Elliptic equations boundary conditions

Proper boundary conditions are generally required for the primary variables like the gas and particle velocities, pressures and volume fractions at all the vessel boundaries as these model equations are elliptic. Moreover, boundary conditions for the granular temperature of the particulate phase is required for the PT, PGT and PGTDV models. For the models including gas phase turbulence, i.e., PGT and PGTDV, additional boundary conditions for the turbulent kinetic energy of the gas phase, as well as the dissipation rate of the gas phase and the gas-particle fluctuation covariance are required. The... 

The governing equations are elliptic so boundary conditions are required at all boundaries. The normal velocity components for both phases are set to zero at the vertical boundaries. The wall boundary conditions for the vertical velocity component, k and e are specified in accordance with the standard wall function approach. The particle phase is allowed to slip along the wall following the boundary condition given by (4.99). 

The equations (5.376)-(5.379) could be considered when t = 0. In this case we see that the obtained equations with the boundary condition (5.380) exactly coincide with the elliptic boundary value problem (5.285)-(5.289). The a priori estimate of the corresponding solution ui, Wi, mi, ni is as follows,... 

We call the nodes, at which equation (1) is valid under conditions (2), inner nodes of the grid uj is the set of all inner nodes and ui = ui + y is the set of all grid nodes. The first boundary-value problem completely posed by conditions (l)-(3) plays a special role in the theory of equations (1). For instance, in the case of boundary conditions of the second or third kinds there are no boundary nodes for elliptic equations, that is, w = w. 

This equation can be solved exactly in terms of elliptic integrals, but this solution is somewhat complex. However, by a slight modification of the boundary condition at the end of the pore, it is possible to obtain a good engineering approximation that is useful for fast reactions. In this approximation, we replace the boundary condition at the end of the pore by... 

The theory for classifying linear, second-order, partial-differential equations is well established. Understanding the classification is quite important to understanding solution algorithms and where boundary conditions must be applied. Partial differential equations are generally classified as one of three forms elliptic, parabolic, or hyperbolic. Model equations for each type are usually stated as... 

The steady-state heat equation (Eq. 3.284) is often used as the model equation for an elliptic partial-differential equation. An important property of elliptic equations is that the solution at any point within the domain is influenced by every point on the boundary. Thus boundary conditions must be supplied everywhere on the boundaries of the solution domain. The viscous terms in the Navier-Stokes equations clearly have elliptic characteristics. 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

Both the Laplace transform and the similarity solution techniques are powerful techniques for partial differential equations in semi-infinite domains. The Laplace transform technique can be used for all linear partial differential equations with all possible boundary conditions. The similarity solution can be used only if the independent variables can be combined and if the boundary conditions in x and t can be converted to boundary conditions in the combined variable. In addition, unlike the Laplace transform technique, the similarity solution technique cannot handle partial differential equations in which the dependent variable appears explicitly. The Laplace transform cannot handle elliptic or nonlinear partial differential equations. The similarity solution can be used for elliptic and for a few nonlinear partial differential equations as shown in section 4.6. There are thirteen examples in this chapter. 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

The program developed for example 6.3 is very general and can be used for any linear elliptic partial differential equation with linear boundary conditions. 

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 solution of the governing reactor model equations are subjected to the boundary conditions specified. The number of boundary conditions required depends on the mathematical properties of the equations (e.g., elliptical, parabolic, hyperbolic, or mixed). 

We formulate the confinement of the hydrogen atom by an elliptical cone defined by xi = Xio- The boundary condition on the complete wave function of Equation (48), with its factors from Equations (39) and (57), is... 

The states of the hydrogen atom, as considered so far, correspond clearly to the elliptic orbits of the old Bohr theory in both cases the electron remains at a finite distance. But in Bohr s theory there are also hyp)erbolic orbits what corresponds to these in quantum mechanics Clearly, solutions of the wave equation which do not disappear at infinity. In order to obtain them we must give up the boundary condition— vanishing of at infinity—and look for solutions which... 

Elhptic equations describe steady-state or equilibrium processes. For such equations, all aux-ihary conditions must be prescribed on the boundary of the region of interest. Initial conditions given to elliptic equations lead to ill-posed problems. Solutions to these ill-posed problems exhibit sensitivity to the initial data. Small changes to the initial data cause large changes in the solution. 

There are three kinds of boundary conditions for elliptic equations. If the values of the unknown function are prescribed on the boundary, then the problem is called the Dirichlet problem. If the derivatives of the unknown function are prescribed on the boundary, then it is called the Neumann problem. If a linear combination of the function values and the derivatives is specified, then it is called the Robin problem. 

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