Coupled Differential Equations Research Articles

SWIRLING FLOW OF FIELD DEPENDENT VISCOUS FERROFLUID OVER A POROUS ROTATING DISK WITH HEAT TRANSFER

The purpose of present study is to investigate the effects of field dependent viscosity on swirling flow of an incompressible electrically non-conducting ferrofluid over a porous rotating disk with suction and heat transfer at the wall. Karman's similarity transformations are used to convert the governing boundary layer equations involved in the problem to a system of nonlinear coupled differential equations. The solution of this system is obtained by using a second-order numerical scheme which combines the features of Finite Difference method and Newton's zero finding algorithms. The flow characteristics including velocity and temperature profiles and boundary layer displacement thickness are studied for various values of MFD (magnetic field dependent) viscosity and suction parameter. Beside these, skin friction coefficients and the rate of heat transfer are also calculated on the surface of the disk. Magnetic field dependent viscosity and suction at the surface of porous rotating disk affect significantly the velocity and temperatures fields, rate of heat transfer and other flow characteristics in the generated ferrofluid boundary layer.

Coupled fractional nonlinear differential equations and exact Jacobian elliptic solutions for exciton–phonon dynamics

Abstract An improved quantum model for exciton–phonon dynamics in an α-helix is investigated taking into account the interspine coupling and the influence of power-law long-range exciton–exciton interactions. Having constructed the model Hamiltonian, we derive the lattice equations and employ the Fourier transforms to go in continuum space showing that the long-range interactions (LRI) lead to a nonlocal integral term in the equations of motion. Indeed, the non-locality originating from the LRI results in the dynamic equations with space derivatives of fractional order. New theoretical frameworks are derived, such that: fractional generalization of coupled Zakharov equations, coupled nonlinear fractional Schrodinger equations, coupled fractional Ginzburg–Landau equations, coupled Hilbert–Zakharov equations, coupled nonlinear Hilbert–Ginzburg–Landau equations, coupled nonlinear Schrodinger equations and coupled nonlinear Hilbert–Schrodinger equations. Through the F-expansion method, we derive a set of exact Jacobian solutions of coupled nonlinear Schrodinger equations. These solutions include Jacobian periodic solutions as well as bright and dark soliton which are important in the process of energy transport in the molecule. We also discuss of the impact of LRI on the energy transport in the molecule.

MHD boundary layer radiative, heat generating and chemical reacting flow past a wedge moving in a nanofluid.

The present study analyzed numerically magneto-hydrodynamics (MHD) laminar boundary layer flow past a wedge with the influence of thermal radiation, heat generation and chemical reaction. This model used for the momentum, temperature and concentration fields. The principal governing equations is based on the velocity u w(x) in a nanofluid and with a parallel free stream velocity u e(x) and surface temperature and concentration. Similarity transformations are used to transform the governing nonlinear boundary layer equations for momentum, thermal energy and concentration to a system of nonlinear ordinary coupled differential equations with fitting boundary conditions. The transmuted model is shown to be controlled by a number of thermo-physical parameters, viz. the magnetic parameter, thermal convective parameter, mass convective parameter, radiation-conduction parameter, heat generation parameter, Prandtl number, Lewis number, Brownian motion parameter, thermophoresis parameter, chemical reaction parameter and pressure gradient parameter. Numerical elucidations are obtained with the legendary Nactsheim-Swigert shooting technique together with Runge–Kutta six order iteration schemes. Comparisons with previously published work are accomplished and proven an excellent agreement.

Copper nanoparticle analysis for peristaltic flow in a curved channel with heat transfer characteristics

In this paper, the mathematical investigation is carried out to discuss the metallic nanoparticle analysis for peristaltic flow in a curved channel. The study of copper nanoparticles in the peristaltic literature with water as base fluid is not explored yet for a curved channel. The equations for the Cu-water nanofluid are formulated for the first time in the literature and simplified under the long-wavelength and low-Reynolds-number assumptions. Exact solutions of nonlinear coupled differential equations are evaluated. The results are discussed graphically for different values of important parameters.

Existence of Solutions of Boundary Value Problems for Coupled Singular Differential Equations on Whole Lines with Impulses

This paper is concerned with boundary value problems of second-order impulsive differential systems on whole lines. By using Banach spaces and nonlinear operators, together with the Schauder fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established.

A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water

In this paper, the analytical approximate traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow‐up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd.

Finite-difference time-domain (fdtd) matlab codes for first- and second-order em differential equations [testing ourselves

A set of two-dimensional (2D) electromagnetic (EM) MATLAB codes, using both first-order coupled differential (Maxwell) equations and second-order decoupled (wave) equations, are developed for both transverse-magnetic (TM) and transverse-electric (TE) polarizations. Second-order MUR type absorbing boundary conditions are used to simulate free space. Metamaterial (MTM) modeling is also included. Performance tests in terms of computational times, memory requirements, and accuracies were done for simple EM scenarios with magnetic field, current, and voltage comparisons. The codes may be used for teaching and research purposes.

Exact Travelling Wave Solutions of two Important Nonlinear Partial Differential Equations

Coupled nonlinear partial differential equations describing the spatio-temporal dynamics of predator-prey systems and nonlinear telegraph equations have been widely applied in many real world problems. So, finding exact solutions of such equations is very helpful in the theories and numerical studies. In this paper, the Kudryashov method is implemented to obtain exact travelling wave solutions of such physical models. Further, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviour. The results reveal that the Kudryashov method is very simple, reliable, and effective, and can be used for finding exact solution of many other nonlinear evolution equations.

Ginzburg-Landau Equations Reduced from Coupled Delay Differential Equations

Ginzburg-Landau Equations Reduced from Coupled Delay Differential Equations

Application of Adomian decomposition method for micropolar flow in a porous channel

In this study the injective micropolar flow in a porous channel is investigated. The flow is driven by suction or injection on the channel walls, and the micropolar model is used to describe the working fluid. This problem is mapped into the system of nonlinear coupled differential equations by using Berman׳s similarity transformation. These are solved for large mass transfer via Adomian decomposition method (ADM). Also the numerical method is used for the validity of this analytical method and excellent agreement is observed between the solutions obtained from ADM and numerical results. Trusting this validity, effects of some other parameters are discussed. It can be seen that increasing in the value of N1 have different results in comparison with N2.

Standing electromagnetic solitons in hot ultra-relativistic electron-positron plasmas

Using a one-dimensional self-consistent fluid model, we investigate standing relativistic bright solitons in hot electron-positron plasmas. The positron dynamics is taken into account. A set of nonlinear coupled differential equations describing the evolution of electromagnetic waves in fully relativistic two-fluid plasma is derived analytically and solved numerically. As a necessary condition for the existence of standing solitons the system should be relativistic. For the case of ultra-relativistic plasma, we investigate non-drifting bright solitary waves. Detailed discussions of the acceptable solutions are presented. New single hump non-trivial symmetric solutions for the scalar potential were found, and single and multi-nodal symmetric and anti-symmetric solutions for the vector potential are presented. It is shown that for a fixed value of the fluid velocity excited modes with more zeros in the profile of the vector potential show a higher magnitude for the scalar potential. An increase in the plasma fluid velocity also increases the magnitude of the scalar potential. Furthermore, the Hamiltonian and the first integral of the system are given.

Influence of ponderomotive force on the microwave and plasma interaction in an elliptical waveguide

The interaction effect of a high-power microwave with the plasma in an elliptical waveguide taking into account the ponderomotive force is presented. Here, we assume the fundamental mode that propagates in an evacuated elliptical waveguide and encounters a plasma, which is filled in another elliptical waveguide of the same size. Here, we consider a balance between the effects of ponderomotive force and the electron pressure and consider the plasma effect through its dielectric permittivity because the electron density distribution of the plasma is modified. The propagation of the mode is described by two nonlinear coupled differential equations obtained using the Maxwell's equations. These equations are solved numerically using fourth order Runge-Kutta method for the field amplitude of the microwave in the waveguide considering the waveguide to be made up of a perfect conductor and filled with homogeneous plasma density distribution. The effects of the electron temperature, the microwave filed, and the frequency on the perturbed density profile are studied.

Possessions of Chemical Reaction on MHD Heat and Mass Transfer Nanofluid Flow on a Continuously Moving Surface

A two-dimensional steady flow of an electrically conducting, viscous incom pressible nanofluid past a continuously moving surface is considered in the presence of uniform transverse magnetic field with chemical reaction. A mathematical governing model has developed for the momentum, temperature and concentration boundary layer. Similarity transformations using to modify the boundary layer equations. Whereas this prominent transformations are used to transform the principal nonlinear boundary layer equations for momentum, thermal energy and concentration to a system of nonlinear or dinary coupled differential equations with fitting boundary conditions. The coupled differential equations are numerically simulated using the famousNactsheim-Swigert shooting technique together with Runge-Kutta six order iteration schemes. Pertinent results with respect to embedded parameters are displayed graphically for the velocity, temperature and concentration profiles and were discussed quantitatively. Skin -friction, Heat transfer rate (Nusselt number) and mass transfer rate (Sherwood number) are il lustrated for the various important parameters entering into the problem separately are discussed with the help of graphs. Finally for the accuracy of the present results a c omparison with previously published research work are accomplished and proven an e xcellent agreement.

Heat Generation Effects on Unsteady Mixed Convection Flow from a Vertical Porous Plate with Induced Magnetic Field

The unsteady mixed convection flow through an impulsively started vertical porous plate with heat generation, thermal radiation, chemical reaction, induced magnetic field, time dependent suction velocity, thermal diffusion, constant heat and mass fluxes have been analysed numerically. The boundary layer equations are transformed into dimensionless nonlinear ordinary coupled differential equations with appropriate transformations. The coupled differential equations are numerically simulated byNactsheim-Swigert shooting technique together with Runge-Kutta six order iteration schemes. Numerical results are presented for velocity, induced magnetic field, temperature and concentration distributions with different parameters entering into the analysisseparately and discussed with the help of graphs and table.

Coupled latent differential equation with moderators: Simulation and application.

Latent differential equations (LDE) use differential equations to analyze time series data. Because of the recent development of this technique, some issues critical to running an LDE model remain. In this article, the authors provide solutions to some of these issues and recommend a step-by-step procedure demonstrated on a set of empirical data, which models the interaction between ovarian hormone cycles and emotional eating. Results indicated that emotional eating is self-regulated. For instance, when people do more emotional eating than normal, they will subsequently tend to decrease their emotional eating behavior. In addition, a sudden increase will produce a stronger tendency to decrease than will a slow increase. We also found that emotional eating is coupled with the cycle of the ovarian hormone estradiol, and the peak of emotional eating occurs after the peak of estradiol. The self-reported average level of negative affect moderates the frequency of eating regulation and the coupling strength between eating and estradiol. Thus, people with a higher average level of negative affect tend to fluctuate faster in emotional eating, and their eating behavior is more strongly coupled with the hormone estradiol. Permutation tests on these empirical data supported the reliability of using LDE models to detect self-regulation and a coupling effect between two regulatory behaviors.

Stochastic Maximum Principle of Near-Optimal Control of Fully Coupled Forward-Backward Stochastic Differential Equation

This paper first makes an attempt to investigate the near-optimal control of systems governed by fully nonlinear coupled forward-backward stochastic differential equations (FBSDEs) under the assumption of a convex control domain. By Ekeland’s variational principle and some basic estimates for state processes and adjoint processes, we establish the necessary conditions for anyε-near optimal control in a local form with an error order of exactε1/2. Moreover, under additional convexity conditions on Hamiltonian function, we prove that anε-maximum condition in terms of the Hamiltonian in the integral form is sufficient for near-optimality of orderε1/2.

Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

Construction of Lyapunov Functionals for Networks of Coupled Delay Differential and Continuous-Time Difference Equations

To address various types of delays including the neutral-type arising in dynamical networks, this paper deals with coupled delay differential and continuous-time difference equations and develops stability and robustness criteria. Subsystems described by differential equations are not required to be input-to-state stable. No assumptions on network topology are made. To tackle networks in such a general formulation, this paper explicitly constructs Lyapunov-type functionals establishing stability and robustness of the overall networks. The construction requires only simple characterizations of subsystems in terms of inequalities with Lyapunov functions and instantaneous norms.

A New Coupled Fractional Reduced Differential Transform Method for the Numerical Solutions of(2+1)-Dimensional Time Fractional Coupled Burger Equations

A very new technique, coupled fractional reduced differential transform, has been implemented to obtain the numerical approximate solution of (2 + 1)-dimensional coupled time fractional burger equations. The fractional derivatives are described in the Caputo sense. By using the present method we can solve many linear and nonlinear coupled fractional differential equations. The obtained results are compared with the exact solutions. Numerical solutions are presented graphically to show the reliability and efficiency of the method.