Third-order ODE Research Articles

Impact of viscous dissipation on MHD flow of Maxwell nanofluid across a linear stretching sheet

The viscous dissipation of Maxwell nanofluid movement on a stretching sheet is examined in the present article. Constructed the modelling equalities with assumptions and emerging parameters such as Magnetic parameters, thermophoresis, Brownian motion, and Biot number etc. Convert those equation to simple third-order ODEs by applying stream functions, MATHEMATICA software used to solve ODE by applying the RK method approach with shooting technique. Presented our outcomes graphically by incorporating the parameters. The elasticity of the Maxwell fluid is directly proportional to temperature, resulting in a reduction in its velocity. The factors that affect fluid flow include viscous dissipation, Lewis number, Brownian motion, and the reversibility of temperature and velocity. Increasing the parameters of Brownian motion leads to significant movement of the nanofluid particles, which in turn increases their kinetic energy and enhances heat generation in the boundary layer. The Lorentz force, which hinders the movement of fluid, leads to a decrease in velocity profiles. This is determined by the magnetic parameter. The heat transfer rate, the velocity decay rate, and the occurrence of viscous dissipation are all occurring in close proximity to the sheet. Also, the model tabular validation presented and current results align well with previously published studies. Cancer treatment and the cooling process in industries, polymer processing, biotechnology and medicine, food industry, cosmetics, oil and gas, textiles, aerospace and automotive, and construction are just a few of the technical and biological uses for it. Industries may enhance material performance, process efficiency, and product quality in a variety of applications by using Maxwell fluid models.

Time-varying stretching velocity analysis for an unsteady flow of Williamson fluid by Hermite wavelet

This study investigates unsteady velocity Uw=ξx/t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${U}_{w}=\\xi x/t$$\\end{document} for a Williamson nanofluid film flowing over a moving surface. This work can be used to outline the effects of an applied angled magnetic-field on liquid film flow, which occurs in numerous real-world solicitations such as coating industries for wire or sheet, labs, painting, and several others. Analyzing williamson nanoliquid film flow over a stretching sheet is the main aim of this investigation. The leading Navier–Stokes models are reduced to third-order nonlinear ODE through similarity transformations that are then undertaken using the Hermite wavelet method (HWM). Both 2-dimensional and axisymmetric film flow circumstances have been analyzed. The moving surface parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi$$\\end{document} is said to have a limited range for which the solution exists. Specifically, ξ≤-1/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\le -1/4$$\\end{document} for axisymmetric flow and ξ≥-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\ge -1/2$$\\end{document} for two-dimensional flow. Before decreasing to the boundary condition, the velocity climbs until it reaches its maximum. By taking into account the stretching (ξ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi >0$$\\end{document}) and shrinking (ξ<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi <0$$\\end{document}) wall conditions, streamlines are also examined for axisymmetric and 2-dimensional flow patterns.

Interaction Characteristics of the Riemann Wave Propagation in the (2+1)-Dimensional Generalized Breaking Soliton System

In this study, the nonlinear (2+1)-dimensional generalized breaking soliton system has been studied by introducing an appropriate traveling wave transformation. The model under consideration is integrable with constant coefficients and demonstrates Riemann wave propagation characteristics, it principally transformed into a third-order nonlinear ODE. The reliable and robust method, namely the modified exponential function method, is applied to the nonlinear system for the first time. The main goal is to investigate and obtain some explicitly exact traveling waves, periodic waves, and soliton solutions. The obtained solutions are in the form of exponential functions, trigonometric hyperbolic functions, and combined structures of the trigonometric hyperbolic with logarithmic functions. Furthermore, the obtained solutions are new and significant in revealing the pertinent features of the physical phenomenon. The results have been expressed in several graphs, including two- and three-dimensional plots, for the best visual assessment of the physical significance and dynamic characteristics. The most potent and efficacious tools are the computer software packages we utilize to derive solutions and graphs.

Exponentially fitted Diagonally Implicit three-stage fifth-order RK Method for Solving ODEs.

The EDITRK5 method. which this paper derives. is an exponentially fitted diagonally implicit RK method for solving ODEs with the equation With the help of the set functions and for exponentially fitting problems. this strategy is designed to integrate precise initial value problems (IVPs). The primary frequency of the issue. is used to increase the method's accuracy. The new approach For the purpose of solving IVPs using exponential functions as solutions. EDITRK5 is a novel three-stage five-order exponentially-fitted diagonally implicit method. When the same issue is reduced to the first-order framework of equations. which can be solved using traditional RK approaches. different forms of third-order ODEs must be constructed using the new system. and numerical comparisons must be made. The numerical results demonstrate that the new strategy is more effective than methods that have already been published.

Generalized Euler and Runge-Kutta methods for solving classes of fractional ordinary differential equations

A third-order fractional ordinary differential equation (FrODE) is very important in the mathematical modelling of physical problems. Generally, the third-order ODE is solved by converting the differential equation to a system of first-order ODEs. However, it is a lot more efficient in terms of accuracy, a number of function evaluations as well as computational time if the problem can be solved directly using numerical methods. In this paper, we are focused on the derivation of the direct numerical methods which are one, two and three-stage methods for solving third-order FrODEs. The RKD methods with two- and three stages for solving third-order ODEs are adapted for solving special third-order FrDEs. Numerical examples have been evaluated to show the effectiveness of the new methods compared with the analytical method. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed methods. Applications of proposed methods are also presented which yield impressive results for the proposed and modified methods. The numerical solutions of the test problems using proposed methods agree well with the analytical solutions. From the numerical results obtained using proposed methods, we can conclude that the proposed methods in which derived or modified in this paper are very efficient.

ALGORITHM TO SOLVE ODES USING LIE SYMMETRIES AND METHODS BASED ON FIRST INTEGRALS

The algorithm is developed in Computer algebra system, open source SageMath software which helps in finding the first integrals of ODEs of order 2 that admit Lie algebra so(3) and the first integrals of third-order ODEs that admit four or more than four symmetry generators. The application of algorithm is illustrated through examples. The advantage of the present algorithm is that, it is universal as SageMath is a free open-source mathematics software system licensed under the GPL. By giving input of values of infinitesimal generators in the interactive window the user get values of Qis and Wijks as output.

Tidal Love numbers of braneworld black holes and wormholes

We study the tidal deformations of various known black hole and wormhole solutions in a simple context of warped compactification -- Randall-Sundrum theory in which the four-dimensional spacetime geometry is that of a brane embedded in five-dimensional Anti-de Sitter spacetime. The linearized gravitational perturbation theory generically reduces to either an inhomogeneous second-order ODE or a homogeneous third-order ODE of which indicial roots associated with an expansion about asymptotic infinity can be related to Tidal Love Numbers. We describe various tidal-deformed metrics, classify their indicial roots, and find that in particular the quadrupolar TLN is generically non-vanishing. Thus it could be a signature of a braneworld by virtue of its potential appearance in gravitational waveforms emitted in binary merger events.

SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY

We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the abovementioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

Determining equations for infinitesimal transformation of second and third-order ODE using algorithm in open-source SageMath

Determining equations for infinitesimal transformation of second and third-order ODE using algorithm in open-source SageMath

An algorithm to convert integrable third-order ODEs to regular higher-order equations near any movable singularities

An algorithm to convert integrable third-order ODEs to regular higher-order equations near any movable singularities

Exponentially-fitted forth-order explicit modified Runge-Kutta type method for solving third-order ODEs

In this paper exponentially-fitted explicit modified Runge-Kutta type method denoted as EFMRKT for solving y′″(x) = f(x, y, y′) is derived. The idea presented is based on the Simos and Berghe approach which exactly integrates initial value problems whose solutions are linear combinations of the set functions ewx and e−wx with w ∈ R the principal frequency of the problem. We developed the new EFMRKT three-stage fourth-order method called EFRKTG4 for solving third-order initial value problems. The numerical results indicate that EFRKTG4 method is more efficient than existing Runge-Kutta methods.

A New Integrator of Runge-Kutta Type for Directly Solving General Third-order ODEs with Application to Thin Film Flow Problem

A New Integrator of Runge-Kutta Type for Directly Solving General Third-order ODEs with Application to Thin Film Flow Problem

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

ABSTRACTThis paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially varying coefficients. Thereby, the ODE is coupled to the PDE in-domain and at the uncontrolled boundary, whereas the ODE is coupled with the latter boundary. For the state feedback design, a two-step backstepping approach is developed, which yields the conventional kernel equations and additional decoupling equations of simple form. In order to implement the state feedback controller, the design of observers for the PDE-ODE systems in question is considered, whereby anti-collocated measurements are assumed. Exponential stability with a prescribed convergence rate is verified for the closed-system pointwise in space. The resulting compensator design is illustrated for a 4 × 4 heterodirectional hyperbolic system coupled with a third-order ODE modelling a dynamic boundary condition.

First integrals and parametric solutions of third-order ODEs admitting

A complete set of first integrals for any third-order ordinary differential equation admitting a Lie symmetry algebra isomorphic to is explicitly computed. These first integrals are derived from two linearly independent solutions of a linear second-order ODE, without additional integration. The general solution in parametric form can be obtained by using the computed first integrals. The study includes a parallel analysis of the four inequivalent realizations of , and it is applied to several particular examples. These include the generalized Chazy equation, as well as an example of an equation which admits the most complicated of the four inequivalent realizations.

Interconnections between various analytic approaches applicable to third-order nonlinear differential equations.

We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ODEs. We establish an important interconnection between the extended Prelle-Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

Remarks on the Jacobi endomorphisms and the Cartan invariants of ODEs

We present new formulae for the Cartan invariants of second- and third-order ODEs in terms of the Jacobi endomorphisms.

Point invariants of third-order ODEs and hyper-CR Einstein–Weyl structures

We characterise Lorentzian three-dimensional hyper-CR Einstein–Weyl structures in terms of invariants of the associated third-order ordinary differential equations.

Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries

The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.

Mixed-mode oscillations and chaotic solutions of jerk (Newtonian) equations

We analyze jerk equations (third-order ODEs) resulting from an underlying prototypical model of mixed-mode oscillations and propose their circuit realizations in this paper. The scalar ODEs and their corresponding circuit realizations are obtained from a system of first-order ODEs with one nonlinearity (third-degree polynomial). One of the jerk equations is Newtonian as it is obtained by computing the time-derivative of the second Newton’s law x″−F/m=0 for a constant mass m and specially designed nonlinear force function F(x,x′,τ). The second jerk equation is non-Newtonian. The two circuits are op-amp RC circuits with interesting dynamical properties, including the mixed-mode and chaotic oscillations. The mixed-mode oscillations follow the rules of Farey arithmetic and the circuits’ dynamics is of a fractal nature.

Exact solutions of a third-order ODE from thin film flow using λ-symmetry method

We obtain exact implicit solutions of the third order ordinary differential equation y'''=y−n for the first time. These exact solutions are obtained using λ-symmetry reduction method, when n=(5/2) and n=(5/4). The obtained solutions are valid for the general initial conditions y(α)=γ,y′(α)=δ,y″(α)=β. It is shown that the contact angle ∅=90° when n=(5/2), and it varies with the initial conditions when n=(5/4).