Riccati Transformation Research Articles

On the Oscillation of Solutions of Differential Equations with Neutral Term

In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples.

Delay Differential Equations of Fourth-Order: Oscillation and Asymptotic Properties of Solutions

In this work, by using the comparison method and Riccati transformation, we obtain some oscillation criteria of solutions of delay differential equations of fourth-order in canonical form. These criteria complement those results in the literature. We give two examples to illustrate the main results. Symmetry plays an essential role in determining the correct methods for solutions to differential equations.

Study on Transfer Matrix Method for the Planar Multibody System With Closed-Loops

Abstract The conventional transfer matrix method for multibody systems (MSTMM) with closed-loops (CLs) has superiority of avoiding the global dynamics equations. However, it requires a transfer equation to link multiple-input single-output (MISO) rigid body with multihinge subset and supplement equations caused CLs. In order to simplify the deduction processing and improve the numerical stability, the Riccati transformation is introduced and the Riccati transfer matrix method for multibody systems (RMSTMM) with CLs is proposed. In a system with CLs, each CL is cut off at the connection point, and the new unknowns generated at the cutoff point are introduced into the Riccati recurrence relation. The numerical results of the conventional MSTMM and the RMSTMM are compared, and the reliability of the RMSTMM is verified. Meanwhile, the constrained Jacobian matrix is used to eliminate the constraint violation of the system. The variations of the constraint violation error are compared to validate necessarily of constraints.

New Criteria for Oscillation of Half-Linear Differential Equations with p-Laplacian-like Operators

In this paper, new oscillatory properties for fourth-order delay differential equations with p-Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our results.

Oscillation Criteria of Solutions of Fourth-Order Neutral Differential Equations

In this paper, we study the oscillation of solutions of fourth-order neutral delay differential equations in non-canonical form. By using Riccati transformation, we establish some new oscillation conditions. We provide some examples to examine the applicability of our results.

Qualitative Property of Third-Order Nonlinear Neutral Distributed-Delay Generalized Difference Equations

This paper investigates the qualitative property of third-order nonlinear neutral distributed-delay generalized difference equations. By utilizing Philos-type technique and Riccati transformation, some oscillation criteria are presented to ensure that every solution of this equation oscillates or converges to zero. To illustrate the significance of our main result, we provide a suitable example.

Oscillation of Nonlinear Fractional Dynamic Equations with Forcing Term

In this paper, interval oscillation criteria for the nonlinear damped dynamic equations with forcing terms on time scales within conformable fractional derivatives are established. Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. Also, we extend the study of oscillation to conformable fractional Euler-type dynamic equation. Examples are presented to emphasize the validity of the main theorems\enleadertwodots.

Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order

New oscillatory properties for the oscillation of unbounded solutions to a class of third-order neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.

Oscillation Criteria of Second-Order Dynamic Equations on Time Scales

In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.

Oscillation analysis of solutions of non-zero continuous linear functional equations

As a mathematical model of mechanical and electronic oscillation, the study and analysis of the oscillation characteristics of the solution of the non-zero continuous linear functional equation are of great significance in theory and practice. In view of the oscillation characteristics of the solutions of the second and third order non-zero continuous functional equations, this paper puts forward a hypothesis, studies the oscillation and asymptotics of the non-zero continuous linear functional differential equations by using the generalized Riccati transformation and the integral average technique, and establishes some new sufficient conditions for the oscillation or convergence to zero of all solutions of the equations, so as to obtain a new theorem for the solutions of the non-zero continuous linear functional equations.

Oscillation Criteria for Second Order Impulsive Delay Dynamic Equations on Time Scale

In this work, we study the oscillation of a kind of second order impulsive delay dynamic equations on time scale by using impulsive inequality and Riccati transformation technique. Some examples are given to illustrate our main results.

Oscillatory Behavior of Solutions for Certain Fractional order Difference Equations

In this paper, new oscillation criteria is obtained for solutions of fractional difference equations by using Riccati transformation technique. Some applications are also presented for the established results.

Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order

In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results.

New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation

In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form (a(ξ)((v(ξ)+b(ξ)v(ϑ(ξ)))′))′+c(ξ)G1(v(κ(ξ)))+d(ξ)G2(v(ς(ξ)))=0 under canonical and non-canonical operators, that is, ∫ξ0∞dξa(ξ)=∞ and ∫ξ0∞dξa(ξ)<∞. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results.

Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem

In this paper, we investigate a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati transformation method. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach’s fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space in an abstract setting. Some financial applications of the proposed result are presented in one-dimensional space.

Philos-Type Oscillation Results for Third-Order Differential Equation with Mixed Neutral Terms

The motivation for this paper is to create new Philos-type oscillation criteria that are established for third-order mixed neutral differential equations with distributed deviating arguments. The key idea of our approach is to use the triple of the Riccati transformation techniques and the integral averaging technique. The established criteria improve, simplify and complement results that have been published recently in the literature. An example is also given to demonstrate the applicability of the obtained conditions.

On the Oscillation Criteria for Fourth-Order p -Laplacian Differential Equations with Middle Term

In this paper, we study the oscillatory properties of the solutions of a class of fourth-order p -Laplacian differential equations with middle term. The new oscillation criteria obtained by using the theory of comparison with first- and second-order differential equations and a refinement of the Riccati transformations. The results in this paper improve and generalize the corresponding results in the literatures. Three examples are provided to illustrate our results.

Oscillation results for a certain class of fourth-order nonlinear delay differential equations

In this work, we study the oscillation of the fourth order neutral differential equations with delay argument. By means of generalized Riccati transformation technique, we obtain new oscillation criteria for oscillation of this equation. An example is given to clarify the main results in this paper.

New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

A computationally efficient modeling method for the vibration analyses of two-dimensional system structures using reduced transfer matrix method for multibody system

Thin two-dimensional plates have been widely used in mechanical structures and systems, and their natural vibration characteristics play an important role in structural design and optimization. In this paper, we develop a novel modelling method for the vibration characteristic analyses of a system having two-dimensional structure components using the Reduced Transfer Matrix Method for Multibody System (MSRTMM). In the proposed modeling method, the Transfer Matrix Method for Multibody System (MSTMM) and the Riccati transformation are integrated to achieve high computational efficiency and stability for the vibration analyses of a system with two-dimensional structures. The principle and procedures of the modeling and vibration analysis method are presented with detailed theoretical derivations. To verify the proposed modeling and vibration analysis method, numerical simulations are performed with three case studies: a rectangular thin plate vibrating in the plane, a parallelogram thin plate vibrating in the plane, and a multi-body system with two-dimensional structure elements vibrating in the plane. The simulation results by MSRTMM well agree with the counterparts by Finite Element Method (FEM). Compared with the existing methods, MSRTMM not only avoids the global dynamics equation, but also has the advantages of high computational speed, highly formalized programmability, and high numerical stability.