System Of Nonlinear Ordinary Equations Research Articles

Artificial neural network approach to simulate the impact of concentration in optimizing heat transfer rate on water-based hybrid nanofluid under slip conditions: A regression analysis

The recent era of science depends upon the efficient performance of the heat transfer rate in several engineering applications for which the role of nanofluids have a greater impact. Therefore, the impact of particle concentration in optimizing heat transfer analysis of water-based hybrid nanofluid is conducted via an artificial neural network. The combined effect of oxide nanoparticles such as MgO and TiO2 in water performs their effective role with various factors involved in the flow phenomena. The magnetized hybrid nanofluid flow over an expanding surface filled with porous material shows its noble behavior on the flow properties. Further, it is not usual to omit the effectiveness of the dissipative heat impact. Therefore, the role of viscous, Joule, and Darcy dissipation is incorporated in the energy profile for which the profiles became coupled. The designed nonlinear model invokes suitable similarity rules that give rise to a system of nonlinear ordinary equations in non-dimensional form. Afterwards, the traditional numerical approach is beneficial to conduct the simulation of the flow profiles. The optimization of the heat transfer rate analysis is obtained by the implementation of an artificial neural network (ANN) for the response of Nusselt number using various factors. The regression analysis for the embedding data shows the effectiveness of the proposed approach in this discussion. The proposed study describes the important outcomes as; an increasing slip produces a thinning in the bounding surface thickness for the case of pure fluid comparing to the nanofluid and hybrid nanofluid. Further, the data set used for the hybrid nanofluid shows an efficient fitting than the case of nanofluid.

Carbon Nanotubes (CNTs) Nanofluids Flow and Heat Transfer under MHD Effect over a Moving Surface

The 2D steady flow of CNTs nanofluids and heat transfer over the moving plate through a uniform free stream under the effect of magnetohydrodynamics (MHD) is studied. The movement of plate is presumed in the opposite or same direction. A mathematical modelling that is governed by a system of partial differential equations (PDEs) subjected to boundary conditions is transformed into a system of nonlinear ordinary equations (ODEs). An attempt at finding the expected outcomes is successfully executed by solving ODEs system using MATLAB bvp4c solver. The effect of various parameters such as magnetic field, CNTs volume friction and moving parameter on velocity and temperature profile, skin friction, Nusselt number and heat transfer rate are investigated numerically. The results are presented using tabular and graphical illustration. From the results, the increment of magnetic field into the flow will increase in both the skin friction and the heat transfer coefficient. Besides that, non-unique solutions are obtained when the plate moves in the range of and . For comparing the performance of base fluid, kerosene works better than water as a base liquid. We are also showing that SWCNTs is more effective both in the skin friction and the heat transfer coefficient compare to MWCNTs.

Effect of suction and MHD induced Navier slip flow due to a non-linear stretching/shrinking sheet

In this study, a problem of a magnetohydrodynamic (MHD) induced Navier slip flow over a non-linear stretching and shrinking sheet with the existence of suction is considered. Similarity transformation is used to transform the governing nonlinear partial differential equations into a system of nonlinear ordinary equations. The transformed ordinary differential equations are then solved by using the Shooting Method in Maple software. Dual solutions are obtained for certain governing parameters. The results obtained show that suction improves skin friction, while the slip parameter reduces shear wall stress. Moreover, it is established that the range of dual solutions for stretching sheet is smaller compared to the shrinking case.

A method to find approximate solutions of first‐order systems of nonlinear ordinary equations

We develop a one‐step matrix method in order to obtain approximate solutions of first‐order nonlinear systems and nonlinear ordinary differential equations, reducible to first‐order systems. We find a sequence of such solutions that converge to the exact solution. We apply the method to different well‐known examples and check its precision, in terms of local error, comparing it with the error produced by other methods. The advantage of the method over others widely used lies on the great simplicity of its implementation.

Dynamics modeling of multicomponent metal ions\u2019 removal onto low-cost buckwheat hulls

The process of adsorption from water solutions containing a ternary system of Cu (II), Zn (II), and Ni (II) ions onto buckwheat hulls as a biosorbent was considered. The sorption capacity for buckwheat hulls was determined in sorption equilibrium batch experiments. The sorption kinetics equation corresponding to the mechanism of metal ions with the adsorbent was assumed. A new method for modeling sorption in a packed column was presented. A system of partial differential equations describing the mass balance, due to the assumption of a properly defined variable, was transformed into a system of ordinary nonlinear equations, which enables the identification of object parameters. The sorption capacity of the sorbent, sorption isotherms, and kinetics equations were used in dynamics modeling.

Radiation and MHD Boundary Layer Stagnation-Point of Nanofluid Flow towards a Stretching Sheet Embedded in a Porous Medium: Analysis of Suction/Injection and Heat Generation/Absorption with Effect of the Slip Model

In existence of the velocity slip model, suction/injection, and heat source/sink, the boundary layer flow near a stagnation-point over a heated stretching sheet in a porous medium saturated by a nanofluid, with effect of the thermal radiation and magnetic field, has been studied. The governing system of partial differential equations was transformed into a system of nonlinear ordinary equations using the appropriate similarity transforms. Then, the obtained system has been numerically solved by the Chebyshev pseudospectral differentiation matrix (ChPDM) approach. It was found that, at some special cases, the current results are in a very good agreement with those presented in the literature. In addition, the flow velocity, surface shear stress, temperature, and concentration are strongly influenced on applying the slip model, which is, therefore, extremely important to predict the flow characteristics accurately in the nanofluid mechanics. It was proved that this velocity slip condition is mandatory and should be taken into account in nanoscale research; otherwise, false results and a spurious physical sight are to be gained. Further, it was deduced that the influence of the stream velocity and shear stress reaches very rapidly the stable manner for both cases of the velocity ratio. However, when this ratio is equal to one, the skin friction coefficient, reduced Nusselt number, and reduced Sherwood number are constant and equal to zero, 0.721082, and 3.06155, respectively. Furthermore, it was proved that the reduced Nusselt number decreases with increase of Brownian motion and thermophoresis; has a very weak effect on increasing Lewis number; increases with increase of Prandtl number; and is higher in the cases of suction, velocity ratio > 1 and heat source in comparison with injection, velocity ratio < 1, and heat sink, respectively. Moreover, the reduced Sherwood number increases with increase of Brownian motion, thermophoresis, and Lewis number; decreases with increase of Prandtl number; is higher in the cases of suction and velocity ratio > 1 in comparison with injection and velocity ratio < 1, respectively; and is approximately the same in the heat source and heat sink cases. Finally, it was shown that the most effective region for radiation effect is[0,1].

Time Domain Inverse Problems in Nonlinear Systems Using Collocation & Radial Basis Functions

In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is de- veloped in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. The duffing optimal control problem with var- ious prescribed initial and final conditions, as well as the orbital transfer Lambert's problem are solved by the proposed RBF collocation method as examples. It is shown that this method is very simple, efficient and very accurate in obtaining the solutions, with an arbitrary solution as the initial guess. Since methods such as the Shooting Method and the Pseudo-spectral Method can be unstable and require an accurate initial guess, the proposed method is advantageous and has promising ap- plications in optimal control and celestial mechanics. The extension of the present study to other optimal control problems, and other orbital transfer problems with perturbations, will be pursued in our future studies.

Water Purification from Heavy Metal Ions in a Packed Column

The process of adsorption from water solutions containing ternary system of Cu(II), Zn(II) and Ni(II) ions in time-variable conditions in a packed column with lyophilized chitosan beads is considered. The system of partial differential equations describing the adsorption column, due to the assumption of a properly defined variable is transformed into a system of ordinary nonlinear equations which enables the identification of object parameters in simple experiments. Analytical solution of that system was possible due to the assumptions made regarding the equation describing sorption kinetics. Using experimental data, the effective diffusion coefficient Deff and the sorption kinetic constants K in the real environment were identified. The sorption capacity q* was obtained in separate sorption equilibrium experiments. Using a mathematical model the influence of initial concentration and solution flow rate on the breakthrough curve was determined. The approach which is presented provides a new method for modeling of sorption in a packed column.

DYNAMIC PULL-IN INSTABILITY AND VIBRATION ANALYSIS OF A NONLINEAR MICROCANTILEVER GYROSCOPE UNDER STEP VOLTAGE CONSIDERING SQUEEZE FILM DAMPING

In this paper, a nonlinear model is used to analyze the dynamic pull-in instability and vibrational behavior of a microcantilever gyroscope. The gyroscope has a proof mass at its end and is subjected to nonlinear squeeze film damping, step DC voltages as well as base rotation excitation. The electrostatically actuated and detected microgyroscopes are subjected to coupled flexural-flexural vibrations that are related by base rotation. In order to detune the stiffness and natural frequencies of the system, DC voltages are applied to the proof mass electrodes in drive and sense directions. Nonlinear integro differential equations of the system are derived using extended Hamilton principle considering nonlinearities in curvature, inertia, damping and electrostatic forces. Afterward, the Gelerkin decomposition method is implemented to reduce partial differential equations of microgyroscope deflection to a system of nonlinear ordinary equations. By using the 4th order Runge–Kutta method, the nonlinear ordinary equations are solved for various values of damping coefficients, air pressures, base rotation and various initial gaps between the proof mass electrodes and the substrates. Results show that the geometric nonlinearity increases the dynamic pull-in voltage and also consideration of the base rotation gives an improved evaluation of the dynamic instability. It is shown that the squeeze film damping has a considerable influence on the dynamic deflection of the microgyroscopes.

A MANIFOLD-BASED EXPONENTIALLY CONVERGENT ALGORITHM FOR SOLVING NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

For solving a non-linear system of algebraic equations of the type: F(subscript i)(x(subscript j)) = 0, i, j = 1, …, n, a Newton-like algorithm is still the most popular one; however, it had some drawbacks as being locally convergent, sensitive to initial guess, and time consumption in finding the inversion of the Jacobian matrix ∂F(subscript i) /∂x(subscript j). Based-on a manifold defined in the space of (x(subscript i), t) we can derive a system of non-linear Ordinary Differential Equations (ODEs) in terms of the fictitious time-like variable t, and the residual error is exponentially decreased to zero along the path of x(t) by solving the resultant ODEs. We apply it to solve 2D non-linear PDEs, and the vector-form of the matrix-type non-linear algebraic equations (NAEs) is derived. Several numerical examples of non-linear PDEs show the efficiency and accuracy of the present algorithm. A scalar equation is derived to find the adjustive fictitious time stepsize, such that the irregular bursts appeared in the residual error curve can be overcome. We propose a future direction to construct a really exponentially convergent algorithm according to a manifold setting.

A Numerical Solution of 2D Buckley-Leverett Equation via Gradient Reproducing Kernel Particle Method

Summary Gradient reproducing kernel particle method (GRKPM) is a meshless technique which incorporates the first gradients of the function into the reproducing equation of RKPM. Therefore, in two-dimensionalspace GRKPM introducesthree types of shape functions rather than one. The robustness of GRKPM’s shape functions is established by reconstruction of a third-order polynomial. To enforce the essential boundary conditions (EBCs), GRKPM’s shape functions are modified by transformation technique. By utilizing the modified shape functions, the weak form of the nonlinear evolutionary Buckley-Leverett (BL) equation is discretized in space, rendering a system of nonlinear ordinary differential equations (ODEs). Subsequently, Gear’s method is applied for temporal discretization of the ODEs. Through numerical experiments, employment of a moderate viscosity seeks the efficacy ofthesolutionwhen thediffusionterm isimportant;moreover, applicationof a small viscosity confirms the potential of the approach for treatment of the problems involving steep gradient regions. The outcomes are verified by performing convergence tests using uniformly spaced particles. Consideration of non-uniform distribution of particles further demonstrates the virtue of the presented methodology in producing smooth profiles in the critical regions near the fronts.

Unsteady MHD convective heat and mass Transfer past a semiinfinite vertical porous plate with variable viscosity and thermal conductivity

In this article, we studied the effects of variable viscosity and thermal conductivity on an unsteady two-dimensional laminar flow of a viscous incompressible electrically conducting fluid past a semi-infinite vertical porous moving plate taking into account the mass transfer. The fluid viscosity is assumed to vary as a linear function of temperature. It is assumed that the porous plate moves with a constant velocity in the direction of fluid flow, and the free stream velocity follows the exponentially increasing small perturbation law. The governing equations for the flow are transformed into a system of non-linear ordinary differential equations by perturbation technique and are solved numerically by using the shooting method. The effects of the various parameters on the velocity, temperature and concentration profiles are presented graphically and the skin-friction coefficient, Nusselt number and Sherwood number at the plate are shown in Tables.

Several new exact solutions for a fast diffusion equation by the differential constraints of the linear determining equations

In this paper, we propose a new approach to develop several new exact solutions to a fast nonlinear diffusion equation. This approach rests on the differential constraints obtained by the method of linear determining equations. The method is capable of reducing the nonlinear diffusion equation to systems of nonlinear ordinary equations. The proposed scheme is presented in a general way so that it can be used in other diffusion processes.

Periodic and solitary wave solutions of the three-wave problem. A different approach

We consider the inverse problem for a three-wave interaction system in a manner different from Zakharovet al. and of Kaup. Our method is an adaptation of the technique due to Date to a 3 × 3 Lax pair. The analysis leads to a system of ordinary nonlinear equations for theμivariables linearizable through a suitable definition of differential on a Riemann surface. Next, in the degenerate case, when theμiare equal in pairs, we prove that such a set of equations is exactly integrable and leads to solitary solutions.

Nonlinear action of Lie groups and superposition principles for nonlinear differential equations

Recently obtained results on superposition principles for systems of ordinary nonlinear equations are reviewed. In particular the general solution of the matrix Riccati equation is expressed in terms of five particular solutions. The study of superposition laws is related to the problem of Bäcklund transformations for nonlinear partial differential equations, to the problem of transforming nonlinear equations into linear ones and to the classification of primitive filtered Lie algebras.

Thermal Convection

The differential equations which apply to slow steady convective motion of a viscous fluid are the standard equations of hydrodynamical theory, combined with equations expressing the law of thermal interchanges in a moving fluid. Approximate solutions are found following the method outlined by Oberbeck (Ann. d. Physik, 1879, p. 271). A comparative study of the exact solution of a special system of non-linear ordinary equations and the Oberbeck approximate solution shows that the magnitude of the constants in the system controls the rate of convergence of the approximation, and often produces divergence. Solutions with graphs of temperature and velocity fields have been computed for special cases of cylindrical and plane boundaries with sinusoidal impressed boundary temperatures. If this temperature has a vertical gradient only, the Oberbeck method produces a null solution, corresponding to equilibrium in stratified layers. Arithmetical solutions for common liquids at ordinary temperatures show that the Oberbeck solutions diverge unless the gradient is much too small to be easily applied in laboratory practice.