Partial Differential Equation systems hyperbolic equations

The partial differential equations system for steady flows of Maxwell type (i.e., with = 0) is of composite type, neither elliptic, nor hyperbolic. This is not surprising, the same being true for instance for the stationary system of ideal incompressible fluids. The new feature, discovered in [8], is that some change of type may occur. In fact an easy but tedious calculation shows that three types of characteristics axe present ... 

Let us consider the genera class of systems described by a system of n nonlinear parabolic or hyperbolic partial differential equations. For simplicity vve assume that we have only one spatial independent variable, z. 

Briefly the idea behind this method is to delineate families of curves in the x-t plane, called characteristic curves, along which the partial differential equations [(123) and (128)] become a system of ordinary differential equations which could then be integrated with greater ease. However, only hyperbolic partial differential equations possess two families of characteristics curves required by the method. 

Rhee, Aris, and Amundson, First-Order Partial Differential Equations Volume 1. Theory and Application of Single Equations Volume 2. Theory and Application of Hyperbolic Systems of Quasi-Linear Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1986,1989. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

The system of hyperbolic and parabolic partial differential equations representing the ID or 2D model of monolith channel is solved by the finite differences method with adaptive time-step control. An effective numerical solution is based on (i) discretization of continuous coordinates z, r and t, (ii) application of difference approximations of the derivatives, (iii) decomposition of the set of equations for Ts, T, c and cs, (iv) quasi-linearization of... 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

The accelerated gradient method is used because of its advantages especially when the control is constrained. The system and its adjoint equations are coupled hyperbolic partial differential equations. They can be solved numerically using the method of characteristics (Lapidus, 1962b Chang and Bankoff, 1969). This method is used with the fourth order Runge-Kutta method (with variable step size to ensure accuracy of the integration) to solve the state and adjoint equations. 

Using the transfer function concept, Koppel (1967) derived the optimal control policy for a heat exchanger system described by hyperbolic partial differential equations using the lumped system approach. Koppel and Shih (1968) also presented a feedback interior control for a class of hyperbolic differential equations with distributed control. In an earlier paper Koppel e/ al. (1968) discussed the necessary conditions for the system with linear hyperbolic partial differential equations having a control which is independent of spatial coordinates. The optimal feedback-feedforward control law for linear hyperbolic systems, whose dynamical response to input variations is characterized by an initial pure time delay, was derived by Denn... 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

Abstract This contribution deals with the modeling of coupled thermal (T), hydraulic (H) and mechanical (M) processes in subsurface structures or barrier systems. We assume a system of three phases a deformable fractured porous medium fully or partially saturated with liquid and a gas which remains at atmospheric pressure. Consideration of the thermal flow problem leads to an extensively coupled problem consisting of an elliptic and parabolic-hyperbolic set of partial differential equations. The resulting initial boundary value problems are outlined. Their finite element representation and the required solving algorithms and control options for the coupled processes are implemented using object-oriented programming in the finite element code RockFlow/RockMech. 

Rhee, H.K., Aris, R., and Amundson, N.R. (1986) First-order partial differential equations, in Theory and Application of Hyperbolic Systems of Quasilinear Equations, vol. I, Prentice-Hall, New Jersey. 

This is a time-dependent, linear (hyperbolic) partial differential equation. The steady state solutions of this equation (defined by 9 /= 0) are of the form g(z, f) = (p z) where cp satisfies Cf(p z) = 0. These are just the first integrals of the ordinary differential equation system z = /(z). 

In order to describe adequately the hydrodynamics of the experimental fixed bed reactor, it is necessary to take into account the axial dispersion in the mathematical model. The time dependent continuity equation including axial dispersion for a fixed bed reactor is given by a partial differential equation (pde) of the parabolic/hyperbolic class. These types of pde s are difficult to solve numerically, resulting in long cpu times. A way to overcome these difficulties is by describing the fixed bed reactor as a cascade of perfectly stirred tank reactors. The axial dispersion is then accounted for by the number of tanks in series. For a low degree of dispersion (Bo < 50) the number of stirred tanks, N, and the Bodenstein number. Bo, are related as N Bo/2 [8].The fixed bed reactor is now described by a system of ordinary differential equations (ode s). No radial gradients are taken into account and a onedimensional model is applied. Mass balances are developed for both the gas phase and the adsorbed phase. The reactor is considered to be isothermal. 

Delay models were discussed in Chapter 10. We repeat here that the most interesting problem is a modeling one. Since the problem is sensitive to how the delay is introduced, care must be taken in the modeling. A physical delay is caused by the physiology of the cell, so model equations must be modified to consider or approximate the cell physiology. Once a model is known, analysis of the corresponding system of equations (either functional differential equations or hyperbolic partial differential equations of a structured model) would be an important contribution. It is likely, however, that the delay will be state-dependent, and the theory for such equations is not well developed. A model with delays due to both cell physiology and diffusion in an unstirred chemostat would also be of interest. 

Mathematical models of catalytic systems in the general form are rather sophisticated. Often, they consist of nonlinear systems of differential equations containing both conventional equations and equations with partial derivatives of parabolic, hyperbolic, and other forms. Efficient simulation is only possible if a well developed qualitative theory of differential equations (mainly, equations with partial derivatives) and high performance programs for computational experiments exist. 

The method of corner boundary functions is well developed also for equations of hyperbolic type [29], for systems of elliptic equations [30], for systems of parabolic equations [31], for partial differential equations in the multidimensional case [32], as well as for difference equations [33]. This method works successfully for a variety of applied problems. 

Such a classification can also be applied to higher order equations involving more than two independent variables. Typically elliptic equations are associated with physical systems involving equilibrium states, parabolic equations are associated with diffusion type problems and hyperbolic equations are associated with oscillating or vibrating physical systems. Analytical closed form solutions are known for some linear partial differential equations. However, numerical solutions must be obtained for most partial differential equations and for almost all nonlinear equations. 

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