System Of Nonlinear Ordinary Differential Equations Research Articles

Mathematical modeling of the propane and propane-propylene cut thermal decomposition process

The article describes the mathematical modeling of the propane pyrolysis process. The developed model is a system of ordinary nonlinear differential equations (SONDE), which describes the changes in the concentrations of the components of the substances involved in the reaction over time. The computer implementation of the mathematical model is based on the MathWorks MATLAB programming environment. The program was used to search for the reaction optimal operating conditions: the temperature and the composition of the raw materials of the pyrolysis process.

Optimal systems and their group-invariant solutions to geodesic equations

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

Wavelet investigation of Soret and Dufour effects on stagnation point fluid flow in two dimensions with variable thermal conductivity and diffusivity

The current analysis is devoted to observing the influence of Soret and Dufour effects on stagnation point fluid flow in two dimensions. The buoyancy effects, variable thermal conductivity and diffusivity are also considered in our study. The governing equations of flow are reduced to a nonlinear ordinary differential equation system via a suitable transformation technique. The numerical simulation is performed by means of a new algorithm based on the Chebyshev wavelet method (MCWM). A detailed assessment of outcomes obtained by MCWM with fourth-order Runge–Kutta technique is available to support our wavelet-based solution. The significant properties and effects of various physical parameters are presented graphically. It is noted that an increase in the values of magnetic parameters cause a decrease in the velocity profile for the assisting case while the behavior of velocity is the opposite for the opposing case. The effects of Soret and Dufour enhanced the temperature and concentration, respectively. The comparison and convergence analysis indicate that the proposed algorithm is an efficient tool and could be extended for other complex nature models in engineering and physics.

Melting effect in MHD stagnation point flow of Jeffrey nanomaterial

Our main focus here is to investigate the melting phenomenon in magnetohydrodynamic flow of Jeffrey nanomaterial by a stretching surface. The mechanism of heat transfer is elaborated via Joule heating and viscous dissipation. Thermophoretic and Brownian motion characteristics are analyzed via the Buongiorno nanofluid model. Additionally, the chemical reaction is studied via activation energy. Further, flow is addressed in the stagnation point region. Flow field expressions (partial differential equations) are converted to ordinary differential equations (ODEs) via the implementation of adequate transformations. These coupled non-linear systems of ODEs are solved via the optimal homotopy analysis method. Velocity, skin friction coefficient, concentration, mass transfer rate (Sherwood number), temperature, and heat transfer rate (Nusselt number) are examined. Flow can be controlled through higher estimations of the Hartman number and ratio of relaxation to the retardation time parameter. The temperature of the fluid intensifies with a larger thermophoresis parameter, Eckert number, Prandtl number, Hartman number, and velocity ratio parameter. The concentration of fluid is higher for a larger estimation of the thermophoresis parameter, Brownian motion parameter, and activation energy parameter. Further, the skin friction coefficient can be reduced via a higher melting parameter, velocity ratio parameter, and the ratio of the relaxation to the retardation time parameter. The Nusselt number is an increasing function of the Deborah number and the thermophoresis parameter. The Sherwood number is larger for a higher Brownian motion parameter and reaction rate parameter.

A physically extended Lorenz system.

The Lorenz system is a simplified model of Rayleigh-Bénard convection, a thermally driven fluid convection between two parallel plates. Two additional physical ingredients are considered in the governing equations, namely, rotation of the model frame and the presence of a density-affecting scalar in the fluid, in order to derive a six-dimensional nonlinear ordinary differential equation system. Since the new system is an extension of the original three-dimensional Lorenz system, the behavior of the new system is compared with that of the old system. Clear shifts of notable bifurcation points in the thermal Rayleigh parameter space are seen in association with the extension of the Lorenz system, and the range of thermal Rayleigh parameters within which chaotic, periodic, and intermittent solutions appear gets elongated under a greater influence of the newly introduced parameters. When considered separately, the effects of scalar and rotation manifest differently in the numerical solutions; while an increase in the rotational parameter sharply neutralizes chaos and instability, an increase in a scalar-related parameter leads to the rise of a new type of chaotic attractor. The new six-dimensional system is found to self-synchronize, and surprisingly, the transfer of solutions to only one of the variables is needed for self-synchronization to occur.

Suction/injection effects on an unsteady MHD Casson thin film flow with slip and uniform thickness over a stretching sheet along variable flow properties

In this investigation the attention is given to a mathematical model of the non-Newtonian Casson liquid over an unsteady stretching sheet under the combined effects of different natural parameters with heat transfer in the presence of suction/injection phenomena. The movement of a laminar thin liquid film and associated heat transfer from a horizontal stretching surface is studied. Magnetic field is proposed perpendicular to the direction of flow, while surface tension is varied quadratically with temperature of the conducting fluid. Further, variable viscosity and thermal conductivity (linear function of temperature) of the flow are examined. The transformation allows to convert the boundary layer model to a system of nonlinear ODEs (ordinary differential equations). Analytical and numerical solutions of the resulting nonlinear ODEs are obtained by using HAM and BVP4C package. Thickness of the boundary layer is investigated by both methods for a classical selection of the unsteadiness parameter. A selection of the parameter ranges is studied for better solution of the problem. Present observation displays the joined effects of magnetic field, surface tension, suction/injection, and slippage at the boundary is to improve the thermal boundary layer thickness. Results for the heat flux (Nusselt number), skin friction coefficient, and free surface temperature are granted graphically and in a table form. Similarly, the effects of natural parameters on the velocity and temperature profiles are investigated.

Impingement jet hydrogen, air and Cu[sbnd]H2O nanofluid cooling of a hot surface covered by porous media with non-uniform input jet velocity

In this paper, a numerical investigation is performed on the flow and thermal performance of a heat sink, covered with an open cell metal foam, under the influences of uniform and non-uniform velocity impinging jets. Hydrogen, air and Cu-water nanofluid are considered as the cooling fluids. Navier-Stokes PDEs including the porous drag induced terms – represented by Darcy-Brinkman-Forchheimer relation – in line with the energy equation, are transformed to a system of ordinary differential equations (ODE) through definition of non-dimensional parameter and similarity variables. The system of non-linear ODEs has been solved numerically and results are validated by comparison with those of a commercial software. Afterward, the influences of hydrodynamic variables, porous medium properties and nanofluid volume fraction on flow and heat transfer performance of the heat sink, have been scrutinized. Results presented in terms of non-dimensional velocity and temperature profiles, as well as the stream function, velocity and temperature contours. Results indicate that increasing the volume fraction of nanofluid have increased the heat transfer rate. In addition, under the constant heat sink inlet mass flow rate, the use of non-uniform impingement jet with decreasing velocity distribution improves the thermal performance of the heat sink.

On Calculation of the First Integrals for Three-dimensional ODE Systems

The search for solutions of nonlinear stationary systems of ordinary differential equations (ODE) is sometimes very complicated. It is not always possible to obtain a general solution in an analytical form. As a consequence, a qualitative theory of nonlinear dynamical systems has been developed. Its methods allow us to investigate the properties of solutions without finding a general solution. Numerical methods of investigation are also widely used.In the case when it is impossible to find an analytically general solution of the ODE system, sometimes, nevertheless, it is possible to find its first integral. There is a number of known results that make it possible to obtain the first integral for certain special cases.The article deals with the method for obtaining the first integrals of ODE systems of the third order, based on the fact of integrability of the involutive distribution.The method proposed in the paper allows us to obtain the first integral of a nonlinear ODE system of the third order in the case when a vector field, which generates an involutive distribution of dimension 2 together with the vector field of the right-hand side of a given ODE system, is known. In this case, the solution of a certain sequence of Cauchy problems allows us to construct a level surface of the function of the first integral containing the given point of the state space of the system. Using the method of least squares, in a number of cases it is possible to obtain an analytic expression for the first integral.The article gives examples of the method application to two ODE systems, namely to a simple nonlinear third-order system and to the Lorentz system with special parameter values. The article shows how the first integrals can be obtained analytically using the method developed for the two systems mentioned above.

Robust Mean-Variance Portfolio Selection with State-Dependent Ambiguity and Risk Aversion: A Closed-loop Approach

This paper studies a class of robust mean--variance portfolio selection problems with state-dependent risk aversion. Model uncertainty, in the sense of considering alternative dominated models, is introduced to the problem to reflect the investor's uncertainty-averse preference. To characterise the robust portfolios, we consider closed-loop equilibrium control and spike variation approaches. Moreover, we show that a closed-loop equilibrium strategy exists and is unique under some technical conditions. This partially addresses open problems left in Bjork et al. (Finance Stoch. 21:331--360, 2017) and Pun (Automatica 94:249--257, 2018). By using a necessary and sufficient condition for the equilibrium, we manage to derive the analytical form of the equilibrium strategy via the unique solution to a nonlinear ordinary differential equation system. To validate the proposed closed-loop control framework, we show that when there is no uncertainty, our equilibrium strategy is reduced to the strategy in Bjork et al. (Math. Finance 24:1--24, 2014), which cannot be deduced under the open-loop control framework.

Impact of Volume Fraction and Hall Effect on Two-Phase Radiative Dusty Nanofluid Flow Over a Stretching Sheet

The behavior of electrically conducting dusty nanofluid through a stretching surface with Hall effect has been investigated through an innovative approach. Impacts of thermal radiation and heat generation/absorption are further considered for current analysis. The nanoparticles are transporter fluid and dust particles are adjourned in it. Radiative term has been involved in energy equation The fluid flow problem has been governed by Partial differential equations and transformed into ordinary ones by means of proper similarity variables. The obtained nonlinear ordinary differential equations system is then tackled homotopy analysis technique. The properties of developing factors are deliberated through diagrams and tables, and talked about in detail. The volume fraction of the dust particles, magnetic field, and mass concentration of the dust particle reduced the axial velocity while the increased Hall parameter increased the axial velocity. The temperature is increased with increased volume fraction of the dust particles.

Determination of the coefficients of nonlinear ordinary differential equations systems using additional statistical information

Abstract. The study of the clinical and epidemiological features of tuberculosis combined with HIV infection (TB + HIV) is one of the priorities in the prevention of infectious diseases and is necessary to improve the quality of medical care for patients. So this article is devoted to the mathematical model of epidemiology, its’ investigation and analysis. The previous works showed what identifiability analysis is and considered methods of performing them, such as orthogonal method, eigenvalue method and etc., for more precise clarification of model parameters. However, the choice of solving the inverse problem to restore unknown parameters is playing a huge role. So here was showed the combination of two numerical algorithms, as stochastic method of simulating annealing to determine the region of the global minimum and gradient method to determine the inverse problem in a region, of solving the inverse problem that will help to create effective treatment plan for the elimination and treatment of the disease.

Nonlinear springs with dynamic frictionless contact

This paper focuses on mathematical and numerical approaches to dynamic frictionless contact of nonlinear viscoelastic springs. This contact model is formulated by a nonlinear ordinary differential equation system and a pair of complementarity conditions. We propose three different numerical schemes in which each of them consists of several numerical methods. As a result, three groups of time-discrete numerical formulations are established. We use the coefficient of restitution to prove convergence of numerical trajectories, passing to the limit in the time step size. The Banach-fixed point theorem is applied to show the existence of global solutions satisfying all conditions. A new form of energy balance is derived, which is verified theoretically and numerically. All of the three schemes are implemented and their numerical results are compared with each other.

Thermodynamics by melting in flow of an Oldroyd-B material

This research reports heat transfer via melting in stagnation point flow of non-Newtonian fluid by a nonlinear stretchable sheet of variable thickness. An incompressible fluid with constant applied uniform magnetic field is inspected. Modeling is based on the constitutive relation of subclass of rate type materials, namely the Oldroyd-B fluid. Heat transfer process also involves the heat source/sink aspect. Nonlinear system of ODEs (ordinary differential equations) is solved via HAM (homotopy analysis method). Interval of convergence for velocity and thermal fields is explicitly determined. Velocity, temperature and Nusselt number are examined under influential variables. Intensification in flow is observed with an increment in melting and wall thickness parameters. Temperature of fluid decays with higher melting, while the opposite trend holds for wall thickness parameter. Small Nusselt number is accounted for higher melting parameter, while it intensifies with larger velocity ratio parameter.

MATHEMATICAL MODELING OF THE PROBLEM OF OPTIMAL CONTROL OF ELECTRIC POWER SYSTEMS

The problems of optimal control of nonlinear ordinary differential equations systems are considered in this paper. The considered mathematical model describes the transient processes in the electric power system. And the problem of optimal control of electric power systems is considered in more detail. Numerical experiments have shown that the control found is optimal for the given problem. Keywords: mathematical model, electric power system, optimal control.

Numerical Simulation of Sloshing Motion in a Rectangular Tank using Differential Quadrature Method

The numerical simulation of sloshing waves for Laplace equation with nonlinear free surface boundary condition in a two-dimensional (2D) rectangular tank is performed using the Differential Quadrature Method (DQM). Application of the DQM to the Laplace equation and the nonlinear free surface boundary condition gives two sets of Ordinary Differential Equations (ODEs) in time. These two sets of ODEs are coupled with each other and can be expressed as a system of nonlinear ODEs which can be further discretized in time using various time integration schemes. The resultant system of nonlinear algebraic equations can then be solved using various iterative methods. In this study, the backward difference time integration scheme (of order six) in conjunction with the Newton-Raphson method is used to solve the resultant system of nonlinear ODEs. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison study with the available solutions in the literature is performed. Numerical results reveal that the DQM can be used as an effective tool for handling nonlinear sloshing problems.

Analytic approximate solutions to the chemically reactive solute transfer problem with partial slip in the flow of a viscous fluid over an exponentially stretching sheet with suction/blowing

Abstract This paper analytically investigates the flow as well as the chemically reactive solute transfer problem in a viscous fluid. The motion equations are reduced to a nonlinear ordinary differential equations system using the similarity transformations. The obtained nonlinear differential system is for the first time approximately solved by means of the Optimal Homotopy Asymptotic Method (OHAM). The effects of the partial slip and suction/blowing parameters are analytically analyzed. Some examples are given; the obtained results provides us with a good agreement with the numerical results and reveal that our procedure is effective, accurate and easy to use.

Self-Similar Solution of Radial Stagnation Point Flow and Heat Transfer of a Viscous, Compressible Fluid Impinging on a Rotating Cylinder

In this study, the radial stagnation point flow of strain rate $$\bar{k}$$ impinging on a cylinder rotating at constant angular velocity ω and its heat transfer are investigated. Reduction in the Navier–Stokes equations and energy equation to primary nonlinear ordinary differential equation systems is obtained by use of appropriate transformations when the angular velocity and wall temperature or wall heat flux all are constant. The impinging free stream is steady and normal to the surface from all sides, and the range of Reynolds number variation ( $$Re = \bar{k}a^{2} /2\upsilon$$ ) is 0.1–1000 in which a and υ are cylinder radius and kinematic viscosity, respectively. Flow results are presented for selected values of compressibility factor and different values of Prandtl numbers along with shear stress and Nusselt number. For all values of Reynolds numbers and surface temperature or surface heat flux, as compressibility factor increases the radial velocity field, the heat transfer coefficient and the wall shear stress increase, whereas the angular velocity field decreases. The rotating movement of the cylinder does not have any effect on the radial component of the velocity, but its increase increases the angular component of the fluid velocity field and the surface shear stress.

Parameters estimation of HIV infection model of CD4+T-cells by applying orthonormal Bernstein collocation method

The HIV infection model of CD4[Formula: see text][Formula: see text]T-cells corresponds to a class of nonlinear ordinary differential equation systems. In this study, we provide the approximate solution of this model by using orthonormal Bernstein polynomials (OBPs). By applying the proposed method, the nonlinear system of ordinary differential equations reduces to a nonlinear system of algebraic equations which can be solved by using a suitable numerical method such as Newton’s method. We prove some useful theorems concerning the convergence and error estimate associated to the present method. Finally, we apply the proposed method to get the numerical solution of this model with the arbitrary initial conditions and values. Furthermore, the numerical results obtained by the suggested method are compared with the results achieved by other previous methods. These results indicate that this method agrees with other previous methods.

Three-dimensional analysis of condensation nanofluid film on an inclined rotating disk by efficient analytical methods

In this study, least square method (LSM) and differential transform method (DTM) applied to solve the three dimensional problem of steady nanofluid deposition on an inclined rotating disk is illustrated. The governing non-linear partial differential equations are reduced to the nonlinear ordinary differential equations system by similarity transform. There is a good agreement between the present analytical and numerical results. Results indicate that increasing nanofluid volume fraction leads to increase in temperature profile and highest temperature obtained for aluminium oxide (Al2O3)-water case.

Application of reaction diffusion model in Turing pattern and numerical simulation

Turing proposed a model for the development of patterns found in nature in 1952. Turing instability is known as diffusion-driven instability, which states that a stable spatially homogeneous equilibrium may lose its stability due to the unequal spatial diffusion coefficients. The Gierer-Mainhardt model is an activator and inhibitor system to model the generating mechanism of biological patterns. The reaction-diffusion system is often used to describe the pattern formation model arising in biology. In this paper, the mechanism of the pattern formation of the Gierer-Meinhardt model is deduced from the reactive diffusion model. It is explained that the steady equilibrium state of the nonlinear ordinary differential equation system will be unstable after adding of the diffusion term and produce the Turing pattern. The parameters of the Turing pattern are obtained by calculating the model. There are a variety of numerical methods including finite difference method and finite element method. Compared with the finite difference method and finite element method, which have low order precision, the spectral method can achieve the convergence of the exponential order with only a small number of nodes and the discretization of the suitable orthogonal polynomials. In the present work, an efficient high-precision numerical scheme is used in the numerical simulation of the reaction-diffusion equations. In spatial discretization, we construct Chebyshev differentiation matrices based on the Chebyshev points and use these matrices to differentiate the second derivative in the reaction-diffusion equation. After the spatial discretization, we obtain the nonlinear ordinary differential equations. Since the spectral differential matrix obtained by the spectral collocation method is full and cannot use the fast solution of algebraic linear equations, we choose the compact implicit integration factor method to solve the nonlinear ordinary differential equations. By introducing a compact representation for the spectral differential matrix, the compact implicit integration factor method uses matrix exponential operations sequentially in every spatial direction. As a result, exponential matrices which are calculated and stored have small sizes, as those in the one-dimensional problem. This method decouples the exact evaluation of the linear part from the implicit treatment of the nonlinear reaction terms. We only solve a local nonlinear system at each spatial grid point. This method combines with the advantages of the spectral method and the compact implicit integration factor method, i.e., high precision, good stability, and small storage and so on. Numerical simulations show that it can have a great influence on the generation of patterns that the system control parameters take different values under otherwise identical conditions. The numerical results verify the theoretical results.