Third-order Differential Equation Research Articles

Asymptotic and Oscillatory Properties of Third-Order Differential Equations with Multiple Delays in the Noncanonical Case

This paper investigates the asymptotic and oscillatory properties of a distinctive class of third-order linear differential equations characterized by multiple delays in a noncanonical case. Employing the comparative method and the Riccati method, we introduce the novel and rigorous criteria to discern whether the solutions of the examined equation exhibit oscillatory behavior or tend toward zero. Our study contributes to the existing literature by presenting theories that extend and refine the understanding of these properties in the specified context. To validate our findings and demonstrate their applicability in a general setting, we offer two illustrative examples, affirming the robustness and validity of our proposed criteria.

Asymptotic behaviour and boundedness of solutions for third-order stochastic differential equation with multi-delay

In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results.

Solvability of an Inverse Problem for an Elliptic-Type Equation

In this study, we consider an inverse problem of determining an unknown source function in the right-hand side of an elliptic equation which is ill-posed in the Hadamard sense. To investigate the solvability of the problem, we reduce it to a Dirichlet problem for a third-order partial differential equation with homogeneous boundary condition. Since the problem is linear, the proof of the uniqueness theorem is based on the Fredholm Alternative Theorem. We prove the existence of the solution to the problem by using the Galerkin method.

MHD boundary layer micropolar fluid flow over a stretching wedge surface: Thermophoresis and brownian motion effect

To investigate the consequence of thermophoresis and Brownian diffusion on convective boundary layer micropolar fluid flow over a stretching wedge-shaped surface. The effects of 
 non-dimensional parameters namely coupling constant parameter (0.01 ≤ B1 ≤ 0.05), magnetic parameter (1.0 ≤ M ≤ 15.0), Grashof number (0.3 ≤ Gr ≤ 0.9), modified Grashof number 
 (0.3 ≤ Gm ≤ 0.8), micropolar parameter (2.0 ≤ G2 ≤ 7.5), vortex viscosity constraint (0.02 ≤ G1 ≤ 0.2), Prandtl number (7.0 ≤ Pr ≤ 15.0), thermal radiation parameter (0.25 ≤ R ≤ 0.50), Brownian motion parameter (0.2 ≤ Nb ≤ 0.62), thermophoresis parameter (0.04 ≤ Nt ≤ 0.10), heat generation parameter (0.1 ≤ Q ≤ 0.5), Biot number (0.65 ≤ Bi ≤ 1.0), stretching parameter (0.2 ≤ A ≤ 0.5), Lewis number (3.0 ≤ Le ≤ 7.0), and chemical reaction parameter (0.2 ≤ K ≤ 0.7) on the steady MHD heat and mass transfer is investigated in the present study. The coupled non-linear partial differential equations are reduced into a set of non-linear ordinary differential equations employing similarity transformation. Furthermore, by using the Runge-Kutta method followed by the shooting technique, the transformed equations are solved. The main goal of this study is to investigate the numerical analysis of nanofluid flow within the boundary layer region with the effects of the microrotation parameter and velocity ratio parameter. The novelty of this paper is to propose a numerical method for solving third-order ordinary differential equations that include both linear and nonlinear terms. To understand the physical significance of this work, numerical analyses and tabular displays of the skin friction coefficient, Nusselt number, and Sherwood number are shown. The new approach of the present study contributes significantly to the understanding of numerical solutions to non-linear differential equations in fluid mechanics and micropolar fluid flow. Micropolar fluids are becoming even more of a focus due to the desire for engineering applications in various fields of medical, mechanical engineering, and chemical processing.

Oscillation criteria for third-order semi-canonical differential equations with unbounded neutral coefficients

In this paper, we investigate the oscillatory behavior of solutions to a class of third-order differential equations of the form Lz(t) + f (t)yβ (σ(t)) = 0, where Lz(t) = (p(t)(q(t)zt(t))t)t is a semi-canonical operator and z(t) = y(t) + g(t)y(τ (t)). The main idea is to convert the semi-canonical operator into canonical form and then obtain some new sufficient conditions for the oscillation of all solutions. The obtained results essentially improve and complement to the known results. Examples are provided to illustrate the main results. Mathematics Subject Classification (2010): 34C10, 34K11, 34K40. Received 19 September 2021; Accepted 20 January 2022

More Effective Criteria for Testing the Oscillation of Solutions of Third-Order Differential Equations

In the current paper, we aim to study the oscillatory behavior of a new class of third-order differential equations. In the present study, we are interested in a better understanding of the relationships between the solutions and their derivatives. The recursive nature of these relationships enables us to obtain new criteria that ensure the oscillation of all solutions of the studied equation. In comparison with previous studies, our results are more general and include models in a wider range of applications. Furthermore, our findings are also significant because no additional restrictive conditions are required. The presented examples illustrate the significance of the results.

Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation

Applying an original megaideal-based version of the algebraic method, we compute the point-symmetry pseudogroup of the dispersionless (potential symmetric) Nizhnik equation. This is the first example of this kind in the literature, where there is no need to use the direct method for completing the computation. The analogous studies are also carried out for the corresponding nonlinear Lax representation and the dispersionless counterpart of the symmetric Nizhnik system. We also first apply the megaideal-based version of the algebraic method to find the contact-symmetry (pseudo)group of a partial differential equation. It is shown that the contact-symmetry pseudogroup of the dispersionless Nizhnik equation coincides with the first prolongation of its point-symmetry pseudogroup. We check whether the subalgebras of the maximal Lie invariance algebra of the dispersionless Nizhnik equation that naturally arise in the course of the above computations define the diffeomorphisms stabilizing this algebra or its first prolongation. In addition, we construct all the third-order partial differential equations in three independent variables that admit the same Lie invariance algebra. We also find a set of geometric properties of the dispersionless Nizhnik equation that exhaustively defines it.

Lagrange formula conjugate third order differential equation

The paper considers a boundary value problem for a third order with no smooth coefficients and pure derivatives. Odds. This is due to the fact to introduce the concept of the conjugate Green's function. It is very difficult to write the form of the conjugate differential operator corresponding to equation in the Lagrange sense. Therefore, in this work, without using strict conditions smoothness under the conditions and boundedness, an explicit form is found conjugate operator since the initial-boundary value problem for integral-differential equations has been studied based on the introduction special conjugate systems in the form of an integral-algebraic equations’ system. In this article, it can be said that Green's function is considered based on Lagrange's formula for the third-order differential equation with boundary conditions and its conjugate.

Integral Representations of Partial Solutions for a Degenerate Third-Order Differential Equation

Integral Representations of Partial Solutions for a Degenerate Third-Order Differential Equation

Artificial neural network with dynamic synapse model

The purpose of this study is to develop and investigate a new short-term memory model based on an artificial neural network without short-term memory effect and a dynamic short-term memory model with astrocytic modulation. Methods. The artificial neural network is represented by a classical convolutional neural network that does not have short-term memory. Short-term memory is modeled in our hybrid model using the Tsodyks-Markram model, which is a system of third-order ordinary differential equations. Astrocyte dynamics is modeled by a mean field model of gliotransmitter concentration. Results. A new hybrid short-term memory model was developed and investigated using a convolutional neural network and a dynamic synapse model for an image recognition problem. Graphs of dependence of accuracy and error on the number of epochs for the presented model are given. The sensitivity metric of image recognition d-prime has been introduced. The developed model was compared with the recurrent neural network and the configuration of the new model without taking into account astrocytic modulation. A comparative table has been constructed showing the best recognition accuracy for the introduced model. Conclusion. As a result of the study, the possibility of combining an artificial neural network and a dynamic model that expands its functionality is shown. Comparison of the proposed model with short-term memory using a convolutional neural network and a dynamic synapse model with astrocytic modulation with a recurrent network showed the effectiveness of the proposed approach in simulating short-term memory.

Oscillatory behavior of semi-canonical third-order delay differental equations with a superlinear neutral term

A class of third-order semi-canonical differential equations with a superlinear neutral term of the type (a(t)[(b(t)(x(t) + p(t)x?(? (t))?)?]?)? + q(t)x?(?(t)) = 0 is considered. Some oscillation conditions are presented which are new in form and complement those already reported in the literature. Some examples illustrating the main results are provided.

Property $ \bar{A} $ of third-order noncanonical functional differential equations with positive and negative terms

Property $ \bar{A} $ of third-order noncanonical functional differential equations with positive and negative terms

On special solutions to the Ermakov–Painlevé XXV equation

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

Matrix factorizations for the generalized Charlier and Meixner orthogonal polynomials

The Cholesky factorization of the moment matrix is considered for the generalized Charlier and generalized Meixner discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre–Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient γn, that happen to be forms of the Painlevé deg-PV in disguise. New Laguerre–Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche.

Ultimate Boundedness of Solutions of Some System of Third-Order Nonlinear Differential Equations

Ultimate Boundedness of Solutions of Some System of Third-Order Nonlinear Differential Equations

On a boundary problem for the fourth order equation with the third derivative with respect to time

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions X (x) and Y (y). To determine X (x),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0,p], and for a Y (y) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0,q]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.

Oscillation of Third-Order Differential Equations with Advanced Arguments

The main objective of this work was to study some oscillatory and asymptotic properties of a new class of advanced neutral differential equations. Using new relations to link the solution and its corresponding function, we introduced new oscillatory criteria that aim to enhance, simplify, and complement some of current results. We provide some examples to demonstrate the significance of our results.

More Effective Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of a Class of Third-Order Functional Differential Equations

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations.

Turing-Hopf bifurcation and bi-stable spatiotemporal periodic orbits in a delayed predator-prey model with predator-taxis

When Turing-Hopf bifurcation occurs, it is naturally worth discussing the types of spatiotemporal patterns and their stabilities. In this paper, for partial functional differential equations with nonlinear diffusion, we establish the explicit formulae for the coefficients of normal form associated with Turing-Hopf bifurcation. This provides an explicit connection between the original system and the third-order ordinary differential equations (i.e. normal form) restricted on center manifold. Subsequently, with the aid of extended formulae, we discuss a delayed predator-prey model with predator-taxis and find that Turing-Hopf bifurcation leads to the emergence of a pair of stable single-peak spatially inhomogeneous steady states, a pair of stable multi-peak spatially inhomogeneous periodic orbits and so on. Also, the parameter regions in which these phenomena arise are given quantitatively. It is further shown that the pair of bistable spatially inhomogeneous periodic orbits arise due to the interaction of predator-taxis and time delay, which won't emerge in the same system without predator-taxis.

A class of single-step hybrid block methods with equally spaced points for general third-order ordinary differential equations

This study presents a class of single-step, self-starting hybrid block methods for directly solving general third-order ordinary differential equations (ODEs) without reduction to first order equations. The methods are developed through interpolation and collocation at systematically selected evenly spaced nodes with the aim of boosting the accuracy of the methods. The zero stability, consistency and convergence of the algorithms are established. Scalar and systems of linear and nonlinear ODEs are approximated to test the effectiveness of the schemes, and the results obtained are compared against other methods from the literature. Significantly, the study shows that an increase in the number of intra-step points improves the accuracy of the solutions obtained using the proposed methods.