Nth-order Differential Equations Research Articles

Boundedness of pseudo-differential operators via coupled fractional Fourier transform

In this article, we obtained some fruitful results of the coupled fractional Fourier transform and its kernel. We defined pseudo-differential operators related to coupled fractional Fourier transform on Schwartz spaces and it is shown that their composition is again a pseudo-differential operator. It made further discussion on the composition of two pseudo-differential operators on Sobolev spaces and derived certain norm inequalities on it. Further, we have successfully applied some of the results of the coupled fractional Fourier transform to investigate the solution of nth-order linear non-homogeneous partial differential equation and wave equation. Lastly, we present some examples involving graphs and tables to illustrate the validity of our theoretical findings.

Analyzing Lower Limb Dynamics in Human Gait Using Average Value-Based Technique

The motivation of this study is to develop effective and economical assistive technologies for people with physical disabilities. The novelty in this manuscript is the application of the average value-based technique to accurately represent the involved biomechanics of the lower limb joints during the human gait cycle. This mathematical formulation of lower limb joints’ biomechanics forms the first objective for modeling and final exoskeleton prototype development. To account for modeling the characteristics of human locomotion, the nth-order linear differential equation with constant coefficients is considered with appropriate modification. The physical characteristics of an individual are represented by the constant coefficients (P0, P1, P2, and P3) of the modified infinite series, which are obtained by processing experimental data collected using an optical technique. The differential terms of the infinite series are replaced by difference terms (δbavg, δ2bavg, and δ3bavg) since the data were captured as a set of digital values. The work presented here is based on the experimental results of individuals suitably categorized according to their physical nature like age and other physical structure. The optically monitored positional values of the lower limb joints of the individual subjects while they are completing the gait cycles are used for getting values of different terms of the model. The data collected through the conduct of experiments are used for finding the values of the terms of the differential equation. The model is effectively validated through experimental results. It was determined that the representation’s accuracy fell within the ±5% acceptable tolerance limit. The model is prepared for healthy as well as disabled persons, through which the disability is quantified. The resulting model can be used to develop assistive devices for people with physical disabilities. This results in the rehabilitation process and thereby helps the reintegration into society, subsequently allowing them to lead a normal life.

Boundary Data Smoothness for Solutions of Nonlocal Boundary Value Problems for nth Order Differential Equations

Boundary Data Smoothness for Solutions of Nonlocal Boundary Value Problems for nth Order Differential Equations

Formula omitted]-Laplace transforms and [formula omitted]-differential equations of the arbitrary-order, theory and applications

This paper provides a framework to study a class of arbitrary-order uncertain differential equations known as arbitrary-order Z+-differential equations. For this purpose, we first present the parametric form of Z+-numbers. Then, we introduce the basic algebraic operations on Z+-numbers, including addition, scalar multiplication, and Hukuhara difference. Definitely, these operations lead us to define the Z+-valued function. Afterward, the limit and continuity concepts of a Z+-valued function are provided under the definition of a metric on the space of Z+-numbers. Furthermore, the concepts of Z+-differentiability, Z+-integral, and Z+-Laplace transform with the convergence theorem for the Z+-valued function and its nth-order derivatives are introduced in detail. Considering all these concepts, a Z+-differential equation (Z+DE) can be expressed in the form of a bimodal differential equation combining a fuzzy initial value problem (FIVP) and a random differential equation (RDE). To this end, we use a combination of “FIVP under strongly generalized Hukuhara differentiability (SGH-differentiability)” and “random differential equation under mean-square differentiability (ms-differentiability)” to define the nth-order differential equations with Z+-number initial values. Further, the existence and uniqueness of the Z+-differential equations are examined by presenting several theorems. Finally, the effectiveness of the approaches is illustrated by solving two examples.

Pursuit differential game problem with integral and geometric constraints.

Abstract We study pursuit differential game problem in which a countable number of pursuers chase one evader. The problem is formulated in a Hilbert space l2 with pursuers’ motions described by nth order differential equations and that of the evader by mth order differential equation. The control functions of the pursuers and evader are subject to integral and geometric constraints respectively.Duration of the game is denoted by the positive number?. Pursuit is said to be completed if there exist strategies uj of the pursuers Pj such that for any admissible control v(·) of the evader E the inequality ky(?) ? xj (?)k ? rj is satisfied for some j ? {1, 2, . . .}. In this paper, sufficient condition for completion of pursuit were obtained and also strategies of the pursuers that ensure completion of pursuit are constructed.

Chaos synchronization using differential equations as extended state observer

In this paper, an extended state observer (ESO) is presented for synchronizing a chaotic master-slave system based on linear differential equations. The universal approximation property enables linear differential equations to estimate uncertainties, consisting of disturbances and unmodeled dynamics. The main contribution of this study is developing a control scheme based on ESO, which is designed using differential equations. In other words, an Nth-order linear differential equation is utilized to model the lumped uncertainty. The controller is proposed based on a model-free approach to eliminate the need for accurate information from the system model and assure a robust tracking performance. Furthermore, a thorough mathematical analysis is given based on the Lyapunov stability theorem to confirm the uniform ultimate boundedness of the observation/tracking approximation errors. To analyze the performance of the ESO-controller scheme in terms of transient response behavior and robustness, the Duffing-Holmes oscillator is considered as the simulation testbed. A set of five different experiments are conducted to evaluate the efficiency of the introduced control approach. The results of ESO are also compared with two powerful approximation methods.

A game problem of pursuit-evasion with nth order differential equations describing players' dynamics

In this paper, we study a pursuit-evasion differential game problem in the Hilbert space Dynamics of countable number of pursuers and evader expressed as nth-order differential equations with geometric constraints on the control functions of the players. The game terminates at a given fixed time which is denoted by ϑ. The game's payoff is the infimum of the distances between the evader and pursuers at the time ϑ. According to the rule of the game, pursuers try to minimise the distance to the evader and the evader tries to maximises it. We found value of the game and constructed players' optimal strategies.

On interactive fuzzy solutions for mechanical vibration problems

Fuzzy initial value problems describing classical mechanical vibrations are the focus of this paper. In particular, this work considers systems described by nth-order linear ordinary differential equations whose initial conditions are uncertain and given by interactive fuzzy numbers. The concept of interactivity arises from the concept of joint possibility distribution (J). An approach based on the sup-J extension principle, which is a generalization of Zadeh’s extension principle, is presented. This theory is applied to two major examples of oscillatory systems: the forced vibration of an uncoupled mass-spring-damper system and the free vibration of a coupled undamped mass-spring system. In both cases, we have that the solution via sup-J extension, where the fuzzy initial conditions are given by linearly correlated fuzzy numbers, is contained in the solution via Zadeh’s extension.

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.

Spectrum assignment in linear systems with several commensurate lumped and distributed delays in state by means of static output feedback

A linear control system defined by a stationary differential equation of nth order with several commensurate lumped and distributed delays in state is considered. In the system, the input is a linear combination of m variables and their derivatives of order not more than n−p and the output is a k-dimensional vector of linear combinations of the state and its derivatives of order not more than p−1. For this system, a spectrum assignment problem by linear static output feedback with commensurate lumped and distributed delays is studied. Necessary and sufficient conditions are obtained for solvability of the arbitrary spectrum assignment problem by static output feedback controller. Corollaries on stabilization of the system are obtained.

Analytical approximation of the first grade MHD squeezing fluid flow with slip boundary condition using a new iterative method

AbstractIn this article, the steady two‐dimensional axisymmetric flow of an incompressible viscous fluid in a porous medium under the influence of a uniform transverse magnetic field with slip boundary condition is analyzed and solved by using a new iterative method (NIM). This NIM depends mainly on the coefficients of a power series resulting from integrating the nth order differential equation with nonlinear operator derivatives. NIM has been used to obtain the approximate analytical solution by varying the pertinent flow parameters. The influence of different parameters on the present solution of the flow is shown through graphs with a discussion. These graphs refer to fact that increasing numbers of Reynolds () and Hartmann () give the attribute of decreasing velocity, with the addition that with an increase in Hartmann number and decrease in Reynolds number the streamlines are stretched towards the x‐axis. Finally, the NIM is effective and highly able to obtain an excellent result.

Oscillation of higher order functional differential equations with an advanced argument

This paper is concerned with the oscillatory behavior of solutions of the nth-order nonlinear differential equations with an advanced argument x(n)(t)+δq(t)xλ(g(t))=0,t≥t0>0, where n≥2 is even or odd and δ=±1. Here q(t)>0, g(t)≥t, and λ is the ratio of odd positive integers. The results are illustrated with an example.

Differential Equations in Banach Algebras

In a complex Banach algebra that is not assumed to be commutative, nth-order linear differential equations with constant coefficients are considered. The corresponding algebraic characteristic equation of the nth degree is assumed to have n distinct roots for which the Vandermonde matrix is invertible. Analogues of Sylvester’s and Vieta’s theorems are proved, and a contour integral of Cauchy type is studied.

Hyers-Ulam stability of n ͭ ͪ order linear differential equation

In this paper, we investigate the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of the homogeneous linear differential equation of nth order with initial and boundary conditions by using Taylor’s Series formula.

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.

A collocation method for differential equations with one‐point Taylor initial conditions

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m−m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(hm). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.

Solving Differential Equation by Modified Genetic Algorithms

Differential equation is a mathematical equation which contains the derivatives of a variable, such as the equation which represent physical quantities, In this paper we introduced modified on the method which proposes a polynomial to solve the ordinary differential equation (ODEs) of second order and by using the evolutionary algorithm to find the coefficients of the propose a polynomial [1] . Our method propose a polynomial to solve the ordinary differential equations (ODEs) of nth order and partial differential equations(PDEs) of order two by using the Genetic algorithm to find the coefficients of the propose a polynomial ,since Evolution Strategies (ESs) use a string representation of the solution to some problem and attempt to evolve a good solution through a series of fitness –based evolutionary steps .unlike (GA) ,an ES will typically not use a population of solution but instead will make a sequence of mutations of an individual solution ,using fitness as a guide[2] . A numerical example with good result show the accuracy of our method compared with some existing methods .and the best error of method it’s not much larger than the error in best of the numerical method solutions.

On the Asymptotics of Solutions of a Singular nth-Order Differential Equation with Nonregular Coefficients

On the Asymptotics of Solutions of a Singular nth-Order Differential Equation with Nonregular Coefficients

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.