Time-dependent Nonlinear Equations Research Articles

Creep and recovery behavior analysis of space mesh structures

The Schapery's nonlinear viscoelastic theory and nonlinear force-density method have been investigated to analyze the creep and recovery behaviors of space deployable mesh reflectors in this paper. Based on Schapery's nonlinear viscoelastic theory, we establish the creep and recovery constitutive model for cables whose pretensions were applied stepwise in time. This constitutive model has been further used for adjustment of cables’ elongation rigidity. In addition, the time-dependent tangent stiffness matrix is calculated by the partial differentiation of the corresponding load vector with respect to the nodal coordinate vector obtained by the nonlinear force-density method. An incremental-iterative solution based on the Newton-Raphson method is adopted for solving the time-dependent nonlinear statics equations. Finally, a hoop truss reflector antenna is presented as a numerical example to illustrate the efficiency of the proposed method for the creep and recovery behavior analysis of space deployable mesh structures.

Combined effect of suction/injection on MHD free-convection flow in a vertical channel with thermal radiation

Abstract Transient hydromagnetic free-convection and thermal radiation flow of viscous, incompressible electrically conducting fluid in the presence of magnetic field in a vertical channel formed by two infinite porous plates is analyzed. The Rosseland diffusion approximation describes the radioactive heat flux in the energy equation. The non-linear time dependent energy and momentum equations under relevant initial and boundary conditions are solved numerically using implicit finite difference. To verify the accuracy of the numerical scheme, steady state version of the problem is solved by perturbation method. A series of numerical experiments show that time required to reach steady state velocity and temperature is directly proportional to the Prandtl number of the working fluid. In the course of numerical computation, an excellent agreement was found between transient and steady state solution at large value of time. The effect of various dimensionless parameters controlling the physical situation is extensively discussed with the aid of graphs.

Axisymmetric flow and heat transfer over an unsteady stretching sheet in power law fluid

An analysis is presented for the unsteady boundary layer flow and heat transfer of power law fluid model over a radially stretching sheet. A uniform magnetic field is applied perpendicular to the direction of the flow. With the aid of new similarity transformations, the governing time dependent nonlinear boundary layer equations are converted into nonlinear ordinary differential equations. The transformed coupled ordinary differential equations are then solved analytically by homotopy analysis method (HAM) and numerically by shooting method. The influence of different parameters like, power law index n, magnetic parameter M, unsteadiness A, suction/injection S, and generalized Prandtl number Pr on velocity and temperature profiles are presented graphically and discussed. It is found that the velocity and the thermal boundary layer thickness are the decreasing function of unsteadiness parameter. The impact of the local skin friction coefficient and local Nusselt number is presented in tabular form.

Progress in many-body theory with the equation of motion method: Time-dependent density matrix meets self-consistent RPA and applications to solvable models

The Bogoliubov-Born-Green-Kirkwood-Yvon or Time-Dependent Density Matrix (TDDM) hierarchy of equations for higher density matrices is truncated at the three body level in approximating the three body correlation function by a quadratic form of two body ones, closing the equations in this way. The procedure is discussed in detail and it is shown in non-trivial model cases that the approximate inclusion of three body correlation functions is very important to obtain precise results. A small amplitude approximation of this time dependent nonlinear equation for the two body correlation function is performed (STDDM*-b) and it is shown that the one body sector of this generalised non-linear second RPA equation is equivalent to the Self-Consistent RPA (SCRPA) approach which had been derived previously by different techniques. It is discussed in which way SCRPA also contains the three body correlations. TDDM and SCRPA are tested versus exactly solvable model cases.

A Numerical Computation of a System of Linear and Nonlinear Time Dependent Partial Differential Equations Using Reduced Differential Transform Method

This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT) method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations.

MHD axisymmetric flow of power-law fluid over an unsteady stretching sheet with convective boundary conditions

This paper examines the boundary layer flow and heat transfer characteristic in power law fluid model over unsteady radially stretching sheet under the influence of convective boundary conditions. A uniform magnetic field is applied transversely to the direction of the flow. The governing time dependent nonlinear boundary layer equations are reduced into nonlinear ordinary differential equations with the help of similarity transformations. The transformed coupled ordinary differential equations are then solved analytically by homotopy analysis method (HAM) and numerically by shooting procedure. Effects of various governing parameters like, power law index n, magnetic parameter M, unsteadiness A, suction/injection S, Biot number γ and generalized Prandtl number Pr on velocity, temperature, local skin friction and the local Nusselt number are studied and discussed. It is found from the analysis that the magnetic parameter diminishes the velocity profile and the corresponding thermal boundary layer thickness.

Soliton and periodic solutions for time-dependent coefficient non-linear equation

In this article, we establish exact solutions for the generalized (3+1)-dimensional variable coefficient Kadomtsev–Petviashvili (GVCKP) equation. Using solitary wave ansatz in terms of functions and the modified sine–cosine method, we find exact analytical bright soliton solutions and exact periodic solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The effectiveness and reliability of the method are shown by its application to the GVCKP equation.

Analysis of shear banding phenomena in non-isothermal flow of fluids governed by the diffusive Johnson–Segalman model

This article investigates phenomena associated with shear banded flows of viscoelastic fluids in both pipe flow and coaxial annular flow. We in particular computationally analyze the effects of stress diffusion, non-isothermal flow conditions and annular gap size on the shear stress path selection (and the selected shear stress value) in shear banded flow of viscoelastic fluids described by the non-local Diffusive Johnson–Segalman (DJS) constitutive model. The DJS model is one of such constitutive models which allows us for a non-zero inter-facial layer between regions of high and low shear rates within the flow field. Temperature effects due to polymer orientation changes, entropic effects (resulting from stress work), conduction heat transfer, Arrhenius chemical kinetics as well as the influence of slip and related frictional heating on the wall are all accounted for. The time dependent model formulated to capture gap effect in the axial annular flow problem (under certain values of the relevant parameters), reduces to the system of equations governing a pipe flow setup. Semi-implicit finite difference methods are employed for the solution process of the coupled nonlinear time dependent partial differential equations governing the flow problem. We discuss with graphical representations the effect of temperature, stress diffusion, wall slip, suction/injection and annular gap size on the related shear rate path selection phenomena for certain material parameter values.

An efficient implicit spectral element method for time-dependent nonlinear diffusion equations by evaluating integrals at one quadrature point

We present an implicit spectral element method to approximate the solution of time-dependent nonlinear diffusion equations in complex geometries. We propose a nodal expansion to approximate derivatives of an unknown function, so integrals involving nonlinearities are evaluated at one quadrature point and discrete form of nonlinear diffusion equations is obtained without forming local and global stiffness matrices. Using the method-of-lines (MoL), the nonlinear partial differential equation is reduced to a nonlinear system of ordinary differential equations such that the MoL formulation is derived in terms of the differentiation matrix instead of the stiffness matrix. Since the arising system is stiff, an implicit method is employed to solve it. Also, with a simple algorithm we obtain a closed analytical form for the Jacobian matrix of the system of nonlinear algebraic equations. In addition, the differentiation matrices for triangular elements and deformed quadrilateral and hexahedral elements are obtained. For deformed elements an isoparametric mapping is used where the expansion coefficients are obtained with the aid of the Gordon–Hall mapping. Several benchmarks, including the scalar and system of these equations, are carried out. As physical applications, the cell motion model over long time, the tumour angiogenesis model on a mixed quadrangle–triangle mesh, and two and three-dimensional p-Laplace equations for various choices of the parameter p are considered.

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

In this paper we study linearized backward differential formula (BDF) type schemes with Galerkin finite element approximations for the time-dependent nonlinear thermistor equations. Optimal $$L^2$$L2 error estimates for the proposed schemes are proved unconditionally. The proof consists of two steps. First, the boundedness of the numerical solution in certain strong norms is obtained by a temporal-spatial error splitting argument. Second, a traditional approach is used to provide an optimal $$L^2$$L2 error estimate for $$r$$r-th order FEMs $$(r \ge 1)$$(r?1). Numerical experiments in both two and three dimensional spaces are conducted to confirm our theoretical analysis and show the high order accuracy and unconditional stability (convergence) of the linearized BDF---Galerkin FEMs.

Numerical investigation on heat and mass transfer effect of micropolar fluid over a stretching sheet through porous media

The present paper deals with the study of unsteady heat and mass transfer characteristics of a viscous incompressible electrically conducting micropolar fluid. The flow past over a stretching sheet through a porous medium in the presence of viscous dissipation. A uniform magnetic field is applied transversely to the direction of the flow. Similarity transformations are used to convert the governing time dependent non-linear boundary layer equations into a system of non-linear ordinary differential equations that are solved numerically by Runge–Kutta fourth order method with a shooting technique. The influence of unsteady parameter (A), Eckert number (Ec), porous parameter (Kp), Prandtl number (Pr), Schmidt number (Sc) on velocity, temperature and concentration profiles are shown graphically. The buoyancy force retards the fluid near the velocity boundary layer in case of opposing flow and is favorable for assisting flow. In case of assisting flow, the absence of porous matrix enhances the flow. The impact of physical parameters on skin friction co-efficient, wall couple stress and the local Nusselt number and Sherwood number are shown in tabular form.

Unsteady MHD boundary layer flow and heat transfer over the stretching sheets submerged in a moving fluid with Ohmic heating and frictional heating

Abstract This paper is devoted to the analysis of the unsteady magnetohydrodynamic (MHD) boundary layer flow and heat transfer on a permeable stretching sheet embedded in a moving incompressible viscous fluid. The combined effects of Ohmic heating, thermal radiation, frictional heating and internal heat absorption/generation are taken into account. The governing time dependent nonlinear boundary layer equations are converted into a systemof nonlinear ordinary differential equations by similarity transformations. Some analytical results that give the characteristics of the velocity field in the boundary layer are presented and proved. The governing equations are then solved by using the shooting technique along with the fourth order Runge-Kutta method. The analytical properties proved in this paper are consistent with those obtained by the numerical method. Furthermore, the effects of the various parameters on the velocity and temperature fields are presented graphically and discussed in detail.

Perfectly matched layer absorbing boundary condition for nonlinear two-fluid plasma equations

Numerical instability occurs when coupled Maxwell equations and nonlinear two-fluid plasma equations are solved using finite difference method through parallel algorithm. Thus, a perfectly matched layer (PML) boundary condition is set to avoid the instability caused by velocity and density gradients between vacuum and plasma. A splitting method is used to first decompose governing equations to time-dependent nonlinear and linear equations. Then, a proper complex variable is used for the spatial derivative terms of the time-dependent nonlinear equation. Finally, with several auxiliary function equations, the governing equations of the absorbing boundary condition are derived by rewriting the frequency domain PML in the original physical space and time coordinates. Numerical examples in one- and two-dimensional domains show that the PML boundary condition is valid and effective. PML stability depends on the absorbing coefficient and thickness of absorbing layers.

The solitary wave solution of the one-dimensional Ginzburg Landau equations using the tanh method

The present paper deals with solving the time dependent nonlinear Ginzburg-Landau equations in one dimension using a solitary wave technique. For this purpose, we use the tanh-function method which is a simple and powerful analytical tool, nding several families of solutions. Mathematics Subject Classication: 35-xx, 34-xx, 76Dxx

Effect of the longitudinal momentum of electrons on their surfatron acceleration by an electromagnetic wave in space plasma

By numerically calculating the second-order nonlinear time-dependent equation for the wave phase on a particle trajectory, the effect of the longitudinal (with respect to the external magnetic field) momentum of electrons on the dynamics of their surfatron acceleration by an electromagnetic wave propagating across the external magnetic field in space plasma is analyzed. It is shown that, for strongly relativistic initial values of the longitudinal component of the electron momentum (the other parameters of the problem being fixed), the electrons are trapped into the ultrarelativistic regime of surfatron acceleration within a definite interval of the initial wave phase Ψ(0) on the particle trajectory. It was assumed in the calculations that Ψ(0) ≤ π. For the initial wave phases lying within the interval of 0 < Ψ(0) ≤ π, the electrons are immediately trapped by the wave, whereas at π ≤ Ψ(0) ≤ 0, no electron trapping is observed even at long computation times. This result substantially simplifies estimates of the wave damping caused by particle acceleration. The dynamics of the velocity components, momentum, and relativistic factor of electrons in the course of their ultrarelativistic acceleration are considered. The obtained results present interest for the development of modern concepts of the mechanisms for the generation of ultrarelativistic particles in space plasma, correct interpretation of experimental data on the flows of such particles, explanation of possible reasons for the deviation of the fast particle spectra observed in the heliosphere from the standard power-law scaling, and analysis of the relation between such deviations and the space weather.

Nucleation and crystal growth kinetics during solidification: The role of crystallite withdrawal rate and external heat and mass sources

A complete analytical solution of the integro-differential model describing the transient nucleation and growth of the crystals at the intermediate stage of phase transitions is constructed. The roles of external heat/mass sources appearing in the balance equations and the crystallite withdrawal rate entering in the Fokker–Planck equation are detailed. An exact analytical solution of the Fokker–Planck equation is found for arbitrary nucleation mechanisms and growth kinetics. Two important cases of the Weber–Volmer–Frenkel–Zel׳dovich and Meirs kinetics are considered in some detail. A non-linear time-dependent integral equation with memory kernel for the metastability level is analytically solved on the basis of the saddle-point method for the Laplace integral in the case of mixed kinetic-diffusion regime of crystal growth, which is of frequent occurrence. It is shown that the desupercooling/desupersaturation rate decreases with increasing the crystal withdrawal rate and intensities of external sources. The density distribution function becomes more and more broad with time. In addition, this function increases with decreasing the crystallite withdrawal rate and with increasing intensities of external sources.

The implicit midpoint rule for nonexpansive mappings

Abstract The implicit midpoint rule (IMR) for nonexpansive mappings is established. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a Hilbert space. Applications to the periodic solution of a nonlinear time-dependent evolution equation and to a Fredholm integral equation are included. MSC:47J25, 47N20, 34G20, 65J15.

Giant-vortex state induced by negatives values of deGennes parameter in a mesoscopic square

Using different negatives values of deGennes parameter in the boundary condition of a square sample superconducting type II is possible to change its vorticity. Due to the proximity effects, the cooper pairs density snapshot show how a giant vortex occurs when the external magnetic field is increased. Solving numerically the coupled nonlinear time dependent Ginzburg-Landau equations we obtain the vorticity, magnetic induction, cooper pair density, magnetization and phase of the order parameter as a function of the external magnetic field for this sample. We found that a giant vortex state occurs when a sample is surrounded by another superconductor at higher critial temperature.

Second-Order Time-Dependent Mild-Slope Equation for Wave Transformation

This study is to propose a wave model with both wave dispersivity and nonlinearity for the wave field without water depth restriction. A narrow-banded sea state centred around a certain dominant wave frequency is considered for applications in coastal engineering. A system of fully nonlinear governing equations is first derived by depth integration of the incompressible Navier-Stokes equation in conservative form. A set of second-order nonlinear time-dependent mild-slope equations is then developed by a perturbation scheme. The present nonlinear equations can be simplified to the linear time-dependent mild-slope equation, nonlinear long wave equation, and traditional Boussinesq wave equation, respectively. A finite volume method with the fourth-order Adams-Moulton predictor-corrector numerical scheme is adopted to directly compute the wave transformation. The validity of the present model is demonstrated by the simulation of the Stokes wave, cnoidal wave, and solitary wave on uniform depth, nonlinear wave shoaling on a sloping beach, and wave propagation over an elliptic shoal. The nearshore wave transformation across the surf zone is simulated for 1D wave on a uniform slope and on a composite bar profile and 2D wave field around a jetty. These computed wave height distributions show very good agreement with the experimental results available.

Unconditionally Optimal Error Estimates of a Crank--Nicolson Galerkin Method for the Nonlinear Thermistor Equations

This paper focuses on unconditionally optimal error analysis of an uncoupled and linearized Crank--Nicolson Galerkin finite element method for the time-dependent nonlinear thermistor equations in $d$-dimensional space, $d=2,3$. In our analysis, we split the error function into two parts, one from the spatial discretization and one from the temporal discretization, by introducing a corresponding time-discrete (elliptic) system. We present a rigorous analysis for the regularity of the solution of the time-discrete system and error estimates of the time discretization. With these estimates and the proved regularity, optimal error estimates of the fully discrete Crank--Nicolson Galerkin method are obtained unconditionally. Numerical results confirm our analysis and show the efficiency of the method.