Solutions Of Higher-order Differential Equations Research Articles

Romanovski-Jacobi spectral schemes for high-order differential equations

We develop direct solution techniques for solving high-order differential equations with constant coefficients using the spectral tau method. The spatial approximation is based on Romanovski-Jacobi polynomials {Rnα,β(x)}n=0N with α>−1, β<−2N−α−1, x∈(0,∞) and n is the polynomial degree. Then, a hybrid approach combining the Romanovski-Jacobi tau method with the Romanovski-Jacobi collocation technique is introduced for the numerical solution of high-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces the problems to those of solving systems of algebraic equations, which greatly simplifies the problem. Finally, the Romanovski-Jacobi collocation method is presented for solving nonlinear high-order initial value problems. Numerical examples are included to demonstrate the validity and applicability of the techniques, and comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

Oscillatory Behavior of Higher-Order Differential Equations with Delay Terms

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two methods for establishing some new conditions for the oscillation of all solutions of higher-order differential equations will be presented. The first method to use the Riccati transformations which differ from those reported in some literature. The second method employs comparison principles with first-order delay differential equations which can easily deduce oscillation of all solutions of studied equations. Not only do the newly proposed criteria improve, extend, and greatly simplify the previously established criteria, but they also have the potential to act as a reference point for the theory of delay differential equations of higher order, which is still in its early stages of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. There will be some examples provided to illustrate the findings.

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.

Dynamics and Solutions of Higher-Order Difference Equations

The invariance method, known as Lie analysis, consists of finding a group of transformations that leave a difference equation invariant. This powerful tool permits one to lower the order, linearize and more importantly, obtain analytical solutions of difference and differential equations. In this study, we obtain the solutions and periodic solutions for some family of difference equations. We achieve this by performing an invariance analysis of this family. Eventually, symmetries are derived and used to construct canonical coordinates required for the derivation of the solutions. Moreover, periodic aspects of these solutions and the stability character of the equilibrium points are investigated.

Using $G_{\alpha}$-transform to Study Higher-order Differential Equations with Polynomial Coefficients

In this study, solutions of higher-order differential equations with polynomial coefficients(HODEPCs) were obtained by applying the $G_{\alpha}$-transform. Based on some characterizations, the solutions of HODEPCs were investigated. With the general solution of the HODEPCs, the curves of the general solution can be shown in several examples.

Asymptotic Properties of the Solutions of Higher-Order Differential Equations on Generalized Hölder Classes

Asymptotic Properties of the Solutions of Higher-Order Differential Equations on Generalized Hölder Classes

Inverse differential quadrature method: mathematical formulation and error analysis.

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions.

СИМВОЛИЧЕСКОЕ ПРЕДСТАВЛЕНИЕ ВЫНУЖДЕННЫХ КОЛЕБАНИЙ РАЗВЕТВЛЕННЫХ МЕХАНИЧЕСКИХ СИСТЕМ

A calculation of dynamics of a mechanical system with n degrees of freedom, including inert bodies and elastic and damping elements, involves the derivation and integration of a system of n second-order differential equations, which are reduced to a differential equation of 2n order. An increase in the degree of freedom of the mechanical system by one increases the order of the resulting differential equation by two. The solution of higher-order differential equations is rather cumbersome and time-consuming. Integration of equations is proposed to be replaced with rather simpler algebraic methods. A number of relevant theorems that relate both active and reactive parameters of mechanical systems in the series and parallel connection of mechanical power consumers are proved. Using parallel-series and series-parallel connections as an example, the calculation methods for branched mechanical systems with any number of degrees of freedom, based on the use of symbolic or complex representation of forced harmonic oscillations, are shown. The phase relationships determining loading conditions and a possibility of its artificial change are considered. The vector diagrams of the amplitudes of forces, velocities and their components in a complex plane at a zero time instant are presented, which give a complete and clear idea of the relationship between these quantities.

On the asymptotic and oscillatory behavior of the solutions of a class of higher-order differential equations with middle term

In this paper, we deal with the asymptotic and oscillatory behavior of the solutions of higher-order differential equations with middle term of the form Lw′+pνfwm−1ν+qνgwσν=0, where Lw≔rνwm−1να. By using generalized Riccati transformations we study the asymptotic behavior and derive a new oscillation criterion. The results obtained here extend and improve some well-known results which have been published recently in the literature. An example is given to illustrate the applicability of the obtained results.

New Comparison Theorems for the Nth Order Neutral Differential Equations with Delay Inequalities

In this work, we present a new technique for the oscillatory properties of solutions of higher-order differential equations. We set new sufficient criteria for oscillation via comparison with higher-order differential inequalities. Moreover, we use the comparison with first-order differential equations. Finally, we provide an example to illustrate the importance of the results.

Instability Effects in F(R)-Modified Gravity and in Gravitational Baryogenesis

The review is devoted to the development and subsequent application of methods for studying the stability of solutions of higher-order differential equations arising in various models of modified gravity and baryosynthesis. Major attention is paid to F(R)-modified theories proposed to describe the accelerated expansion of the universe and to baryon asymmetry generation in the framework of the spontaneous and gravitational baryogenesis.

A control parameter analysis method based on a transfer function matrix of hybrid multi-terminal HVDC system with flexible adaptability for different operation modes

To achieve better controller performance, the impact of control parameters on the stability and dynamic performance is essential to be clarified mathematically for hybrid multi-terminal HVDC (Hybrid-MTDC) systems considering different operation modes (OMs). In this paper, a generic small-signal model (SSM) is established, which can be modified flexibly for different OMs. Then the SSM can be converted into a transfer function matrix (TF-Matrix), which avoids deriving the transfer function separately for each controller of the system, and greatly improves the work efficiency. By inputting a step into the TF-Matrix, the time-domain mathematical model for the all system state variables can be obtained, and it is more efficient than iterative solutions of high-order differential equations. To this end, a quantitative control parameter analysis method based on a TF-Matrix is proposed. Utilizing the proposed method, a typical Hybrid-MTDC system is investigated, and the results demonstrate the practicality and accuracy of the proposed method.

Stabilization of Structured Populations via Vector Target-Oriented Control.

In contrast with unstructured models, structured discrete population models have been able to fit and predict chaotic experimental data. However, most of the chaos control techniques in the literature have been designed and analyzed in a one-dimensional setting. Here, by introducing target-oriented control for discrete dynamical systems, we prove the possibility to stabilize a chosen state for a wide range of structured population models. The results are illustrated with introducing a control in the celebrated LPA model describing a flour beetle dynamics. Moreover, we show that the new control allows to stabilize periodic solutions for higher-order difference equations, such as the delayed Ricker model, for which previous target-oriented methods were not designed.

Numerical solution of high-order differential equations by using periodized Shannon wavelets

In this paper, periodized Shannon wavelets are applied as basis functions in solution of the high-order ordinary differential equations and eigenvalue problem. The first periodized Shannon wavelets are defined. The second the connection coefficients of periodized Shannon wavelets are related by a simple variable transformation to the Cattani connection coefficients. Finally, collocation method is used for solving the high-order ordinary differential equations and eigenvalue problem. Some equations are solved in order to find out advantage of such choice of the basis functions.

State Maps from Integration by Parts

We develop a new approach to the construction of state vectors for linear time-invariant systems described by higher-order differential equations. The basic observation is that the concatenation of two solutions of higher-order differential equations results in another (weak) solution once their remainder terms resulting from (repeated) integration by parts match. These remainder terms can be computed in a simple and efficient manner by making use of the calculus of bilinear differential forms and two-variable polynomial matrices. Factorization of the resulting two-variable polynomial matrix defines a state map, as well as a state map for the adjoint system. Minimality of these state maps is characterized. The theory is applied to three classes of systems with additional structure, namely self-adjoint Hamiltonian, conservative port-Hamiltonian, and time-reversible systems. For the first two classes it is shown how the factorization leading to a (minimal) state map is equivalent to the factorization of another two-variable polynomial matrix, which is immediately derived from the external system characterization, and defines a symplectic, respectively, symmetric, bilinear form on the minimal state space.

Remarks on multiple symmetric positive solutions of higher-order difference equations with Lidstone boundary conditions

This paper is concerned with the existence of multiple symmetric positive solutions for higher-order difference equations with Lidstone boundary conditions. The main contribution is correcting the mistake on the definition of the Green's functions in related papers about this subject and giving the correct results for mentioned Boundary Value Problems (BVPs).

Existence of periodic solutions of higher-order differential equations

We established some sufficient conditions for the existence of solutions of high-order periodic boundary value problem for n ≥ 2 ( E ) u ( n ) ( t ) + D ( t , u ( t ) , … , u ( n − 2 ) ( t ) ) u ( n − 1 ) ( t ) + g ( t , u ( t ) , … , u ( n − 2 ) ( t ) ) = h ( t ) , for t ∈ [ 0 , T ] , ( BC ) { u ( i ) ( 0 ) = 0 , i = 0 , 1 , 2 , … , n − 3 , ( BVP ) u ( n − 2 ) ( 0 ) = u ( n − 2 ) ( T ) , u ( n − 1 ) ( 0 ) = u ( n − 1 ) ( T ) , where h ε L 1(0, T), D ε C([0, T]× R n−1, R), and g:[0, T] × R n−1 → R be a Carathéodory function, T-periodic in the first variable.

A note on the numerical solution of high-order differential equations

Numerical solution of high-order differential equations with multi-boundary conditions is discussed in this paper. Motivated by the discrete singular convolution algorithm, the use of fictitious points as additional unknowns is proposed in the implementation of locally supported Lagrange polynomials. The proposed method can be regarded as a local adaptive differential quadrature method. Two examples, an eigenvalue problem and a boundary-value problem, which are governed by a sixth-order differential equation and an eighth-order differential equation, respectively, are employed to illustrate the proposed method.

Unbounded positive solutions of higher-order difference equations with singular nonlinear term

In this paper, several necessary and/or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form Δ mX n-1+σf(n,x n−τ)=0, nεN 0 , where m ≥ 2, σ = +1, or σ = −1, and f : N 0 × R → R is a real-valued continuous function such that f( n, x) ≥ 0 for all ( n, x) ∈ N 0 × (0, ∞), where N 0 = (0, 1,) ….

Oscillation of higher-order nonlinear difference equations of neutral type

We shall establish some new criteria for the oscillation of all solutions of higher-order difference equations of the form δ m(x n-x n-r)+q nf(x n-g=0, m⩾1