = y(n)(z)+gn-1(z)y(n-1)(z)+. . .+g1(z)y(1)(z)+g0(z)y(z) = 0 be a differential equation of nth order with analytic in circle |z| < R coefficients. We will call above equation by equation of Chebyshev type in
 |z| < R, if fundamental system of its solution is a Chebyshev system in circle |z| < R . In present
 paper the conditions with the fulfillment of which the equation L[y] = 0 is of Chebyshev type.

Oscillation and Asymptotic Behavior for nth-order Nonlinear Neutral Delay Dynamic Equations on Time Scales

In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equations $$\begin{array}{rcl}&&\left\{a(t)\Psi(x(t))\left[|(x(t)+p(t)x(\tau(t)))^{\Delta ^{n-1}}|^{\alpha-1}(x(t)+p(t)x(\tau(t)))^{\Delta^{n-1}}\right]^{\gamma}\right\}^{\Delta}\\[12pt]&&{}\quad +\lambda F(t,x(\delta(t)))=0,\end{array}$$ on time scales, where α>0 is a constant, γ>0 is a quotient of odd positive integers and λ=±1. Our results in this paper not only extend and improve some known results but also present a valuable unified approach for the investigation of oscillation and asymptotic behavior of nth-order nonlinear neutral delay differential equations and nth-order nonlinear neutral delay difference equations. Examples are provided to show the importance of our main results.

On oscillatory solutions of certain forced Emden–Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x ″ ( t ) + a ( t ) | x ( t ) | λ sign [ x ( t ) ] = e ( t ) , where t ⩾ t 0 ⩾ 1 and λ > 1 , that decay to 0 when t → + ∞ as O ( t − μ ) for μ > 0 as close as desired to the “critical quantity” μ ⋆ = 2 λ − 1 . For this class of equations, we have lim t → + ∞ E ( t ) = 0 , where E ( t ) < 0 and E ″ ( t ) = e ( t ) throughout [ t 0 , + ∞ ) . We also establish that for any μ > μ ⋆ and any negative-valued E ( t ) = o ( t − μ ) as t → + ∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o ( t − μ ) . In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722–732].

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2 nth-order linear differential equations

Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2 nth-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional 2 nth-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.

Existence and uniqueness of periodic solutions for a kind of nonlinear nth order differential equations with delays

By applying the continuation theorem of coincidence degree theory, we establish new results on the existence and uniqueness of 2π-periodic solutions for a class of nonlinear nth order differential equations with delays.