Third-order Ordinary Differential Equation Research Articles

Determination of Nonchaotic Behavior for Some Classes of Polynomial Jerk Equations

In this work, by using an algebraic criterion presented by us in an earlier paper, we determine the conditions on the parameters in order to guarantee the nonchaotic behavior for some classes of nonlinear third-order ordinary differential equations of the form [Formula: see text] called jerk equations, where [Formula: see text] is a polynomial of degree [Formula: see text]. This kind of equation is often used in literature to study chaotic dynamics, due to its simple form and because it appears as mathematical model in several applied problems. Hence, it is an important matter to determine when it is chaotic and also nonchaotic. The results stated here, which are proved using the mentioned algebraic criterion, corroborate and extend some results already presented in literature, providing simpler proofs for the nonchaotic behavior of certain jerk equations. The algebraic criterion proved by us is quite general and can be used to study nonchaotic behavior of other types of ordinary differential equations.

Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations

In this paper, an asymptotic numerical method based on a fitted finite difference scheme and the fourth-order Runge–Kutta method with piecewise cubic Hermite interpolation on Shishkin mesh is suggested to solve singularly perturbed boundary value problems for third-order ordinary differential equations of convection diffusion type with a delay. An error estimate is derived using the supremum norm and it is of almost first-order convergence. A nonlinear problem is also solved using the Newton’s quasi linearization technique and the present asymptotic numerical method. Numerical results are provided to illustrate the theoretical results.

On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.

Numerical study of third-order ordinary differential equations using a new class of two derivative Runge-Kutta type methods

This study introduces new special two-derivative Runge-Kutta type (STDRKT) methods involving the fourth derivative of the solution for solving third-order ordinary differential equations. In this regards, rooted tree theory and the corresponding B-series theory is proposed to derive order conditions for STDRKT methods. Besides, explicit two-stages fifth order and three-stages sixth order STDRKT methods are derived and stability,consistency and convergence of STDRKT methods are analysed in details. Accuracy and effectiveness of the proposed techniques are validated by a number of various test problems and compared to existing methods in the literature.

An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method

A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

Taylor series solution for a third order boundary value problem arising in Architectural Engineering

Abstract Many architectural systems are modelled by boundary value problems. Though there are many numerical and analytical methods to solve such problems, this paper suggests a simple but effective way to the third-order ordinary differential equations by the Taylor series technology. The governing equation is differentiated to obtain high order derivatives of a certain point, and solution is expanded into a Taylor series of the point, the boundary conditions are then used to identify the unknowns involved in the approximate solution. An example is given to elucidate the solution process, the maximal error is less 2%, and its accuracy can be further improved if a higher order series is solved. The approximate solution is compared with that by the variational iteration method, showing the present method is simpler and more straightforward. The comparison with the numerical solution culminates in its reliability and effectiveness. The method can be extended to fractal calculus.

On the generalized Blasius equation

The Blasius equation is a well known third-order nonlinear ordinary differential equation, which arises in certain boundary layer problems in the fluid dynamics. In this paper we prove the existence and the uniqueness of the solution of a generalized Blasius equation using nonstandard analysis techniques.

General Solutions of Some Complex Third-order Differential Equations

According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs with entire coefficient in the neighborhood of z0.

Non-adiabatic membrane voltage fluctuations driven by two ligand-gated ion channels.

In this paper, we construct a piecewise deterministic Markov process to model the membrane voltage fluctuations driven by two ligand-gated channels. The series-solution method is applied to the third-order ordinary differential equations to get its general solutions. Also, the stationary probability density function (PDF) is just the special solution that satisfies certain boundary conditions. The bifurcation conditions of the PDF at the boundary are obtained analytically. As an application, the PDF is used to calculate the power dissipation of the ionic currents in the (nonequilibrium) steady states.

The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

This paper introduces the improved LS-SVM algorithms for solving two-point and multi-point boundary value problems of high-order linear and nonlinear ordinary differential equations. To demonstrate the reliability and powerfulness of the improved LS-SVM algorithms, some numerical experiments for third-order, fourth-order linear and nonlinear ordinary differential equations with two-point and multi-point boundary conditions are performed. The idea can be extended to other complicated ordinary differential equations.

A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock–Paper–Scissors

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models.

Existence Theory for Nonlinear Third-Order Ordinary Differential Equations with Nonlocal Multi-Point and Multi-Strip Boundary Conditions

We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely on the modern tools of functional analysis and are well illustrated with the aid of examples. An analogue problem involving non-separated integro-multi-point boundary conditions is also discussed.

Linearization criteria for two-dimensional systems of third-order ordinary differential equations by complex approach

Linearization criteria for two-dimensional systems of second-order ordinary differential equations (ODEs) have been derived earlier using complex symmetry analysis. For such systems, the linearizable form, linearization criteria and symmetry group classification are presented. In this paper, we extend the complex approach to obtain a complex-linearizable form of two-dimensional systems of third-order ODEs. This form leads to a linearizable class and linearization criteria of these systems of ODEs.

Existence theory for coupled nonlinear third-order ordinary differential equations with nonlocal multi-point anti-periodic type boundary conditions on an arbitrary domain

In this paper, we derive existence and uniqueness results for a coupled system of nonlinear third order ordinary differential equations equipped with nonlocal multi-point anti-periodic type coupled boundary conditions. Leray-Schauder alternative and Banach contraction mapping principle are the main tools of our study. Examples are constructed for illustrating the obtained results. Under appropriate conditions, our results correspond to the ones for an ant-periodic boundary value problem of nonlinear third order ordinary differential equations.

Hybrid Numerical Method with Block Extension for Direct Solution of Third Order Ordinary Differential Equations

This paper focuses on the development of a hybrid method with block extension for direct solution of initial value problems (IVPs) of general third-order ordinary differential equations. Power series was used as the basis function for the solution of the IVP. An approximate solution from the basis function was interpolated at some selected off-grid points while the third derivative of the approximate solution was collocated at all grid and off-grid points to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was implemented with five test problems including the Genesio equation to confirm its accuracy and usability. The rate of convergence (ROC) reveals that the method is consistent with the theoretical order of the proposed method. Comparison of the results with some existing methods shows the superiority of the accuracy of the method.

Numerical Stiffness Analysis for Solid Oxide Fuel Cell Real-Time Simulation Applications

Real-time simulation is important for the fuel cell online diagnostics and hardware-in-the-loop tests before industrial applications. However, it is hard to implement real-time multidimensional, multiphysical fuel cell models due to the model numerical stiffness issues. In this paper, the numerical stiffness of a tubular solid oxide fuel cell real-time model is first analyzed to identify the perturbation ranges related to the fuel cell electrochemical, fluidic, and thermal domains. Some of the commonly used ordinary differential equation (ODE) solvers are then tested for the real-time simulation purpose. At last, a novel two-stage third-order parallel stiff ODE solver is proposed to improve the stability and reduce the multidimensional real-time fuel cell model execution time. To verify the proposed model and the ODE solver, real-time simulation experiments are carried out in a common embedded real-time platform. The experimental results show that the execution speed satisfies the requirement of real-time simulation. The solver stability under strong stiffness and the high model accuracy are also validated. The proposed real-time fuel cell model and the stiff ODE solver can also help to design the online diagnostic control method.

Formulating Mathematica pseudocodes of block-Milne's device for accomplishing third-order ODEs

Formulating Mathematica pseudocodes for carrying out third-order ordinary differential equations (ODEs) is of essence necessary for proficient computation. This research paper is prepared to formulate Mathematica Pseudocodes block Milne’s device (FMPBMD) for accomplishing third-order ODEs. The coming together of Mathematica pseudocodes and proficient computing using block Milne’s device will bring about ease in ciphering, proficiency, acceleration and better accuracy. Side by side estimation and extrapolation is considered with successive function approximation gives rise to FMPBMD. This FMPBMD turns out to bring about the star local truncation error thereby finding the degree of the scheme. FMPBMD will be implemented on some numerical examples to corroborate the superiority over other block methods established by employing fixed step size and handled computation.

A series solution of the boundary value problem arising in the application of fluid mechanics

PurposeThis paper aims to study the two-dimensional steady magneto-hydrodynamic flow of a second-grade fluid in a porous channel using the homotopy perturbation method (HPM).Design/methodology/approachThe governing Navier–Stokes equations of the flow are reduced to a third-order nonlinear ordinary differential equation by a suitable similarity transformation. Analytic solution of the resulting differential equation is obtained using the HPM. Mathematica software is used to visualize the flow behavior. The effects of the various parameters on velocity field are analyzed through appropriate graphs.FindingsIt is found that x component of the velocity increases with the increase of the Hartman number when the transverse direction variable ranges from 0 to 0.2 and the reverse behavior is observed when transverse direction variable takes values between 0.2 and 0.5. It is noted that the y component of the velocity increases rapidly with the increase of the transverse direction variable. The y component of the velocity increases marginally with the increase of the Hartman number M. The effect of the Reynolds number R on the x and y components of the velocity is quite opposite to the effect of the Hartman number on the x and y components of the velocity and the effect of the parameter on the x and y components of the velocity is similar to that of the Reynolds number.Originality/valueTo the best of the author’s knowledge, nobody had tried before two-dimensional steady magneto-hydrodynamic flow of a second-grade fluid in a porous channel using the HPM.

Rational Chebyshev collocation method for the similarity solution of two dimensional stagnation point flow

In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equation of a boundary value problem with a semi-infinite domain by using similarity transformation. The rational Chebyshev method reduces this nonlinear ordinary differential equation to a system of algebraic equations. This technique is a powerful type of the collocation methods for solving the boundary value problems over a semi-infinite interval without truncating it to a finite domain. We also present the comparison of this work with others and show that the present method is more accurate and efficient.

Direct Integrators for the General Third-Order Ordinary Differential Equations with an Application to the Korteweg–de Vries Equation

We construct a new class of implicit continuous linear multistep methods (LMMs) which are used as boundary value methods for the numerical integration of the general third order initial and boundary value problems in ordinary differential equations, including the Korteweg–de Vries equation. The boundary value methods obtained from these continuous LMMs are weighted the same and are used to simultaneously generate approximate solutions to the exact solutions in the entire interval of integration. We established the convergence analysis of the methods and several numerical examples are given to show the performance of the methods.