Nth Order Ordinary Differential Equations Research Articles

On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.

A differential game of pursuit-evasion with constrained players’ energy

In the Hilbert space l2, we investigate a pursuit-evasion differential game involving countable number of pursuers and one evader. Players move in agreement with certain nth order ordinary differential equations with control functions of players satisfying integral constraints. The period of the game, which is denoted as θ, is fixed. During the game, pursuers want to minimize the distance to the evader and the evader want to maximize it. The game's payoff is the distance between evader and closest pursuer at time θ. Independent of relationship between energy resources of the players, we provide formula that defines value of the game and constructed players' optimal strategies.

Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

This paper deals with the existence of ω-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. Lnu(t)=f(t,u(t),u(t−τ)),t∈R, where Lnu(t):=u(n)(t)+∑i=0n−1aiu(i)(t), ai∈R, i=0,1,⋯,n−1, are constants, f(t,x,y):R×E×E⟶E is continuous and ω-periodic with respect to t, τ>0. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.

First Integrals of Differential Operators from SL(2,ℝ) Symmetries

The construction of first integrals for SL(2,R)-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra sl(2,R). Firstly, we provide for n=2 an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of sl(2,R), and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for n>2, provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of sl(2,R) is known. Several examples illustrate the procedures.

Evaluation of the One-Dimensional Lp Sobolev Type Inequality

This study applies the extended L 2 Sobolev type inequality to the L p Sobolev type inequality using Hölder’s inequality. The sharp constant and best function of the L p Sobolev type inequality are found using a Green function for the nth order ordinary differential equation. The sharp constant is shown to be equal to the L p norm of the Green function and to the pth root of the value of the origin of the best function.

Closed-form higher-order numerical differentiators for differentiating noisy signals

In the paper, nth-order differential of a noisy signal is recast as an nth-order ordinary differential equation with an unknown right-hand side, which is an inverse problem to recover the forcing term. We derive weak-form methods to solve the inverse problem, with sinusoidal functions as test functions. By exploring the orthogonality of sinusoidal functions, the expansion coefficients in the trial functions of weak-form numerical differentiators can be determined analytically. Several examples verify the efficiency, accuracy and robustness of the weak-form numerical differentiators for computing the higher-order differentials of noisy data. Moreover, the applications of the weak-form numerical differentiators are also demonstrated, to recover the external forces of nonlinear dynamical systems with single or multiple degrees of freedoms, which are evaluated under the pollution of large noise on the measured data of displacements.

Acoustic analog computing system based on labyrinthine metasurfaces

Acoustic computing devices, including switches, logic gates, differentiator and integrator, have attracted extensive attentions in both academic research and engineering. However, no scheme of acoustic computing device with more complex functionality has been proposed, such as ordinary differential equation (ODE) solver. Here, we propose an acoustic analog computing (AAC) system based on three cascaded metasurfaces to solve the nth-order ODEs. The metasurfaces are constructed with layered labyrinthine units featuring broad amplitude and phase modulation ranges. The simulated transmitted pressure of the AAC system agrees well with the theoretical solution of ODE, demonstrating the excellent functionality. Unlike the optical ODE solver based on differentiator or integrator, whose geometry becomes more complicated for solving higher order ODE, the proposed AAC system with fixed geometry can be designed for arbitrary nth-order ODE in principle. The proposal may find applications in various scenarios such as acoustic communication, analog computing and signal processing.

An Approach for the Generation of an Nth-Order Chaotic System with Hyperbolic Sine.

Chaotic systems with hyperbolic sine nonlinearity have attracted the attention of researchers in the last two years. This paper introduces a new approach for generating a class of simple chaotic systems with hyperbolic sine. With nth-order ordinary differential equations (ODEs), any desirable order of chaotic systems with hyperbolic sine nonlinearity can be easily constructed. Fourth-order, fifth-order, and tenth-order chaotic systems are taken as examples to verify the theory. To achieve simplicity of the electrical circuit, two back-to-back diodes represent hyperbolic sine nonlinearity without any multiplier or subcircuits. Thus, these systems can achieve both physical simplicity and analytic complexity at the same time.

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).

Five Points Mono Hybrid Point Linear Multistep Method for Solving Nth Order Ordinary Differential Equations Using Power Series Function

Five Points Mono Hybrid Point Linear Multistep Method for Solving Nth Order Ordinary Differential Equations Using Power Series Function

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.

Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation

Abstract A hyperjerk system is a dynamical system, which is modelled by an nth order ordinary differential equation with n ⩾ 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system of n first order ordinary differential equations with n ⩾ 4. In this research work, a 4-D novel hyperchaotic hyperjerk system has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel hyperjerk system are obtained as L1 = 0.1448, L2 = 0.0328, L3 = 0 and L4 = −1.1294. The Kaplan-Yorke dimension of the novel hyperjerk system is obtained as DKY= 3.1573. Next, an adaptive backstepping controller is designed to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve global hyperchaos synchronization of the identical novel hyperjerk systems with three unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using SPICE is presented in detail to confirm the feasibility of the theoretical hyperjerk model.

Smoothness of solutions with respect to multi-strip integral boundary conditions for nth order ordinary differential equations

Under certain conditions, solutions of the boundary value problem y(n) = f(x,y,y',...,y(n-1)), a < x < b, y(i-1)(x1) = yi, i=1,...,n-1, y(x2) ∑ i=1mγi ∫ ξiηiy(x)dx=yn, a<x1<ξ1<η1<ξ2<η2<...<ξm<ηm<x2<b, are differentiated with respect to the boundary conditions.

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.

The [formula omitted]-symmetry reduction method and Jacobi last multipliers

For nth order ordinary differential equations, it is studied the role of a Jacobi last multiplier (JLM) in the reduction processes that arise from the existence of either a k parametric symmetry group or a λ-symmetry. For the reduction derived from a λ-symmetry, JLMs are inherited as integrating factors of the auxiliary equations. Several ways that have appeared recently to solve the determining equations of the λ-symmetries are also analysed. Two examples illustrate the combined use of λ-symmetries and JLMs to obtain the complete solution of the equations.

Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced Nth order ordinary differential equation (ODE), N = 2, 3, 4, …, in this chain yields (N − 1)th order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced Nth order ODE, N = 2, 3, 4, …, in the Abel chain always ends at the (N − 1)th order ODE in the Riccati chain. We describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry.

A modified block method for the direct solution of second order ordinary differential equations

The direct solution of general second order ordinary differential equation is considered in this paper. The method is based on the collocation and interpolation of the power series approximate solution to generate a continuous linear multistep method. We modified the existing block method in order to accommodate the general nth order ordinary differential equation. The modified block is adopted to generate the independent solution at the grid points. This method was found to be efficient when tested on second order ordinary differential equation

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.

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.

Positive solutions of boundary value problems for nth order ordinary differential equations

In this paper, we investigate the problem of existence and nonexistence of positive solutions for the nonlinear boundary value problem: u (n) (t) + �a(t)f(u(t)) = 0, 0 < t < 1,