Partial differential equations dimensions

For this case the partial differential equation in one dimension becomes (HI 8) ... 

Gill and Nunge (G 16) solved the equation for diffusion accompanied by simultaneous chemical reaction with a changing concentration of the bulk liquid. They assume a film of thickness / in which the diffusion and simultaneous reaction take place, and suppose the liquid bulk outside this film to be completely mixed and to have a constant and uniform concentration. Their partial differential equation in one dimension is... 

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) ... 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

This is Fick s second law of diffusion, the equation that forms the basis for most mathematical models of diffusion processes. The simple form of the equation shown above is applicable only to diffusion in one dimension (x) in systems of rectangular geometry. The mathematical form of the equation becomes more complex when diffusion is allowed to occur in more than one dimension or when the relationship is expressed in cylindrical or spherical coordinate geometries. Since the simple form shown above is itself a second-order partial differential equation, the threat of added complexity is an unpleasant proposition at best. 

A comparison of the benefits and drawbacks of common numerical solution techniques for complex, nonlinear partial differential equation models is given in Table II. Note that it is common and in some cases necessary to use a combination of the techniques in the different dimensions of the model. 

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. 

We now make the ansatz that the form of this partial differential equation does not change if V = V(x, t) rather than a constant. Generalizing eqn (2.28) to three dimensions, we find the time-dependent Schrodinger equation,... 

The remaining two chapters of Part IV set the basis for the more advanced environmental models discussed in Part V. Chapter 21 starts with the simple one-box model already discussed at the end of Chapter 12. One- and two-box models are combined with the different boundary processes discussed before. Special emphasis is put on linear models, since they can be solved analytically. Conceptually, there is only a small step from multibox models to die models that describe the spatial dimensions as continuous variables, although the step mathematically is expensive as the model equations become partial differential equations, which, unfortunately, are more complex than the simple differential equations used for the box models. Here we will not move very far, but just open a window into this fascinating world. 

Axial symmetric representation is typically considered for single tubular fuel cell modeling because it allows one to design, mesh and compute only a portion of the cell taking advantage of the symmetry condition. The partial differential equations are written in cylindrical coordinates and solved in two dimensions (radial and axial). This assumption is possible when there are no significant circumferential variations in the boundary conditions. Such a hypothesis is reasonable when the... 

Errors and confusion in modelling arise because the complex set of coupled, nonlinear, partial differential equations are not usually an exact representation of the physical system. As examples, first consider the input parameters, such as chemical rate constants or diffusion coefficients. These input quantities, used as submodels in the detailed model, must be derived from more fundamental theories, models or experiments. They are usually not known to any appreciable accuracy and often their values are simply guesses. Or consider the geometry used in a calculation. It is often one or two dimensions less than needed to completely describe the real system. Multidimensional effects which may be important are either crudely approximated or ignored. This lack of exact correspondence between the model adopted and the actual physical system constitutes the basic problem of detailed modelling. This problem, which must be overcome in order to accurately model transient combustion systems, can be analyzed in terms of the multiple time scales, multiple space scales, geometric complexity, and physical complexity of the systems to be modelled. 

Even the states of systems with infinite dimension, like systems described by partial differential equations, may lie on attractors of low dimension. The phase space of a system may also have more than one attractor. In this case the asymptotic behavior, i.e., the attractor at which a trajectory ends up, depends on the initial conditions. Thus, each attractor is surrounded by an attraction basin, which is the part of the phase space in which the trajectories from all initial conditions end up. 

The process was considered as continuous and compartmental models were used to approximate the continuous systems [335]. For such applications, there is no specific compartmental model that is the best the approximation improves as the number of compartments is increased. It order to put compartmental models of continuous processes in perspective it may help to recall that the first step in obtaining the partial differential equation, descriptive of a process continuous in the space variables, is to discretize the space variables so as to give many microcompartments, each uniform in properties internally. The differential equation is then obtained as the limit of the equation for a microcompartment as its spatial dimensions go to zero. In approximation of continuous processes with compartmental models one does not go to the limit but approximates the process with a finite compartmental system. In that case, the partial differential equation... 

As stated in the introduction, dimension reduction of the governing partial differential equations describing reactors is necessary for the purpose of design, control, and optimization of chemical processes, and is typically achieved by three different approaches, as illustrated in Fig. 5. 

The one-dimensional model is by no means descriptive of everything that goes on in the reactor, because it provides calculated temperatures, concentrations, pressures, and so on only in one dimension — lengthwise, down the axis of the tube. Actually, transport processes and diffusion cause variations and gradients not only axially but also radially within tubes and within individual catalyst pellets. Furthermore, the reactor may not actually operate at steady-state, and so time might also be included as a variable. All of these factors can be described quite easily by partial differential equations in as many as four dimensions (tube length, tube radius, pellet radius, and time). 

Other than for the classical case of the finite dimension, it is known that relation (21) is true for compact or analytic semi-groups. This covers the case of Newtonian flows, where the associated system of partial differential equations is parabolic. In this case, one also has the implication (S3) (Si) . (See [58] for instance.)... 

Deterministic analysis Coupled biochemical systems Reaction kinetics are represented by sets of ordinary differential equations (ODEs). Rates of activation and deactivation of signaling components are dependent on activity of upstream signaling components. Spatially specified systems Reaction kinetics and movement of signaling components are represented by partial differential equations (PDEs). Useful for studies of reaction-diffusion dynamics of signaling components in two or three dimensions. (64-70)... 

Adsorption beds are essentially transient, spatially distributed sterns, where the properties in the solid and gas phases varying over time in one or more spatial dimensions. The mathematical description of adsorption beds is usually described by a series of partial differential equations and a ebraic equatimis. In this paper, distinguishable features of the CSS model are briefly given. 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

Equation (14.55) is a partial differential equation in two-dimensions that can be solved by following the method described in Section 13.2.2. The first step is to use a set of dimensionless variables that can be defined as a dimensionless normal distance from the electrode = y/d x) and an x-dependent dimensionless frequency K = o S x)/D. Equation (14.55) can be written as... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. In this chapter, we describe how one can arrive at the semianalytical solutions (solutions are analytical in the time variable and numerical in the spatial dimension) for linear parabolic partial differential equations using Maple, the method of lines and the matrix exponential. 

In this chapter, we describe how one can arrive at the semianalytical solutions (solutions are analytical in the y variable and numerical in the spatial dimension) for linear elliptic partial differential equations using Maple and the matrix exponential method. 

Partial differential equations in time and one space dimension ... 

Partial differential equations in two or more spatial dimensions (and possibly time, too). 

PARTIAL DIFFERENTIAL EQUATIONS IN TIME AND ONE SPACE DIMENSION 317... 

From a mathematical point of view, the treatment of spatiotemporal dynamics on disk electrodes is considerably more difficult than that of the (infinitesimally thin) ring electrode. Of course, on the one hand this is due to the additional spatial dimension. Since the direction into the electrolyte has also to be considered, the problem is spatially three-dimensional. However, even if this complication is neglected by considering, in a first step, only the radial and axial directions (i.e., neglecting possible structures in the azimuthal direction), solving the resulting partial differential equations is still a challenging task. This is due to the mixed boundary... 

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