Differential equations parabolic form

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

Figure 8 depicts our view of an ideal structure for an applications program. The boxes with the heavy borders represent those functions that are problem specific, while the light-border boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically either systems of nonlinear algebraic equations, ordinary differential equation initial or boundary value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the user must write the subroutine that defines his particular system of equations. However, that subroutine should be able to make calls to problem-independent software to return many of the components that are needed to assemble the governing equations. Specifically, such software could be called to return in-... 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

These conditions together with those concentrations X, (/ = N - r,..., N) whose value is maintained constant inside V constitute the constraints applied to the system by the environment. Only for some particular set of values of these constraints is an equilibrium state realized between V and its external world. Although we refer here only to chemical systems, the class of phenomena obeying parabolic differential equations of the form (12) is much broader. A discussion of or references to self-organization phenomena in other fields (e.g., ecology, laser theory, or neuronal networks) can be found in Ref. 2. 

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... 

Finally, substitution of the reformulated expressions for ion fluxes in material balance Eq. (2) results, for the spherically symmetric particle, in the two nonlinear parabolic type partial differential equations that follow in reduced form ... 

The Laplace operator V2 takes the form given in equations (2.11), (2.12) and (2.13) for the different coordinate systems. The differential equations (2.15) and (2.16) for steady-state temperature fields, are linear and elliptical, as long as W is independent of, or changes linearly with d. This leads to different methods of solution than those used in transient conduction where the differential equations are parabolic. 

Consider a general linear homogeneous parabolic partial differential equation in dimensionless form... 

In this chapter semianalytical solutions (solutions analytical in t and numerical in x) were obtained for parabolic PDEs. In section 5.1.2, the given homogeneous parabolic PDE was converted to matrix form by applying finite differences in the spatial direction. The resulting matrix differential equation was then integrated analytically in time using Maple s matrix exponential. This methodology helps us solve the dependent variables at different node points as an analytical function of time. This is a powerful technique and is valid for all linear parabolic PDEs. This... 

Parabolic partial differential equations with homogenous boundary conditions are solved in this section. The dependent variable u is assumed to take the form u = XT, where X is a function of x alone and T is a function of t alone. This leads to separate differential equations for X and T. This methodology is best illustrated using an example. 

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. 

In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form, Eq. (4.1), specification of the Laplacian operator is required. Although this specification is immediate for spaces of integral dimension, it is less straightforward for spaces of intermediate or fractal dimension [47,55,56]. As examples of problems in chemical kinetics where the relevance of an approach based on Eq. (4.1) is open to question, one can cite the avalanche of work reported over the past two decades on diffusion-reaction processes in microheterogeneous media, as exemplified by the compartmentalized systems such as zeolites, clays and organized molecular assemblies such as micelles and vesicles (see below). In these systems, the (local) dimension of the diffusion space is often not clearly defined. 

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. 

Based upon the above assumptions, fundamental differential equations are obtained. Laplace transformation method was also used to solve the simultaneous differential equations under the given initial and boundary conditions. Analytical solutions are obtained in the form of dimensionless concentrations which involve error functions concerning time and the depth from the surface. For the progress of neutralization, the parabolic law involving constant terms was derived as follows ("X, neutralization depth of concrete) ... 

The governing equations, along with the appropriate constitutive relations, completely describe the fluid flow within a given geometry. However, the mathematical model forms a system of partial differential equations obeying mixed elliptic-parabolic behaviour which cannot be solved unless we specify the boundary conditions for the problem. Mathematically they fix the integration constants yielded upon integration. From a physical point of... 

The initial-boundary value problem represented by Eq. 7.1 can be transmuted into a boundary integral equation by several different methods. Brebbia and Walker (1980) and Curran et al. (1980) approximated the time derivative in the equation in a finite difference form, thus changing the original parabolic partial differential equation to an elliptical partial differential equation, for which the standard boundary integral equation may be established. 

Let us assume the parabolic partial differential equation to be of the following form ... 

Mathematically, the two-dimensional model (balance Equations 5.194 and 5.209) forms a system of parabolic partial differential equations. The best way to numerically solve this system is to convert the partial differential equation... 

The existence of truncation errors in finite difference approximations to differential equations is discussed in numerical analysis texts with respect to round-off error and computational instabilities (Roache, 1972 Richtmyer and Morton, 1957), but Lantz (1971) was among the first to address the form of the truncation error as it related to diffusion. Lantz considered a linear, convective, parabolic equation similar to 9u/9t + U 9u/9x = e S u/Sx and differenced it in several ways. He showed that the effective diffusion coefficient was not 8, as one might have suggested analytically, but 8 + 0(Ax, At) (so that the actual diffusion term appearing in computed solutions is the modified coefficient times c2u/9x2) where the 0(Ax,At) truncation errors, being functions of u(x,t), are comparable in magnitude to 8. Because this artificial diffusion necessarily differs from the actual physical model, one would expect that the entropy conditions characteristic of the computed results could likely be fictitious. 

Prom the mathematical point of view equations (8.5.1) and (8.5.10) with definitions (8.5.3)-(8.5.5) form a system of nonlinear parabolic partial differential equations of the Reaction -I- Diffusion t3q>e, describing a wide range of physical phenomena that are of a great importance for thin films growth dynamics. 

Linear second-order partial differential equations in two independent variables are further classified into three canonical forms elliptic, parabolic, and hyperbolic. The general form of this class of equations is... 

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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