Third-order Nonlinear Differential Equation Research Articles

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

On the rule of trajectory structure for a third-order nonlinear difference equation using semi-cycle analysis method

On the rule of trajectory structure for a third-order nonlinear difference equation using semi-cycle analysis method

On the existence of a positive solution to a boundary value problem for a third-order nonlinear functional differential equation with an integral boundary condition at one of the ends of the segment

The article studies the existence of positive solutions on the segment [0,1] of a two-point boundary value problem for one nonlinear third-order functional differential equation with an integral boundary condition at one of the ends of the segment. Using the Go–Krasnoselsky fixed point theorem and some properties of the Green's function of the corresponding differential operator, sufficient conditions for the existence of at least one positive solution to the problem under consideration are obtained. An example is given to illustrate the results obtained.

Stochastic neuro-swarming intelligence paradigm for the analysis of magneto-hydrodynamic Prandtl–Eyring fluid flow with diffusive magnetic layers effect over an elongated surface

In recent years, the integration of stochastic techniques, especially those based on artificial neural networks, has emerged as a pivotal advancement in the field of computational fluid dynamics. These techniques offer a powerful framework for the analysis of complex fluid flow phenomena and address the uncertainties inherent in fluid dynamics systems. Following this trend, the current investigation portrays the design and construction of an important technique named swarming optimized neuro-heuristic intelligence with the competency of artificial neural networks to analyze nonlinear viscoelastic magneto-hydrodynamic Prandtl–Eyring fluid flow model, with diffusive magnetic layers effect along an extended sheet. The currently designed computational technique is established using inverse multiquadric radial basis activation function through the hybridization of a well-known global searching technique of particle swarm optimization and sequential quadratic programming, a technique capable of rapid convergence locally. The most appropriate scaling group involved transformations that are implemented on governing equations of the suggested fluidic model to convert it from a system of nonlinear partial differential equations into a dimensionless form of a third-order nonlinear ordinary differential equation. The transformed/reduced fluid flow model is solved for sundry variations of physical quantities using the designed scheme and outcomes are matched consistently with Adam's numerical technique with negligible magnitude of absolute errors and mean square errors. Moreover, it is revealed that the velocity of the fluid depreciates in the presence of a strong magnetic field effect. The efficacy of the designed solver is depicted evidently through rigorous statistical observations via exhaustive numerical experimentation of the fluidic problem.

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.

Iterative oscillation criteria of third-order nonlinear damped neutral differential equations

<p>Using comparison principles, we examine the asymptotic characteristics of a third-order nonlinear damped neutral differential equation. Our results substantially generalize numerous previously established results as well as drastically improving them. To illustrate the relevance and effectiveness of our results, we use numerical examples.</p>

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

An Algorithm for Solving Nonlinear Third-Order Differential Equations Using Exponential Spline Functions

This research dealt with the study of the boundary values associated with differential equations, which are of the non-linear and third-order type. A new algorithm was created that uses exponential spline functions to study and address boundary value problems of a general nature. We have demonstrated that the numerical approach used, which was built using exponential spline functions, gives us good and accurate results for these problems, which have been compared to existing numerical methods. We found that the proposed method is accurate and effective compared to other Spalline methods.

SOME QUALITATIVE PROPERTIES OF SOLUTIONS OF CERTAIN NONLINEAR THIRD-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAY

This study considered certain nonlinear third-order stochastic differential equations with delay. The third-order equation is reduced to an equivalent system of first-order differential equations and used to construct the desired complete Lyapunov-Krasovski\v{\i} functional. Standard conditions guaranteeing stability when the forcing term is zero, boundedness of solutions when the forcing term is non-zero, and lastly the existence and uniqueness of solutions are derived. The obtained results indicated that the adopted technique is effective in studying the qualitative behaviour of solutions. The obtained results are not only new but extend the frontier of knowledge of the qualitative behaviour of solutions of nonlinear stochastic differential with delay. Finally, two special cases are given to illustrate the derived theoretical results.

On the Size of the Safety Area around the Launch Trajectory of a Rocket

The safety zone around the flight path of a rocket is determined by the fall of debris in the case of an accidental explosion or commanded termination. The trajectory of a tumbling body in a vertical plane is determined by specifying the velocity, flight path angle and angle of attack as functions of time. This involves the lift, drag and pitching moment coefficients as functions of the angle of attack over a full circle—0 to 360 degrees—to account for the tumbling motion. The problem is reduced to a third-order non-linear differential equation for the angle of attack by using the approximation of free fall coordinates. The analytical and numerical solutions show that two types of tumbling fall are possible, one with rotation and the other with oscillation. The tumbling trajectories are plotted and discussed for a variety of initial conditions, mass and aerodynamic properties of the tumbling body.

Nonlinear third-order differential equations with distributed delay: Some new oscillatory solutions

We consider a certain class of third order nonlinear delay differential equations in this work. The results that we obtained are an improvement and extension of some results mentioned in previous literature, as the criteria we obtained are less restrictive compared to the previous results reported in literature. An example is provided to illustrate new results.

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.

Existence of multiple positive solutions for a third-order boundary value problem with nonlocal conditions

We study the existence of multiple positive solutions for a nonlinear third-order differential equation subject to various nonlocal boundary conditions. The boundary conditions that we study contain Stieltjes integral and include the special cases of m-point conditions and integral conditions. The main tool in the proof of our result is Krasnosel’skii’s fixed point theorem. To illustrate the applicability of the obtained results, we consider examples.

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 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.

Asymptotic Behavior and Oscillation of Third-Order Nonlinear Neutral Differential Equations with Mixed Nonlinearities

In this paper, we investigate the asymptotic properties of third-order nonlinear neutral differential equations with mixed nonlinearities using the comparison principle. Our results not only vastly improve upon but also broadly generalize many previously known ones. Examples demonstrating the applicability and efficacy of our results are provided.

Asymptotic behavior of solutions of the third-order nonlinear advanced differential equations

<abstract><p>The aim of this work is to study some asymptotic properties of a class of third-order advanced differential equations. We present new oscillation criteria that complete, simplify and improve some previous results. We also provide many different examples to clarify the significance of our results.</p></abstract>

On Novel Spline Function Method for Solving Third-Order Boundary Value Problems

In this paper, a numerical method is suggested for solving general a nonlinear third order boundary value problem (BVP). In this method, the given nonlinear third-order BVP will be transformed into two third-order initial value problems (IVPs), then spline function approximations are applied to both two IVP for finding the Spline solution and its derivatives up to third order of the given BVP. The study shows that the spline solution of the BVP is existent and unique, and the convergence order of the spline method is fourth with a local truncation error . The presented algorithm is designed for solving a general BVP, where it is applied to some types of nonlinear third-order differential equations. Comparisons of the results obtained by spline method with other methods show the efficiency and highly accurate of the proposed method.

Исследование в окрестности подвижной особой точки некоторого класса нелинейного дифференциального уравнения третьего порядка для комплексной области

В настоящей работе проводится исследование некоторого класса дифференциальных уравнений с подвижными особенностями. Приводится обобщение полученных ранее результатов исследования, а именно, теоремы существования и единственности решения рассматриваемого класса уравнений на комплексную область. Получена структура аналитического приближенного решения и априорные оценки погрешности. Данные оценки были оптимизированы с помощью апостериорных. Теоретические положения были протестированы с помощью численного эксперимента. In this paper, we study a certain class of differential equations with moving singularities. A generalization of the previously obtained results of the study, namely, the existence and uniqueness theorem for the solution of the considered class of equations, to the complex domain is given. The structure of the analytical approximate solution and a priori error estimates are obtained. These estimates were optimized using post hoc estimates. Theoretical provisions were tested using a numerical experiment.