Explicit solution method

A first course on DEs typically studies how to solve equations of the form y (t) = f(t,y(t)). Such a course develops methods to find explicit solutions for specific (theoretically solvable) classes of differential equations and besides, it studies the behavior of solutions of DEs for which there are or are not any known explicit solution methods. In a nutshell, such a course looks at DEs and their solutions both quantitatively and qualitatively. 

The constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

If an analytical solution is available, the method of nonlinear regression analysis can be applied this approach is described in Chapter 2 and is not treated further here. The remainder of the present section deals with the analysis of kinetic schemes for which explicit solutions are either unavailable or unhelpful. First, the technique of numerical integration is introduced. 

This system of i + 2 equations is nonlinear, and for this reason probably has not received attention in the least-squares method (207). We are able to give an explicit solution (163) for the particular case when Xy = xj and m,- = m for all values of i that is, when all reactions of the series are studied at a set of temperatures, not necessarily equidistant, but the same for all reactions. Let us introduce... 

The price of using implicit methods is that one now has a system of equations to solve at each time step, and the solution methods are more complicated (particularly for nonlinear problems) than the straightforward explicit methods. Phenomena that happen quickly can also be obliterated or smoothed over by using a large time step, so implicit methods are not suitable in all cases. The engineer must decide if he or she wants to track those fast phenomena, and choose an appropriate method that handles the time scales that are important in the problem. 

In principle, concentration-time curves can be constructed for successive reactions of different orders. For the plug flow or batch reactor and for the mixed reactor explicit solutions are difficult to obtain thus, numerical methods provide the best tool for treating such reactions. 

Equation (2.53) can be solved numerically by the Runge-Kutta method (see Appendix 2A). Explicit solutions of (2.53) are given here. 

Although not recommended for practical use, the classical Euler extrapolation is a convenient example to illustrate the basic ideas and problems of numerical methods. Given a point (tj y1) of the numerical solution and a step size h, the explicit Euler method is based on the approximation (yi+1 - /( i+l " 4 dy/dt to extrapolate the solution... 

To compare the explicit and implicit Euler methods we exploited that the solution (5.3) of (5.2) is known. We can, however, estimate the truncation error without such artificial information. Considering the truncated Taylor series of the solution, for the explicit Euler method (5.7) we have... 

Fig. 15.3 Illustration of a stable and unstable solution to the model problem (Eq. 15.5) by the forward (explicit) Euler method.

Figure 15.3 illustrates the performance of the explicit Euler method on the model problem, Eq. 15.5. In both panels, the time step is h = 0.1, but the left-hand panel has X = 10 and the right-hand panel has X = 30. The heavy lines show the y = t2 + 1 solution and the course of the numerical solution. The thinner lines show solution trajectories from different initial conditions. 

Compared to the explicit Euler method (Eq. 15.9), note that the right-hand side is evaluated at the advanced time level tn+1- If f(t, ) is nonlinear then Eq. 15.22 must be solved iteratively to determine yn+. Despite this complication, the benefit of the implicit method lies in its excellent stability properties. The lower panel of Fig. 15.2 illustrates a graphical construction of the method. Note that the slope of the straight line between y +i and yn is tangent to the nearby solution at tn+, whereas in the explicit method (center panel) the slope is tangent to the nearby solution at t . 

When r(n) and g(n) are both linear in n it is usually impossible ) to give an explicit solution of the master equation other than the stationary solution. An approximate treatment is given in chapter VIII and a systematic approximation method will be developed in chapter X. We here merely list a few typical examples. 

Equation (5.62) for the current-potential response in CV has been deduced by assuming that the diffusion coefficients of species O and R fulfill the condition Do = >r = D. If this assumption cannot be fulfilled, this equation is not valid since in this case the surface concentrations are not constant and it has not been possible to obtain an explicit solution. Under these conditions, the CV curves corresponding to Nemstian processes have to be obtained by using numerical procedures to solve the diffusion differential equations (finite differences, Crank-Nicholson methods, etc. see Appendix I and ([28])3. 

To date, there has been no explicit solution for this problem for p > 3, since the surface concentrations of electroactive species O and R are time dependent and therefore the Superposition Principle cannot be applied (see also Sect. 4.3) [1,5]. In these conditions, a non-explicit integral solution has been deduced using the Laplace transform method (see Appendix H). 

A solution methodology of the above, a nonlinear differential equation, will be shown. In essence, this solution method serves the mass-transfer rate and the concentration distribution in closed, explicit mathematical expression. The method can be applied for Cartesian coordinates and cylindrical coordinates, as will be shown. For the solution of Equation 14.2, the biocatalytic membrane should be divided into M sublayers, in the direction ofthe mass transport, that is perpendicular to the membrane interface (for details see e.g., Nagy s paper [40]), with thickness of A8 (A8 = 8/M) and with constant transport parameters in every sublayer. Thus, for the mth sublayer ofthe membrane layer, using dimensionless quantities, it can be obtained ... 

The above results in the case of the binary copolymerization can be easily obtained from the analysis of the expression (5.4). However, for the copolymerization of more than two monomers such an analysis is not possible since the proper explicit solution of Eqs. (5.2) is not available yet. The traditional methods of dynamic systems theory are considered the most effective for establishing the common qualitative peculiarities of the trajectory behavior [204-205]. 

The kinetic theory can also be used for polyfunctional systems with unequal reactivities of groups and substitution effects, but an explicit solution of the partial differential equation corresponding to Eq. (23) derived for the equireactive system is not possible. One can use, however, the method of moments for derivation of certain averages as was explained in... 

Two broad classes of technique are available for modeling matter at the atomic level. The first avoids the explicit solution of the Schrodinger equation by using interatomic potentials (IP), which express the energy of the system as a function of nuclear coordinates. Such methods are fast and effective within their domain of applicability and good interatomic potential functions are available for many materials. They are, however, limited as they cannot describe any properties and processes, which depend explicitly on the electronic structme of the material. In contrast, electronic structure calculations solve the Schrodinger equation at some level of approximation allowing direct simulation of, for example, spectroscopic properties and reaction mechanisms. We now present an introduction to interatomic potential-based methods (often referred to as atomistic simulations). 

The authors in this paper present an explicit symplectic method for the numerical solution of the Schrodinger equation. A modified symplectic integrator with the trigonometrically fitted property which is based on this method is also produced. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential. 

In ref. 165 a S5miplectic explicit RKN method for Hamiltonian systems with periodical solutions is obtained. The characteristics of this new method are ... 

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