Differential Equations with Complex Variables

Remember 2.4 The method for numerical solution of differential equations with complex variables is to separate the equations into real and imaginary parts and to solve them simultaneously. 

K. F. Riley, M. P. Hobson, and S. I. Bence, Mathematical Methods for Physics and Engineering A Comprehensive Guide (Cambridge University Press, Cambridge, 1998) I. R. Ockendon, S. Howison, J. Ockendon, A. Lacey, and A. Movchan, Applied Parital Differential Equations (Oxford University Press, Oxford, 2003) H. F. Weinberger, First Course in Partial Differential Equations with Complex Variables and Transform Methods (Dover, Dover edition New York, 1995). 

The Schrodinger equation for even a single /V-electron atom is a partial differential equation with 3N variables, and to make matters worse, the interelectron interaction causes the solutions to be true 3/V-dimensional functions that cannot simply be broken down into smaller constituent parts. Nevertheless, despite the staggering complexity of even small-sized systems, quantum theory has yielded great success in calculating useful properties of complex systems and in producing... 

Solutions of assemblies of ordinary differential equations with time as the independent variable are ideally suited for solution by analog computation. Hence complex kinetics equations of the type considered in this section may conveniently be solved with an analog computer. This is illustrated in the problems at the end of this chapter. 

In Eq. (8.8) the components of the velocity field are generally very complex functions. Partial differential equations with variable coefficients are difficult to solve. The problem is simplified by approximating the velocity components within the diffusion layer by taking into account... 

Equation (2.12) is a differential equation with respect to time t that is able to take the effects of complex thermal loading (thermal loading at variable heating rates) into account. As any thermal loading procedure is also a function of time, and based on a finite difference method, the temperature at each finite time step can be approximated as a constant At a time step, J, with a constant heating rate, fip Eq. 2.12 can be converted to ... 

Chapters 10 through 13 are devoted to the solution of nonlinear differential equations using numerical techniques. The chapters build in complexity from differential equations in single variables to coupled systems of nonlinear differential equations to nonlinear partial differential equations. Both initial value and boundary value differential equations are discussed with many examples. The discussion emphasizes the reuse of computer code developed in previous chapters in solving systems of coupled nonlinear equations. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

These three equations (11), (12), and (13) contain three unknown variables, ApJt kn and sr The rest are known quantities, provided the potential-dependent photocurrent (/ph) and the potential-dependent photoinduced microwave conductivity are measured simultaneously. The problem, which these equations describe, is therefore fully determined. This means that the interfacial rate constants kr and sr are accessible to combined photocurrent-photoinduced microwave conductivity measurements. The precondition, however is that an analytical function for the potential-dependent microwave conductivity (12) can be found. This is a challenge since the mathematical solution of the differential equations dominating charge carrier behavior in semiconductor interfaces is quite complex, but it could be obtained,9 17 as will be outlined below. In this way an important expectation with respect to microwave (photo)electro-chemistry, obtaining more insight into photoelectrochemical processes... 

Equations of gas dynamics with heat conductivity. We are now interested in a complex problem in which the gas flow is moving under the heat conduction condition. In conformity with (l)-(7), the system of differential equations for the ideal gas in Lagrangian variables acquires the form... 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

References Ablowitz, M. J., and A. S. Fokas, Complex Variables Introduction and Applications, Cambridge University Press, New York (2003) Asmar, N., and G. C. Jones, Applied Complex Analysis with Partial Differential Equations, Prentice-Hall, Upper Saddle River, N.J. (2002) Brown, J. W., and R. V Churchill, ComplexVariables and Applications, 7th ed., McGraw-Hill, New York (2003) Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003) Kwok, Y. K., Applied Complex Variables for Scientists and Engineers, Cambridge University Press, New York (2002) McGehee, O. C., An Introduction to Complex Analysis, Wiley, New York (2000) Priestley, H. A., Introduction to Complex Analysis, Oxford University Press, New York (2003). 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

Process models are unfortunately often oversold and improperly used. Simulations, by definition, are not the actual process. To model the process, assumptions must be made about the process that may later prove to be incorrect. Further, there may be variables in the material or processing equipment that are not included in the model. This is especially true of complex processes. It is important not to confuse virtual reality with reality. The claim is often made that the model can optimize a cure cycle. The complex sets of differential equations in these models cannot be inverted to optimize the multiple properties they predict. It is the intelligent use of models by an experimenter or an optimizing routine that finds a best case among the ones tried. As a consequence, the literature is full of references to the development of process models, but examples of their industrial use in complex batch processes are not common. 

The remaining two chapters of Part IV set the basis for the more advanced environmental models discussed in Part V. Chapter 21 starts with the simple one-box model already discussed at the end of Chapter 12. One- and two-box models are combined with the different boundary processes discussed before. Special emphasis is put on linear models, since they can be solved analytically. Conceptually, there is only a small step from multibox models to die models that describe the spatial dimensions as continuous variables, although the step mathematically is expensive as the model equations become partial differential equations, which, unfortunately, are more complex than the simple differential equations used for the box models. Here we will not move very far, but just open a window into this fascinating world. 

It is worth noting that the derivation outlined above is more generally applicable since, for many types of diffusional mass transfer (spherical, cylindrical, bounded, etc.), it is possible to rewrite the original differential equations in the time domain in terms of new variables in such a way that the second diffusion law is of the same form as eqn. (19b), with appropriate formulation of the boundary conditions [22, 75]. However, in finding the inverse transforms, difficulties may arise because of the more complex meaning of the time domain variables. 

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