Terms Of Nonlinear Differential Equations Research Articles

Momentum and thermal transport analysis in MHD nanofluid through the thermally heated square conduit: Finite element method

ObjectiveThe purpose of this investigation is to report the thermal performance of water-based nanofluids with two different types of hybrid nanofluids for the cooling process of the radiator inside the thermally heated square cavity under the addition of thermal radiation and magnetic field by using the finite element method. Research gapThe heat transfer analysis of MHD nanofluid with the suspension of titanium alloy Ti6A14V and aluminum alloy AA7075 in cavity flow problems under the effects of thermal radiation and thermal boundary conditions by using the Galerkin finite element method is not reported before. NoveltyIn this numerical investigation, the novel mathematical model of two types of titanium alloy Ti6A14V (class of titanium) and aluminum alloy AA7075 is proposed first time in the cavity flows under the influence of a uniform magnetic field and non-uniform thermally heated bottom wall. The momentum and energy equations are updated with terms of magnetic field and thermal radiation, respectively. The cavity is filled with AA7075- Ti6A14V/water nanofluids. Two parallel vertical walls of the cavity are set at constant temperatures (Tc) and the top one is adiabatic while the temperature at the lower wall is taken as the variable. Method of solutionThe physical model is composed in terms of nonlinear differential equations and transformed into a dimensionless form by using suitable non-dimensional quantities. The Galerkin finite element method along with iterative Newton’s Raphson method is selected to simulate the problem and obtain the numerical solution with a high degree of accuracy. The results are illustrated for fixed Prandtl number Pr=10.0 throughout the calculation, Rayleigh number 1000≤Ra≤100000, nanoparticle volume friction 0.0≤ϕ≤0.05 Hartman number 0≤Ha≤10 and thermal radiation parameter 1.0≤Ra≤10.0. Both grid independence tests and comparison tests are directed to get the accuracy of the established model. Important findingsThe computational results revealed that the aluminum alloy (AA7075) alloys nanoparticles are more productive as compared to titanium (Ti6Al4V) alloy nanoparticles in terms of enhancing the thermal features of water-based nanofluid. The aluminum alloy nanofluid and titanium alloy nanofluid provide 3.9 % & 5.7 % and 2.9 % & 4.2 % more heat transfer at the bottom and side walls as compared to regular fluid. ApplicationsThe current nanofluid models are more efficient than the conventional model to control the heat transfer rate and flow phenomena. The computational results revealed that if the radiator is assembled by using the modern concept of hybrid nanofluid then that will increase the heat transfer of the engine and enhance the lifespan of the machinery.

Elastic Responses of a Composite Shell Structure Subjected to Impact Loading

Received 17 December 2019; accepted 17 June 2020
 This work investigated the elastic responses of a composite laminate shell subjected to a transverse low-velocity impact. The governing equation based on the equations of motion of both the impactor and target was developed to detetrmine the impact force. The displacement of the shell subjected to unit impulse loading was solved using the finite element method. A non-linear differential equation in terms of the indentation depth was derived by incorporating the Hertzian contact law and theory of convolution. Runge-Kutta method was employed to solve the non-linear integro-differential equation, leading to the determination of the impact force at the point of contact between the impactor and the composite shell. The elastic responses including the displacement and stress of the composite laminate shell were evaluated using the finite element method by exerting the impact force on the apex of the composite shell. Present approach was verified with the analytical, experimental and numerical results reported in the existing literatures. The influences of stacking sequence of the composite laminate shell on the impact responses were examined through a series of parametric studies. In addition, impact responses of the spherical shells with different materials such as steel, aluminum and glass were studied.

Chebyshev collocation-optimization method for studying the Powell–Eyring fluid flow with fractional derivatives in the presence of thermal radiation

In this work, a mathematical model of the fractional-order in non-Newtonian Powell–Eyring fluid flow (PEFF) and heat transfer is presented and solved numerically. The set of nonlinear differential equations in terms of velocity, temperature which describes our proposed problem is tackled through the spectral collocation method based on Chebyshev polynomials of the third kind. This method reduces the presented model to a nonlinear system of algebraic equations. This system is constructed as a constrained optimization problem and optimized to get the unknown coefficients of the series solution. The effects of the thermal radiation, Powell–Eyring parameters; suction parameter and Prandtl number on the PEFF are discussed. The numerical values of the dimensionless velocity and temperature are depicted graphically. The results show that the given procedure is an easy and efficient tool to investigate the solution of such models. Some of findings of this important work help to govern the velocity and the rate of heat transportation through the boundary layer. At the same time, this study highlights many applications in fields of engineering and industry, where the quality of the desired product depends on the rate of heat transfer, thermal radiation phenomenon, and the composition of the material used, and manufacturing processes.

Stochastic Response of Hysteresis System Under Combined Periodic and Stochastic Excitation Via the Statistical Linearization Method

AbstractA statistical linearization approach is proposed for determining the response of the single-degree-of-freedom of the classical Bouc–Wen hysteretic system subjected to excitation both with harmonic and stochastic components. The method is based on representing the system response as a combination of a harmonic and of a zero-mean stochastic component. Specifically, first, the equation of motion is decomposed into a set of two coupled non-linear differential equations in terms of the unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization method are used for the determination of the Fourier coefficients of the deterministic component, and the variance of the stochastic component, respectively. This yields a set of coupled algebraic equations which can be solved by any of the standard apropos algorithms. Pertinent numerical examples demonstrate the applicability, and reliability of the proposed method.

Study of Injection Pulling of Oscillators in Phase-Locked Loops

The injection pulling of oscillators in phase-locked loop (PLL) circuits is analyzed by the linear approximation of nonlinear differential equations in terms of the oscillator phase. Applying this analysis to injection pulling in an injection-locked PLL (IL-PLL) and mutual injection between oscillators in a cascaded PLL, it is shown that the feedback loop becomes unstable due to the injection and the transfer function exhibits a large peak gain at the edge of the PLL bandwidth. These observations are confirmed by simulations based on linear model. It is also found that coupling from the second stage to the first stage can control the peak gain of transfer characteristics. Measurements on the test circuits of IL-PLL and cascaded PLL validate the proposed linear model. In particular, it reveals the dependence of jitter generation on the timing difference between the injection and oscillator. Additionally, in the cascaded PLL, it is shown that a properly controlled injection pulse mitigates the jitter generation caused by mutual injection pulling. This study presents important viewpoints on self and mutual interference in stand-alone and individual PLLs.

Internal resonances in nonlinear fractionally damped plates of the Uflyand-Mindlin type

In the present paper, the non-linear free vibrations of fractionally damped plates are investigated for the case when its equations of motion involve the rotary inertia and shear deformations via five coupled nonlinear differential equations in terms of three mutually orthogonal displacements and two angles of rotation. The damping features of the surrounding medium are described by the fractional detivative Kelvin-Voigt model. The procedure resulting in decoupling linear parts of governing equations has been proposed, what allows one to utulyze effectively the fractional derivative expansion method for solving the non-linear equations of motion. The occurrence of the internal or combinational resonances has been revealed and classified for plates of the Uflyand-Mindlin type.

CLASSIFICATION OF INTERNAL RESONANCES IN NONLINEAR FRACTIONALY DAMPED UFLYAND-MINDLIN PLATES

In the present paper, the nonlinear free vibrations of fractionally damped plates are studied, equations of motion of which take the rotary inertia and shear deformations into account and involve fivecoupled nonlinear differentialequations in terms of three mutually orthogonal displacements and two angles of rotation. The procedure resulting in decoupling linear parts of equations has been proposed with further utilization of the generalized method of multiple time scales for solving nonlinear governing equations of motion, in so doing the amplitude functions have been expanded into power series in terms of the small parameter and depend on differenttime scales. The occurrence of the internal or combinational resonances in Uflyand-Mindlinplates has been revealed and classified

Nonlinear vibration of full-filled fluid corrugated sandwich functionally graded cylindrical shells

This article proposes a semianalytical approach for the nonlinear free and forced asymmetric vibration of corrugated sandwich functionally graded cylindrical shells containing fluid under harmonic radial load. The corrugation is considered with two types: trapezoidal corrugation and round corrugation. By using the Donnell shell theory, taking into account the geometrical nonlinearity of von Kármán and an equivalent model for corrugated structures, the governing equations of shell are established. The nonlinear motion equations are reduced to nonlinear differential equation in terms of time by applying the Galerkin procedure. The numerical investigations for the nonlinear dynamic response of cylindrical shells are obtained by using the fourth-order Runge–Kutta method. Numerical results show the very large effects of corrugation and fluid on the natural frequency and nonlinear vibration behavior of shells.

Analytical Solution for Nonlinear Three-Dimensional Guidance With Impact Angle and Field-of-View Constraints

An analytical 3-D guidance law with impact angle and field-of-view (FOV) constraints considering nonlinear coupled dynamics is proposed. As a stepping stone, the guidance model is transformed to a set of nonlinear differential equations in terms of the relative range variable. To meet the desired impact angles in the pitch and yaw planes, two cubic polynomials including eight coefficients are developed with respect to the relative range for creating reference line-of-sight (LOS) profiles. The unknown coefficients are explicitly solved by initial and terminal conditions on the LOS angles and LOS rates in the mutually orthogonal planes. Then, the analytical 3-D impact angle guidance (IAG) law is derived via formulating the second-order LOS dynamics of the transformed model. Moreover, the relation between the seeker's look angle and the reference LOS profiles is developed, such that the achievable impact angle set can be obtained to handle the FOV limit. Numerical simulations with comparison study and a realistic model are conducted to verify effectiveness and robustness of the guidance law. Its feasibility is additionally validated by applying it to the guidance of unmanned aerial vehicles landing on surface moving carriers.

Thermal Conductivity Equations via the Improved Adomian Decomposition Methods

Several mathematical models that explain natural phenomena are mostly formulated in terms of nonlinear differential equations. Many problems in applied sciences such as nuclear physics, engineering, thermal management, gas dynamics, chemical reaction, studies of atomic structures and atomic calculations lead to singular boundary value problems and often only positive solutions are vital. However, most of the methods developed in mathematics are used in solving linear differential equations. For this reason, this research considered a model problem representing temperature distribution in heat dissipating fins with triangular profiles using MATLAB codes. MADM was used with a computer code in MATLAB to seek solution for the problem involving constant and a power law dependence of thermal conductivity on temperature governed by linear and nonlinear BVPs, respectively, for which considerable results were obtained. A problem formulated dealing with a triangular silicon fin and more examples were solved and analyzed using tables and figures for better elaborations where appreciable agreement between the approximate and exact solutions was observed. All the computations were performed using MATHEMATICA and MATLAB.

Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations

Numerical analysis is a subject that is concerned with how to solve real life problems numerically. Numerical methods form an important part of solving differential equations emanated from real life situations, most especially in cases where there is no closed-form solution or difficult to obtain exact solutions. The main aim of this paper is to review some numerical methods for solving initial value problems of ordinary differential equations. The comparative study of the Third Order Convergence Numerical Method (FS), Adomian Decomposition Method (ADM) and Successive Approximation Method (SAM) in the context of the exact solution is presented. The methods will be compared in terms of convergence, accuracy and efficiency. Five illustrative examples/test problems were solved successfully. The results obtained show that the three methods are approximately the same in terms of accuracy and convergence in the case of first order linear ordinary differential equations. It is also observed that FS, ADM and SAM were found to be computationally efficient for the linear ordinary differential equations. In the case of the non-linear ordinary differential equations, SAM is found to be more accurate and converges faster to the exact solution than the FS and ADM. Hence, It is clearly seen that the ADM is found to be better than the FS and SAM in the case of non-linear differential equations in terms of computational efficiency.

Analysis of nonlinear thermal dynamic responses of sandwich functionally graded cylindrical shells containing fluid

Based on the classical shell theory, taking into account the nonlinear geometry of von Karman-Donnell, this article deals with the nonlinear dynamic analysis of Functionally Graded Material (Sandwich-FGM) cylindrical shells containing fluid under mechanical and thermal loads. By using the Galerkin method, the nonlinear dynamic equation is transformed into nonlinear differential equation in terms of time. The investigation of nonlinear dynamic response of sandwich-FGM cylindrical shells containing fluid is established. Numerical results show effect of temperature, fluid, geometric parameters of structure and material parameters (coefficient k) on the dynamic response of structure.

Solving inhomogeneous nonlinear differential equations in terms of orthogonal functions using the Adomian decomposition functions

Solving inhomogeneous nonlinear differential equations in terms of orthogonal functions using the Adomian decomposition functions

Bacterial gliding fluid dynamics on a layer of non-Newtonian slime: Perturbation and numerical study

Gliding bacteria are an assorted group of rod-shaped prokaryotes that adhere to and glide on certain layers of ooze slime attached to a substratum. Due to the absence of organelles of motility, such as flagella, the gliding motion is caused by the waves moving down the outer surface of these rod-shaped cells. In the present study we employ an undulating surface model to investigate the motility of bacteria on a layer of non-Newtonian slime. The rheological behavior of the slime is characterized by an appropriate constitutive equation, namely the Carreau model. Employing the balances of mass and momentum conservation, the hydrodynamic undulating surface model is transformed into a fourth-order nonlinear differential equation in terms of a stream function under the long wavelength assumption. A perturbation approach is adopted to obtain closed form expressions for stream function, pressure rise per wavelength, forces generated by the organism and power required for propulsion. A numerical technique based on an implicit finite difference scheme is also employed to investigate various features of the model for large values of the rheological parameters of the slime. Verification of the numerical solutions is achieved with a variational finite element method (FEM). The computations demonstrate that the speed of the glider decreases as the rheology of the slime changes from shear-thinning (pseudo-plastic) to shear-thickening (dilatant). Moreover, the viscoelastic nature of the slime tends to increase the swimming speed for the shear-thinning case. The fluid flow in the pumping (generated where the organism is not free to move but instead generates a net fluid flow beneath it) is also investigated in detail. The study is relevant to marine anti-bacterial fouling and medical hygiene biophysics.

A FEM approach to biomagnetic fluid flow in multiple stenosed channels

The unsteady and laminar biomagnetic fluid flow of a viscous, incompressible, Newtonian and electrically conducting fluid is numerically investigated. Specifically, the two-dimensional flow driven in an infinite channel containing multiple stenosis and subject to a spatially varying external magnetic field is considered. The numerical method is based on the use of finite element method (FEM) in spatial discretization and unconditionally stable backward finite difference scheme for the time integration, for solving coupled nonlinear differential equations in terms of stream function, vorticity and temperature. Due to nonlinearity, an iterative process is employed between the equations. The study focuses on the alteration of the flow and its temperature behaviors due to the irregular constrictions along the channel. Also, the effects of the external magnetic field, its location and its intensity on the biomagnetic fluid flow are analyzed. The numerical results are presented in terms of streamlines, equivorticity and temperature contours for several values of magnetic field intensity in irregularly and multiply stenosed channels. It is shown that the lengths of the vortices appeared due to the stenosis and magnetic source, increase with an increase in the intensity of the magnetic field and the severity of the constriction.

A method for solution of the multi-objective inverse problems under uncertainty

We describe a method to solve multi-objective inverse problems under uncertainty. The method was tested on non-linear models of dynamic series and population dynamics, as well as on the spatiotemporal model of gene expression in terms of non-linear differential equations. We consider how to identify model parameters when experimental data contain additive noise and measurements are performed in discrete time points. We formulate the multi-objective problem of optimization under uncertainty. In addition to a criterion of least squares difference we applied a criterion which is based on the integral of trajectories of the system spatiotemporal dynamics, as well as a heuristic criterion CHAOS based on the decision tree method. The optimization problem is formulated using a fuzzy statement and is constrained by penalty functions based on the normalized membership functions of a fuzzy set of model solutions. This allows us to reconstruct the expression pattern of hairy gene in Drosophila even-skipped mutants that is in good agreement with experimental data. The reproducibility of obtained results is confirmed by solution of inverse problems using different global optimization methods with heuristic strategies.

Group theoretical modeling of thermal explosion with reactant consumption

Today engineering and science researchers routinely confront problems in mathematical modeling involving nonlinear differential equations. Many mathematical models formulated in terms of nonlinear differential equations can be successfully treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of the symmetrical heating of an exothermally reacting medium with approximations to the body’s temperature distribution similar to those made by Thomas [17] and Squire [15]. The quantitative results were found to be in a good agreement with Adler and Enig in [1], where the authors were comparing the integral curves corresponding to the critical conditions for the first-order reaction. Further development of the modeling by including the critical temperature is proposed. Overall, it is shown, in particular, that the application of Lie group analysis allows one to extend the previous analytic results for the first order reactions to nth order ones.

Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.

Models from experiments: combinatorial drug perturbations of cancer cells

We present a novel method for deriving network models from molecular profiles of perturbed cellular systems. The network models aim to predict quantitative outcomes of combinatorial perturbations, such as drug pair treatments or multiple genetic alterations. Mathematically, we represent the system by a set of nodes, representing molecular concentrations or cellular processes, a perturbation vector and an interaction matrix. After perturbation, the system evolves in time according to differential equations with built-in nonlinearity, similar to Hopfield networks, capable of representing epistasis and saturation effects. For a particular set of experiments, we derive the interaction matrix by minimizing a composite error function, aiming at accuracy of prediction and simplicity of network structure. To evaluate the predictive potential of the method, we performed 21 drug pair treatment experiments in a human breast cancer cell line (MCF7) with observation of phospho-proteins and cell cycle markers. The best derived network model rediscovered known interactions and contained interesting predictions. Possible applications include the discovery of regulatory interactions, the design of targeted combination therapies and the engineering of molecular biological networks.