Analytical closed-form solution methods

Achieving a complete solution of the set of equations above is difficult, as pointed out earlier. In addition to the numerical solution (33), Pearson (35) proposed a heuristic approach. Insight into the nature of melting with drag-forced removal can be obtained, however, by considering some special cases that lead to analytical, closed-form solutions. These simplified cases per se represent very useful solutions to the modeling of processing methods. 

If we were to change the kinetics so that the first reaction was second order in A and the second reaction was first order in B, then we would see largely the same picture emerging in the graphs of dimensionless concentration versus time. There would of course be differences, but not large departures in the trends from what we have observed for this all first-order case. But what if the reactions have rate expressions that are not so readily integrable What if we have widely differing, mixed-order concentration dependencies In some cases one can develop fully analytical (closed-form) solutions like the ones we have derived for the first-order case, but in other cases this is not possible. We must instead turn to numerical methods for efficient solution. 

X he most commonly encountered mathematical models in engineering and science are in the form of differential equations. The dynamics of physical systems that have one independent variable can be modeled by ordinary differential equations, whereas systems with two, or more, independent variables require the use of partial differential equations. Several types of ordinary differential equations, and a few partial differential equations, render themselves to analytical (closed-form) solutions. These methods have been developed thoroughly in differential calculus. However, the great majority of differential equations, especially the nonlinear ones and those that involve large sets of... 

Unfortunately, there exists no closed-form solution for the transform St a + i(j)). This directly implies that we need a new method for the approximation of the single exercise probabilities Tlj a [ ] assuming a multi-factor model with more than one payment date. On the other hand, the transform Et (n) can be solved analytically for nonnegative integer numbers n. This special solutions of Et z) can be used to compute the n-th moments of the underlying random variable V To Ti ) under the Ti forward measure. Then, by plugging these moments in the lEE scheme we are able to obtain an excellent approximation of the single exercise probabilities (see e.g. section (5.3.3) and (5.3.4)). 

To practitioners in reservoir engineering and well test analysis, the state-of-the-art has bifurcated into two divergent paths. The first searches for simple closed-form solutions. These are naturally restricted to simplified geometries and boundary conditions, but analytical solutions, many employing method of images techniques, nonetheless involve cumbersome infinite series. More recent solutions for transient pressure analysis, given in terms of Laplace and Fourier transforms, tend to be more computational than analytical they require complicated numerical inversion, and hence, shed little insight on the physics. 

It is clear that our calculations produce results that make physical sense. Of course, in the present problem where an analytical solution is available, there is no need to resort to numerical methods. But the solution is useful because it allows us to study the effects of grid selection, that is, the role of Ax and At in affecting computed solutions. We emphasize that the above calculations provide the time scales characteristic of the displacement flows. Both fronts start at the midpoint of the core, and both simulations terminate near the end of the core. Their total transit times are obviously different. These time scales, as our earlier closed-form solution... 

Directly solving the 13 simultaneous algebraic equations is too complicated in the context of obtaining the closed-form solutions. By using the method in Luo and Tong (2002), two sets of three simultaneous equations with three unknowns can be obtained respectively. Therefore, explicit analytical solutions can be derived. The details can be found in Luo and Tong (2002) and are not discussed further here due to the space limitation. 

Delale F, Erdogan F, Aydinoglu MN (1981) Stresses in adhesively bonded joints-a closed form solution. J Compos Mater 15 249-271 Fernlund G, Spelt JK (1991) Failure load prediction of structural adhesive joints, Part 1 analytical method. Int J Adhes Adhes ll(4) 213-220 Fernlund G, Papini M, McGammond D, Spelt JK (1994) Fracture load predictions for adhesive joints. Compos Sci Technol 51(4) 587-600 Goland M, Reissner E (1944) The stresses in cemented joints. JAppl Mech 11 A17-A27 Gleich DM, Van Tooren MJL, Beukers A (2001) Analysis and evaluation of bondline thickness effects on failure load in adhesively bonded structures. J Adhes Sci Technol 15(9) 1091-1101... 

For the study of time-resolved processes such as discussed in Chapter 6, but with many strongly coupled states in the manifold ipi, a closed-form solution cannot be carried through analytically and approximate treatments are necessary. This is not the case when solving Equation 6.58 for many states, which can always be integrated by numerical methods. All require modern, high-speed computers for their execution and the development of numerical recipes to handle large determinants. Furthermore, numerical solutions may often be obtained much more easily than closed-form solutions and may be sufficiently accurate, as the physical situation warrants. On the other hand, the closed-form solution given by Equation 6.58 serves as a convenient introduction to pursue much more difficult problems when possible. 

If/(x) has a simple closed-form expression, analytical methods yield an exact solution, a closed form expression for the optimal x, x. Iff(x) is more complex, for example, if it requires several steps to compute, then a numerical approach must be used. Software for nonlinear optimization is now so widely available that the numerical approach is almost always used. For example, the Solver in the Microsoft Excel spreadsheet solves linear and nonlinear optimization problems, and many FORTRAN and C optimizers are available as well. General optimization software is discussed in Section 8.9. 

Because of this importance of stress localizations many investigations have been dedicated both to the laminate free-edge effect, starting with the finite difference analyses of Pipes and Pagano 1970 [2] and the matrix crack problem. For both cases, closed-form analytical solutions are of a more or less approximate character. On the other hand, numerical methods as the finite elements (FEM) require a high discretizational effort because of... 

To guide model development, the observed data were first examined graphically to determine general characteristics and to look for trends with respect to dose, time, and the impact of anti-mAb antibodies. Models were developed using NONMEM (Version 5). Two different model types were developed the first model (MODEL 1, see Appendix 45.1) used an analytical solution (closed-form) where the nonlinearity was accounted for by allowing the model parameters to be a function of mAb dose and the titer of anti-mAb antibody, while the second model (MODEL 2, see Appendix 45.2) used differential equations to allow a more mechanistic approach to characterize the nonlinearity. For each model, three estimation methods were evaluated first-order (FO), first-order conditional estimation (FOCE), and FOCE with interaction. Various forms of between-subject variability models were evalu-... 

The interested reader may wish to consult Sadhal and Johnson for the details of solving those two equations by using the method of Collins. For present purposes, it is not essential to follow the remainder of the analysis. What is important is that a closed-form analytic solution can be obtained with the constants... 

The traditional methods of solution of many of the soluble problems of non-relativistic quantum mechanics employ a wide variety of analytical and algebraic methods, and their closed-form eigensolutions are usually expressed in terms of many different higher mathematical functions. However, most of these diverse functions can also be expressed quite conveniently in terms... 

All the Hongen-Watson models prior to this section have been presented analytically (i.e., in closed form) because solution of equation (14-121) for vacant-site fraction y is trivial when aU (Pi = I, which is consistent with the fact that each gas in the reactive mixture exhibits single-site adsorption. Numerical methods are required to calculate y if one or more gases adsorbs on several adjacent active sites without dissociation. 

In this chapter, we develop analytical solution methods, which have very close analogs with methods used for linear ODEs. A few nonlinear difference equations can be reduced to linear form (the Riccati analog) and the analogous Euler-Equidimensional finite-difference equation also exists. For linear equations, we again exploit the property of superposition. Thus, our general solutions will be composed of a linear combination of complementary and particular solutions. 

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