Nth-order Differential Equations Research Articles

On linear equations with general polynomial solutions

We provide necessary and sufficient conditions for which an nth-order linear differential equation has a general polynomial solution. We also give necessary conditions that can directly be ascertained from the coefficient functions of the equation.

Non-linear problems with non-monotonic blow-up solutions: Non-local transformations, test problems, exact solutions, and numerical integration

The method of non-local transformations is proposed for numerical integration of non-linear Cauchy problems having non-monotonic blow-up solutions. In such problems there exists a singular point whose position is unknown in advance (for this reason, the standard numerical methods for solving blow-up problems can lead to significant errors). In addition, the non-monotonic behavior of the solution excludes a possibility of applying the hodograph transformation and some other methods that are used for numerical investigation of simpler problems having monotone blow-up solutions. In this paper, the method is described for numerical integration of similar problems for non-linear nth-order ordinary differential equations xt(n)=g(t,x,xt′,…,xt(n−1)), based on the introduction of a new non-local independent variable ξ, which is related to the original variables t and x by the equation ξt′=g(t,x,xt′,…,xt(n−1),ξ), and the subsequent transformation of the original problem to the Cauchy problem for the corresponding system of first-order differential equations. With a suitable choice of the regularizing function g, the proposed method leads to problems whose solutions are presented in parametric form and do not have blowing-up singular points; therefore the transformed problems allow the application of the standard fixed-step numerical methods. A number of test problems with non-monotonic blow-up is constructed for differential equations of the first, second, third, and fourth orders, which have exact solutions expressed in elementary functions. The numerical integration of test problems has shown high efficiency of methods based on non-local transformations of a special type (and the practical inapplicability of the arc-length transformation for numerical solving blow-up problems with ODEs of high order). In addition to problems for single ODEs, Cauchy problems for systems of coupled equations also are considered. Some recommendations are given on a suitable choice of a variable step in direct adaptive numerical methods in which the transformations of equations are not used. It is important to note that the method of non-local transformations can also be used to numerically integrate other problems with large solution gradients (including problems of boundary-layer type).

On global asymptotic stability of [formula omitted] with [formula omitted] not persistently exciting

We study global convergence to zero of the solutions of the nth order differential equation ẋ=ϕ(t)ϕ⊤(t)x. We are interested in the case when the vector ϕ is not persistently exciting, which is a necessary and sufficient condition for global exponential stability. In particular, we establish new necessary conditions on ϕ(t) for global asymptotic stability of the zero equilibrium of the “unexcited” system. A new sufficient condition, that is strictly weaker than the ones reported in the literature, is also established. Unfortunately, it is also shown that this condition is not necessary.

On the Existence of Positive Solution for nth Order Differential Equation for Boundary Value Problems

We considering the problem of solving a nonlinear differential equation in the Banach space of real functions and continuous on a bounded and closed interval. By means of the fixed point theory for a strict set contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem.

Comment on “On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator” [J. Math. Phys. 48, 032701 (2007)

In a remarkable paper Chandrasekar et al. showed that the (second-order constant-coefficient) classical equation of motion for a damped harmonic oscillator can be derived from a Hamiltonian having one degree of freedom. This paper gives a simple derivation of their result and generalizes it to the case of an nth-order constant-coefficient differential equation.

On higher-order nonlinear boundary value problems with nonlocal multipoint integral boundary conditions

In this paper, we study the existence of solutions for nonlinear nth-order ordinary differential equations and inclusions with nonlocal multipoint integral boundary conditions. Fixed point theorems due to Schaefer and Banach are employed to prove the existence results for the single-valued case, whereas the existence of solutions for the multivalued problem is established by means of a nonlinear alternative for Kakutani maps and Covitz–Nadler fixed point theorem. The obtained results are well explained by examples. We extend our discussion to some new problems.

Advantage of gH-difference on the nth- fuzzy linear differential equations with constant coefficients

In this article using the concept of generalized Hukuhara difference, the nth-order fuzzy differential equation problem with constant coefficients will be solved. Then the solution will be compared in two cases. The first case is the one where displacement is not performed. It means that right side of nth-order fuzzy differential equation not transfer to the left side. The second case is the one where displacements is performed by applying generalized Hukuhara difference concept. We show that forcing term under gH-difference dose not exist when a displacement is not performed and the solution of bout fuzzy differential equations is the same considering gH- difference. Finally, we present more examples for the second-order fuzzy differential equation with constant coefficients.

Positive solutions of nth-order impulsive differential equations with integral boundary conditions

Abstract This paper is concerned with the existence and nonexistence of positive solutions of nth-order impulsive boundary value problem with integral boundary conditions. The fixed point theorem of cone expansion and compression is used to investigate the existence of at least one positive solution. Also, an example is given to illustrate the effective of our results.

Repeated quantization method and its applications to the construction of asymptotics of solutions of equations with degeneration

We study the asymptotics of solutions of homogeneous nth-order differential equations with a cusp degeneration for the case in which the principal symbol has multiple roots. We describe a new method for constructing the asymptotics, which we call the repeated quantization method. Examples of application of the method are given.

Asymptotics and oscillation of nth-order nonlinear differential equations with p-Laplacian like operators

This paper is concerned with nth-order nonlinear differential equations of the form $(a(t)|x^{(n-1)}(t)|^{p-2}x^{(n-1)}(t) )^{\prime}+ r(t)|x^{(n-1)}(t)|^{p-2}x^{(n-1)}(t)+q(t)|x(g(t))|^{p-2}x(g(t))=0 $ with $n\ge2$ . By discussing the signs of ith-order derivatives of eventually positive solutions, for $i=1,\ldots,n-1$ , and using the generalized Riccati technique and integral averaging technique, we derive new criteria for oscillation and asymptotic behavior of the equation. Our results generalize and improve many existing results in the literature.

Invariance of Hyers-Ulam stability of linear differential equations and its applications

We prove that the generalized Hyers-Ulam stability of linear differential equations of nth order (defined on I) is invariant under any monotone one-to-one correspondence $\tau: I \to J$ which is n times continuously differentiable. Moreover, using this result, we investigate the generalized Hyers-Ulam stability of the linear differential equation of second order and the Cauchy-Euler equation.

On asymptotic equivalence of nth order nonlinear differential equations

Abstract This paper is devoted to the problem of asymptotic equivalence of nth order differential equations with exponentially equivalent right-hand sides. With the help of this result asymptotic behaviour of solutions to nonhomogeneous differential equations is described

Solvability of the periodic problem for higher-order linear functional differential equations

We obtain necessary and sufficient conditions for the unique solvability of the periodic boundary value problem for a family of nth-order linear functional differential equations with pointwise constraints on the functional operators.

On the generalized superstability of nth-order linear differential equations with initial conditions

We establish the generalized superstability of differential equations of nth-order with initial conditions and investigate the generalized superstability of differential equations of second order in the form of y??(x) + p(x)y?(x)+q(x)y(x) = 0 and the superstability of linear differential equations with constant coefficients with initial conditions.

A Method of Moment Approach in Solving Boundary Value Problems

Several available methods, known in literatures, are available for solving nth order differential equations and their complexities differ based on the accuracy of the solution. A successful method, known to researcher in the area of computational electromagnetic and called the Method of Moment (MoM) is found to have its way in this domain and can be used in solving boundary value problems where differential equations are resulting. A simplified version of this method is adopted in this paper to address this problem, and two differential equations examples are considered to clarify the approach and present the simplicity of the method. As illustrated in this paper, this approach can be introduced along with other methods, and can be considered as an attractive way to solve differential equations and other boundary value problems.

A New Method to Find Fuzzy Nth Order Derivation and Applications to Fuzzy Nth Order Arithmetic Based on Generalized H-Derivation

In this paper, fuzzy nth-order derivative for n in N is introduced. To do this, nth-order derivation under generalized Hukuhara derivative here in discussed. Calculations on the fuzzy nth-order derivative on fuzzy functions and their relationships, in general, are introduced. Then, the fuzzy nth-order differential equations is solved, for n in N.

Asymptotic behavior of solutions of nth-order ordinary differential equations with regularly varying nonlinearities

We obtain existence conditions and asymptotic, as t → ω (ω ≤ +∞), representations of a certain class of monotone solutions of an nth-order differential equation whose right-hand side is a sum of terms with regularly varying nonlinearities.

Modified Method of Semi-Analytical Transition Matrix on the Analysis of a Coupled Vibration System

The coupled bending-extensional in-plane vibration of a rotating curved beam is considered. The dynamic system is governed by two coupled differential equations and six boundary conditions. The conventional method of transition matrix is usually used to solve the system composed of one nth order differential equation and n boundary conditions. In this study, the system of a rotating curved beam is different from that of the conventional one. The modified method of a semi-analytical transition matrix is developed to study this system. Finally, several physical observations about the rotating curved beam are manifested.

An efficient analytical treatment of twelve order boundary value problems

Purpose – In this article, the author proposed a new analytical solution procedure for 12 order differential equation. The new analytical technique namely homotopy analysis transform method is efficient and effective for solving higher order differential equations. The results indicate the effectiveness and stability of the proposed algorithm. The paper aims to discuss these issues. Design/methodology/approach – The author designed a new solution methodology for higher order differential equations which is a combination of Laplace transformation and homotopy deformation theory. Findings – The author mainly discussed two numerical applications with error analysis of analytical scheme which shows high order accuracy of the projected technique which will serve as a milestone for solving higher order differential equations in engineering with no efforts. Originality/value – The author proposed an innovative and original idea for nth-order differential equations in this article.

Review of Five Sets of Piecewise Constant Orthogonal Functions for Function Approximation, Integration and Solution of First Order Differential Equation Using These Function Sets

This paper reviews five types of piecewise constant orthogonal functions namely Walsh, block-pulse, sample-and-hold, triangular and hybrid functions. Approximation, integration of the functions are performed and compared with the actual result. By performing integration we get the operational matrix for each individual function set. With the help of integration operational matrices we can solve an nth order differential equation which is of importance for solving control engineering problems although a first order ordinary differential equation has been dealt with in this paper.