Linear differential forms, exact

A linear differential form is exact if it corresponds to the total derivative of some function z = z(.y ). Therefore, an exact linear differential form may be written... 

In thermodynamics, we deal with a number of linear differential forms, some of which are exact. The second law of thermodynamics is concerned with finding an integrating factor for the linear differential form dq which transforms it into the exact differential form dS = dqjT for reversible processes. 

It can be shown that Eq. (A-35) is a sufficient as well as a necessary condition for the exactness of a linear differential form. We specialize Eq. (A-33) to the case of two independent variables and use Gauss s theorem to obtain... 

X2° = X30 = 0 assumed to be known exactly. The only observed variable is = x. Jennrich and Bright (ref. 31) used the indirect approach to parameter estimation and solved the equations (5.72) numerically in each iteration of a Gauss-Newton type procedure exploiting the linearity of (5.72) only in the sensitivity calculation. They used relative weighting. Although a similar procedure is too time consuming on most personal computers, this does not mean that we are not able to solve the problem. In fact, linear differential equations can be solved by analytical methods, and solutions of most important linear compartmental models are listed in pharmacokinetics textbooks (see e.g., ref. 33). For the three compartment model of Fig. 5.7 the solution is of the form... 

This is a linear differential equation with constant coefficients that can be solved by conventional techniques. In this case, the coefficients are A2 and A1 Ni. This equation has exactly the same form as that which results when describing series reactions, and its solution was presented in Section 2.4. After assuming a solution of the form... 

Since the properties/and/" are state functions and the definition (6.2.1) is a linear combination of state functions, the difference/ is also a state function. This means/ forms exact differentials, so (6.2.1) can be written as... 

The examples of computational accuracy so far have been for linear differential equations. This is reasonable since for such equations the exact solution of such linear equations is known. It would also be instructive to explore similar results for nonlinear equations. In general, nonlinear differential equations do not have closed form solutions so one must extrapolate to such equations. There are, however, a small set of nonlinear differential equations for which an exact solution can be obtained and one such equation is the second order equation ... 

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

For purely intramolecular equilibria the system of differential equations (72) is already linear and the procedure described above transforms the system into the form of equation (104) exactly without any approximations. For such equilibria further simplification of equations (72) and (104) is possible by deleting all those quantities which refer to empty sets of nuclei. 

Methods for solving mass and heat transfer problems. The convective diffusion equation (3.1.1) is a second-order linear partial differential equation with variable coefficients (in the general case, the fluid velocity depends on the coordinates and time). Exact closed-form solutions of the corresponding problems can be found only in exceptional cases with simple geometry [79,197, 270, 370, 516]. This is especially true of the nonlinear equation (3.1.17). Exact solutions are important for adequate understanding of the physical background of various phenomena and processes. They can serve as test solutions to verify whether the problem is well-posed or to estimate the accuracy of the corresponding numerical, asymptotic, and approximate methods. 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

Because of the complicated nature of biomolecular geometries and charge distributions, the PB equation (PBE) is usually solved numerically by a variety of computational methods. These methods typically discretize the (exact) continuous solution to the PBE via a finite-dimensional set of basis functions. In the case of the linearized PBE, the resulting discretized equations transform the partial differential equation into a linear matrix-vector form that can be solved directly. However, the nonlinear equations obtained from the full PBE require more specialized techniques, such as Newton methods, to determine the solution to the discretized algebraic equation. ... 

The expressions (9.251) and (9.246) form a nonlinear set of integro-differential equations in P t) and 0(t). An exact solution to this set is not considered here. We seek instead the solution to the corresponding linearized equations. For this purpose we introduce... 

The exact solution of the non linearized Poisson-Boltzmaim equation 2.52 can be obtained explicitly under an analytical form, this property being closely related to the one dimensional character of the present problem. By multiplying both sides by (Hi/dj. one obtains exact differentials which, after integrating twice and applying the boundary conditions yields... 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (PIDE). In the PBE (12.308) the size property variable ranges from 0 to oo. In order to apply a numerical scheme for the solution of the equation a first modification is to fix a finite computational domain. The conventional approximation is to truncate the equation by substitution of the infinite integral limits by the finite limit value max- The function/(, r, t) denotes the exact solution of the exact equation. It might be assumed that the solution of the truncated PBE is sufficiently close to the exact equation so that the two solutions are practically equal. Hence, the solutions of both forms of the PBE are denoted by fiC, r, t). 

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