Nth Order Linear 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.

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.

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.

The refinement-schemes-based unified algorithms for certain nth order linear and nonlinear differential equations with a set of constraints

We first present a generalized class of binary interpolating refinement schemes and their properties. Then the refinement-schemes-based unified algorithms for the solution of certain nth order linear and nonlinear differential equations with a set of constraints are presented. Moreover, several algorithms based on the refinement schemes for solving differential equations are the special cases of our algorithms.

Boundary value conditions for linear differential equations with power degenerations

On the interval [0,1] we consider the nth order linear differential equation, the coefficient of the highest derivative of which is equivalent to the power function t^{mu } when trightarrow 0. The main aim of the paper is to pose “generalized” Cauchy conditions for the given equation at the point of singularity t=0, which would be correct for any mu >0.

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.

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.

The Binomial Theorem Procured From the Solution of an ODE

SummaryWe obtain the binomial theorem as a unique solution of an nth order linear nonhomogeneous ordinary differential equation with constant coefficients and given initial conditions.

The distributional continuous fractional generalized Hankel-Clifford transformation

The main objective of this paper is to study the fractional generalized Hankel-Clifford transformation and some of their basic properties. The continuous fractional generalized Hankel-Clifford transformation, its inversion formula are also studied. The finite continuous fractional generalized Hankel-Clifford transformation is also studied in Zemanian space. Applications of the finite fractional generalized Hankel-Clifford transformation in solving generalized nth order linear non-homogeneous partial differential equation is given.

Hyers--Ulam stability of nth order linear differential equations

Hyers--Ulam stability of nth order linear differential equations

Hyers-Ulam-Rassias stability of nth order linear ordinary differential equations with initial conditions

In this paper, we investigate the stability of nth order linear ordinary differential non-homogeneous equation with initial conditions in the Hyers-Ulam-Rassias sense.

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.

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

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.

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.

Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.

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.