Parabolic equation

Smoothed data presented at rounded temperatures, such as are available in Tables 6.2 and 6.4, plus the C° values at 298 K listed in Table 6.1 and 6.3, are especially suitable for substitution in the foregoing parabolic equations. The use of such a parabolic fit is appropriate for interpolation, but data extrapolated outside the original temperature range should not be sought. 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation... 

The canonical form of a grid equation of common structure. The maximum principle is suitable for the solution of difference elliptic and parabolic equations in the space C and is certainly true for grid equations of common structure which will be investigated in this section. 

Monotone schemes for parabolic equations of general form. It is required to find a solution of the following problem foor a parabolic equation... 

Example 2 The statement of the first boundary-value problem for the parabolic equation with mixed derivatives in the parallelepiped Go = 0 < a < L, a =1,2,..., p is... 

The problem here consists of finding a continuous in Qt solution of the system of parabolic equations... 

The problem statement. First of all, it should be noted that it is impossible to generalize directly the alternating direction method for three and more measurements as well as for parabolic equations of general form. Second, economical factorized schemes which have been under consideration in Section 2 of the present chapter are quite applicable under the assumption that the argument x = (xq, x, ., Xp) varies within a parallelepiped. 

A locally one-dlinensional scheme (LOS) for the heat conduction equation in an arbitrary domain. The method of summarized approximation can find a wide range of application in designing economical additive schemes for parabolic equations in the domains of rather complicated configurations and shapes. More a detailed exploration is devoted to a locally one-dimensional problem for the heat conduction equation in a complex domain G = G -f F of the dimension p. Let x — (sj, 2,..., a- p) be a point in the Euclidean space R. ... 

As usual, it is preassumed in a common setting that the problem concerned is uniquely solvable and its solution u = u x,t) possesses all necessary derivatives which do arise in all that follows. The domain of interest G is still subject to the same conditions as we imposed in Section 5 for parabolic equations. Also, let — jr, j = 0,1,... be a uniform... 

Additive schemes for a system of equations. Later in this section we will survey some devices that can be used in trying to produce additive schemes for systems of parabolic equations. With this aim, problem (50) we have completely posed in Section 2 will serve as a basis for the up-to-date presentation of tools and techniques, their theory and applications. In this connection we may attempt the operator L in the form L — L + with triangle operators L, L+, the associated matrices of which arrange themselves as sums = k +, where k = and... 

Karetkina, N. (1980) Nonconditionally stable difference schemes for parabolic equations with the first derivatives. Zh. Vychisl. Mat. i Mat. Fiz., 20, 236-240 (in Russian) English transl. in USSR Comput. Mathem. and Mathem. Physics. 

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it... 

See Partial Differential Equations. ) If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. 

Like the oxidation of hydrocarbons, the autocatalytic oxidation of polymers is induced by radicals produced by the decomposition of the hydroperoxyl groups. The rate constants of POOH decomposition can be determined from the induction period of polymer-inhibited oxidation, as well as from the kinetics of polymer autoxidation and oxygen uptake. The initial period of polymer oxidation obeys the parabolic equation [12]... 

Oxidation of polymer in the presence of dioxygen acceptors is limited by the diffusion of dioxygen into the polymer bulk. The lifetime of polymer does not depend on the acceptor concentration at [acceptor] > [acceptor]min. The lifetime of a polymer sample t depends on p02, A, D, and the thickness of the sample / according to the parabolic equation [14-18]... 

The response curve to an impulse input has the parabolic equation,... 

During the time intervals between random eddy events, (4.37) is solved numerically using the scalar fields that result from the random rearrangement process as initial conditions. A standard one-dimensional parabolic equation solver with periodic boundary conditions (BCs) is employed for this step. The computational domain is illustrated in Fig. 4.3. For a homogeneous scalar field, the evolution of t) will depend on the characteristic length... 

The Baffin Bay picrites (Francis, 1985) show a very good covariation of FeO, MgO, and Ni. Defined from the twelve XRF data listed in Table 1.12, the variables x = ln(FeO/MgO) and y = ln(Ni/MgO) have been fitted by the parabolic equation... 

The parabolic equations derived in a slowly varying envelope approximation that describe the second harmonic generation (SHG) of ultrashort pulses in media with locally inhomogeneous wave-vector mismatch, have the form ... 

Solving the full problem for arbitrary values of coefficients is costly and one would like to find the effective (or averaged) values of the dispersion coefficient and the transport velocity and an effective corresponding ID parabolic equation for the effective concentration. 

In order to get a parabolic equation for we choose Qx such that dffC and dxf4 I ot appear in the effective equation. Then Cox is of the form Cox = fl5f4 + dxC% and after a short calculation we find that... 

P. S. Hagan, Travelling wave and multiple traveling wave solutions of parabolic equations, SIAM J. Math. Anal., 18 (1982), pp. 717-738. 

L. Rubinstein, Free boundary problem for a nonlinear system of parabolic equations, including one with reversed time, Ann. Mat. Pura Appl., 135 (1983), pp. 29-42. 

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