Third-order Nonlinear Ordinary Differential Equation Research Articles

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.

Exact traveling wave solutions for (2+1)-dimensional Konopelchenko-Dubrovsky equation by using the hyperbolic trigonometric functions methods

Abstract In this research, the extended rational sinh-cosh method and the modified extended tanh-function method for mathematically constructing traveling wave solutions to the (2+1)-dimensional integro-differential Konopelchenko-Dubrovsky evolution equation are successfully employed to obtain specific appropriate solutions for the first time. A traveling wave transformation was utilized to turn the provided model into a third-order nonlinear ordinary differential equation. Solitary and periodic wave solutions for the model under investigation are obtained in terms of various complex hyperbolic trigonometric and rational functions. Several of the aforementioned solutions have been represented by two- and three-dimensional graphics with appropriate arbitrary parameters to highlight their physical implications. Two-dimensional graphs have presented the influence of time evolution on the solution’s structures.

ELASTIC TRANSFORMATION AND ITS INVERSE TRANSFORMATION FOR SOLVING FIRST- AND THIRD-ORDER NONLINEAR VARIABLE COEFFICIENT ORDINARY DIFFERENTIAL EQUATIONS

Here elastic transformation and its inverse transformation are used to solve two kinds of first- and third-order nonlinear ordinary differential equations (ODEs) with variable coefficients. First, elastic transformations are used to upgrade and reduce the first- and third-order nonlinear ODEs to a class of second-order linear homogeneous ODEs. Then the general solutions of the first- and third-order ODEs are obtained separately by using the general solution of the obtained second-order linear homogeneous ODEs, elastic transformation, inverse transformation of elastic transformation and related operation rules. Second, the steps for solving these two kinds of ODEs are summarized. Finally, examples and parametric analysis are given to prove that it is effective, simple and feasible to solve two kinds of first- and third-order nonlinear ODEs by elastic transformation and its inverse transformation. Elastic transformation and its inverse transformation provide a new method for solving ODEs. The research in this paper expands the solvable classes of ODEs.

On the conservation laws, Lie symmetry analysis and power series solutions of a class of third-order polynomial evolution equations

In the present paper, we consider a special class of third-order polynomial evolutionary equations. These equations, via Lie theory admit the same one-parameter point transformations which leave the equations invariant. Reductions with these invariant functions lead to highly nonlinear third-order ordinary differential equations. We use a power series to establish interesting solutions to the reduced equations, whereby recurrence relations occur and convergence of the series may be tested. Finally, the conserved vectors of the class are constructed.

ON SOLUTIONS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS BASED ON ELASTIC TRANSFORMATION METHODS

We consider difficult problems in solving nonlinear ordinary differential equations with variable coefficients. We use elastic transformation methods (elastic upgrading transformation and elastic reduced transformation) to transform the first and third order equations into the associated Chebyshev equation. Then, according to the general solution of the associated Chebyshev equation, we obtain the general solutions of the first-order and third-order nonlinear ordinary differential equations with variable coefficients, giving curves of general solution. The elastic transformation method provides a new idea and expands the solvable classes for solving nonlinear ordinary differential equation with variable coefficients.

A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations

In this paper, we present a method of deriving extended Prelle-Singer method's quantifiers from Darboux Polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and its cofactors, we extract the extended Prelle-Singer method's quantities without evaluating the Prelle-Singer method's determining equations. We consider three different cases of known Darboux polynomials. In the first case, we prove the integrability of the given third-order nonlinear equation by utilizing the Prelle-Singer method's quantifiers from the two known Darboux polynomials. If we know only one Darboux polynomial, then the integrability of the given equation will be dealt as case $2$. Likewise, case $3$ discuss the integrability of the given system where we have two Darboux polynomials and one set of Prelle-Singer method quantity. The established interconnection not only helps in deriving the integrable quantifiers without solving the underlying determining equations. It also provides a way to prove the complete integrability and helps us in deriving the general solution of the given equation. We demonstrate the utility of this procedure with three different examples.

Existence theory for a third-order ordinary differential equation with non-separated multi-point and nonlocal Stieltjes boundary conditions

<abstract><p>This study develops the existence of solutions for a nonlinear third-order ordinary differential equation with non-separated multi-point and nonlocal Riemann-Stieltjes boundary conditions. Standard tools of fixed point theorems are applied to prove the existence and uniqueness of results for the problem at hand. Further, we made use of the fixed point theorem due to Bohnenblust-Karlin to discuss the existence of solutions for the multi-valued case. Lastly, we clarify the reported results by means of examples.</p></abstract>

Existence of positive solutions for third-order semipositone boundary value problems on time scales

In this paper, we consider the existence of positive solutions for a semipositone third-order nonlinear ordinary differential equation on time scales. In suitable growth conditions, by considering the properties on time scales and establishing a special cone, some new results on the existence of positive solutions are established when the nonlinearity is semipositone.

Jeffery–Hamel flow of a shear-thinning fluid that mimics the response of viscoplastic materials

The flow between two divergent plane walls with a source at the vertex (Jeffery–Hamel flow) of a shear-thinning fluid, that mimics the response of a class of seemingly viscoplastic materials, is studied. The semi-inverse approach is used to obtain the governing equations for the velocity profile. The third-order non-linear ordinary differential equation governing the flow is solved numerically, and the effects of the material moduli that characterize the fluid, the angle between plane walls, and the inertial term, on the velocity are reported and discussed. Results show that there is a critical angle beyond which flow reversal occurs, and the number and angular extent of inflow regions varies based on the material moduli and the inertial term.

Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations

<abstract><p>Aiming at the initial value problems of variable coefficient nonlinear ordinary differential equations, this paper introduces the elastic transformation method into the process of solving the initial value problems of nonlinear ordinary differential equations with variable coefficients. A class of first-order and a class of third-order nonlinear ordinary differential equations with variable coefficients can be transformed into Chebyshev equations through elastic upgrading transformation and elastic reduction transformation respectively. According to the properties of Chebyshev polynomials and the initial conditions, the solutions to the initial value problems of the original first-order and third- order differential equations can be obtained through the elastic inverse transformation, and then the curves of the solutions can be drawn. The introduction of the elastic transformation method not only provides a new idea for solving the initial value problems of nonlinear differential equations, but also expands the solvable classes of ordinary differential equations.</p></abstract>

Machine learning methods for probabilistic locked-mode predictors in tokamak plasmas

A rotating tokamak plasma can interact resonantly with the external helical magnetic perturbations, also known as error fields. This can lead to locking and then to disruptions. We leverage machine learning (ML) methods to predict the locking events. We use a coupled third-order nonlinear ordinary differential equation model to represent the interaction of the magnetic perturbation and the plasma rotation with the error field. This model is sufficient to describe qualitatively the locking and unlocking bifurcations. We explore using ML algorithms with the simulation data and experimental data, focusing on the methods that can be used with sparse datasets. These methods lead to the possibility of the avoidance of locking in real-time operations. We describe the operational space in terms of two control parameters: the magnitude of the error field and the rotation frequency associated with the momentum source that maintains the plasma rotation. The outcomes are quantified by order parameters that completely characterize the state, whether locked or unlocked. We use unsupervised ML methods to classify locked/unlocked states and note the usefulness of a certain normalization of the order parameters. Three supervised ML classifiers are used in suite to estimate the probability of locking in the region of control parameter space with hysteresis, i.e., the set of control parameters for which both locked and unlocked states can exist. The results show that a neural network gives the best estimate of the locking probability. An analogy of the present locking model with the van der Waals equation of state is also provided.

Optimal Homotopy Asymptotic Method for Investigation of Effects of Thermal Radiation, Internal Heat Generation, and Buoyancy on Velocity and Heat Transfer in the Blasius Flow

In this study, analytical examination of effects of internal heat generation, thermal radiation, and buoyancy force on flow and heat transfer in the Blasius flow over flat plate has been presented. The governing nonlinear partial differential equations of the problem are transformed into a set of coupled nonlinear third-order ordinary differential equations by the similarity variable method and have been systematically solved using the optimal homotopy asymptotic method. The main aim of the present study is to inspect the effects of various physical parameters in the flow model on velocity and heat transfer in steady two-dimensional laminar boundary layer flow with convective boundary conditions. The influences of the Grashof number, internal heat generation, the Biot number, radiation parameter, and the Prandtl number on the skin-friction coefficient, the fluid velocity profile, and temperature distribution have been determined and discussed in detail through several plots. The finding revealed that the fluid velocity and temperature delivery upsurge with snowballing in the values of the Biot number and internal heat generation parameters. The temperature profile of the fluid declines contrary to the value of the Grashof number and the Prandtl number but increases with thermal radiation. Moreover, it is found that the skin-friction coefficient and the rate of heat intensify with the Grashof number, internal heat generation, the Biot number, and thermal radiation parameter. The obtained result is likened with the previously published numerical results in a limited case of the problem and shows an excellent agreement.

Investigation of Effects of Thermal Radiation, Magnetic Field, Eckert Number, and Thermal Slip on MHD Hiemenz Flow by Optimal Homotopy Asymptotic Method

Analytical investigation of thermal radiation, Prandtl number, Eckert number, permeability parameter, magnetic field, velocity, and thermal slip effects on magnetohydrodynamic Hiemenz flow over a permeable plate with forced convection has been presented. Similarity variable conversion method has been applied to transmute the fundamental governing equations of the fluid dynamics in flow into a pair of nonlinear third-order ordinary differential equations and is analytically solved by the optimal homotopy asymptotic method (OHAM). The influences of several relevant physical parameters in the model on velocity and temperature of the fluid have been studied and analysed profoundly by use of graphs and tables. It is detected that, with mounting value of suction/blowing parameter and magnetic field parameter, the skin friction coefficient enhances. Likewise, it is seen that the Nusselt number increases with enhancing value of magnetic parameter. It is also witnessed that the velocity increases as the Eckert number, blowing/suction parameter, and permeability parameter increase, but it decays against magnetic field and velocity slip parameter. Moreover, the result reveals that the fluid temperature upsurges along with snowballing the radiant heat, magnetic field parameter, and the Eckert number. However, it descends against thermal slip parameter, Prandtl number, wall temperature exponent, and velocity slip parameter. A comparison with previous studies has been made, and the result shows an excellent agreement.

The discontinuous Galerkin method for general nonlinear third-order ordinary differential equations

In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is L2-stable and provides the optimal (p+1)-th order of accuracy for smooth solutions when using piecewise p-th degree polynomials. Moreover, we prove that the derivative of the DG solution is superclose with order p+1 toward the derivative of Gauss-Radau projection of the exact solution. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Several numerical results are provided to confirm the convergence of the proposed scheme.

Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

Differential equations are commonly used to model several engineering, science, and biological applications. Unfortunately, finding analytical solutions for solving higher-order Ordinary Differential Equations (ODEs) is a challenge. Numerical methods represent a leading candidate for solving such ODEs. This work presents an innovated adaptive technique that uses polynomials to solve linear or nonlinear third-order ODEs. The proposed technique adapts the coefficients of the polynomial to obtain an explicit analytical solution. A signed least mean square algorithm is exploited to enhance the adaptation process and decrease both computational requirements and time. The efficiency of the proposed Adaptive Polynomial Method (APM) is illustrated through six well-known examples. The proposed technique is compared with recent analytical and numerical methods to validate its effectiveness in terms of Mean Square Error (MSE) and computation time. An application in a thin film flow system is modeled to a third-order ODE. The proposed technique is compared with recent numerical and analytical methods in solving the thin film flow equation, and it achieves better results. Furthermore, the proposed technique provides an analytical solution with an increased dynamic range and much lower computational time than those of the conventional numerical methods.

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.

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.

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.

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.

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.