Solving Differential Equations

There is still another technique, in which dimension analysis is used. The differential equations of transport can be set up sometimes easily, but it is too difficult to solve them under the conditions required. However, if the equations are made dimensionless, then the relevant combinations of dimensionless quantities can be found out. 

We follow here the derivation given by Bird [31, pp. 97-98]. The equation of continuity for a Newtonian fluid with constant density and viscosity is given by 

This set of equations can be made dimensionless by using the quantities 

Here we use the star ( ), following the notation of Bird [31] to indicate that we are dealing with a reduced and dimensional physical quantity. The result is 

Re is the Reynolds number and Fr is the Froude number. Numerous flow problems can be solved by applying similarity relations. For the individual problems, the various scaling parameters given in Eqs. (11.8) to (11.9) must be chosen in a proper way. For example. Bird [31, pp. 101-102] exemplifies how the depth of a vortex in an agitated tank can be found out by similarity calculations. 

---

In solving differential equations such as the Schrodinger equation involving two or more variables (e.g., equations that depend on three spatial coordinates x, y, and z or r, 0,... 

The z -transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. 

Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. 

The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations. Suppose we have a grid of equidistant points... 

Notiee that this method of solving differential equations yields the desired particular solution, the initial conditions being introdueed early in the procedure. 

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

Computer tools can contribute significantly to the optimization of processes. Computer data acquisition allows data to be more readily collected, and easy-to-implement control systems can also be achieved. Mathematical modeling can save personnel time, laboratory time and materials, and the tools for solving differential equations, parameter estimation, and optimization problems can be easy to use and result in great productivity gains. Optimizing the control system resulted in faster startup and consequent productivity gains in the extruder laboratory. 

Preliminary comments. By applying approximate methods the problem of solving differential equations leads to the systems of linear algebraic equations ... 

Schemes on non-equidistant grids. Quite often, in practical implementations difference schemes on non-equidistant grids are in common usage for solving differential equations. In Chapter 2, Section 1 we have produced for the simplest equation u" = —/ a difference scheme on the non-equidistant grid... 

We first provide an impetus of solving differential equations in an approach unique to control analysis. The mass balance of a well-mixed tank can be written (see Review Problems) as... 

Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or linearized differential equation, dy/dt = f(y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y(s). The time domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. 

The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations [Gottlieb, D., and S. A. Orszag, Numerical Analysis of Spectral Methods Theory and Applications, SIAM, Philadelphia (1977) Trefethen, L. N., Spectral Methods in Matlab, SIAM, Philadelphia (2000)]. Suppose we have a grid of equidistant points... 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

When PSpice runs a Transient Analysis, it solves differential equations to find voltages and currents versus time. The time between simulation points is chosen to be as large as possible while keeping the simulation error below a specified maximum. In some cases, where PSpice can take large time steps, you may get a graph that does not look sinusoidal ... 

The Runga-Kutta method is an effective numerical method of solving differential equations. Consider the following first-order differential equation... 

In this section we deal with estimating the parameters p in the dynamical model of the form (5.37). As we noticed, methods of Chapter 3 directly apply to this problem only if the solution of the differential equation is available in analytical form. Otherwise one can follow the same algorithms, but solving differential equations numerically whenever the computed responses are needed. The partial derivations required by the Gauss - Newton type algorithms can be obtained by solving the sensitivity equations. While this indirect method is... 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

To gain facility in setting up and solving differential equations using standard software packages, solve the initial-value problem discussed in the text (i.e., Eqs. 12.196, 12.197, and 12.200). 

---