Computer software Mathematica

By applying symbolic computation of Mathematica software, a mathematical mechanization of ADM based on the principle of parameterization, was carried out to reduce the solution for approximate expressions of model equations (A.l) and (A.2). The code of parameterized ADM is listed below. 

Finally, because the world is not an ideal gas, equations of state or activity coefficients frequently must be used to describe a real system of interest. This results in considerable mathematical complexity. Consequently, many problems are best solved using the computer software I have provided that is discussed in the previous appendix, or with MATHCAD, MATLAB, MATHEMATICA, POLYMATH, or equation-solving software and programs you develop on your own. 

The Routh-Hurwitz conditions are well known and can be used to determine, in principle, the stability properties of the steady state of any n-variable system. This advantage is, however, balanced by the fact that in practice their use is very cumbersome, even for n as small as 3 or 4. The evaluation, by hand, of all the coefficients Cl of the characteristic polynomial and the Hurwitz determinants A constitutes a rather arduous task. It is for this reason that in the past this tool of linear stability analysis could hardly be found in the literature of nonlinear dynamics. The situation changed with the advent of computer-algebra systems or symbolic computation software. Software such as Mathematica (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario) makes it easy to obtain exact, analytical expressions for the coefficients C/ of the characteristic polynomial (1.12) and the Hurwitz determinants A . 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions... 

In practice, the solution of polynomial equations is problematic if no simple roots are found by trial and error. In such circumstances the graphical method may be used or, in the cases of a quadratic or cubic equation, there exist algebraic formulae for determining the roots. Alternatively, computer algebra software (such as Maple or Mathematica, for example) can be used to solve such equations... 

By calculating ms using Eq. 18, the color change that passes through the gray point can be obtained. QxQy, J, and ms values can be calculated, for instance, by the Mathematica software (Wolfram Research, Champaign, IL, USA) with a personal computer. 

Mathematica, version 5.1 software for technical computation Wolfram Research Champaign, IL 2004. 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

As demonstrated by Mikhailov and Cotta [9] the eigenvalues could be computed with specified working precision by using Mathematica software system [10], but the Mathematica rule given in [9] needs a small modification to by applicable for high-order eigenvalues. 

The roots of (63) gives the desired eigenvalues. The FindRoot function of Mathematica software system calculates these roots starting from the values given by the asymptotic formula on p.ll3 of the book [20]. Fig. 5 shows the seconds per eigenvalue spend on 3 Gz computer to find 100 roots of a slightly modified eq.(63). The first 50 roots are computed much faster than the last 50 roots. 

A second option is to use existing packages for numerical methods. The software libraries by IMSL and NAG have a wide variety of state-of-the-art numerical integrators. These libraries are well documented, reliable, and flexible, and can be found at most university computing centers or networks. The packages Matlab, Mathematica, and Maple are more interactive and also have programs for solving ordinary differential equations. 

Most of the Mathematica symbols are the same as those used in Excel or various computer programming languages such as BASIC except for the use of a blank space for multiplication. Excel and BASIC use only the asterisk for multiplication. In ordinary formulas, placing two symbols together without a space between them can stand for multiplication. In Mathematica, if you write xy, the software will think you mean a variable called xy, and not the product of x and y. However, you can write either 2x or 2 x for 2 times x, but not x2. It is probably best to use the asterisk ( ) for multiplication rather than a space in input statements. Watch for the use of the blank space in output statements. Complex arithmetic is done automatically, using the capital letter I for. Several constants are available by using symbols Pi, E, I, Infinity, and Degree stand for n,e,i =, oo, and... 

Before closing this section it should be mentioned that most of the methods discussed above can be solved using commercially available programs such as symbolic computation (Mathematica, Maple, or Mathcad 3.0), equation solvers (TK Solver Plus or Mathcad 3.0), spreadsheets (e.g., Lotus 123, Quattro Pro, Microsoft Excel, or Wingz), simulators (Extend or Stella), or Microsoft FORTRAN 5.0. For example, one can introduce the idea of molecular mechanics on model systems using a spreadsheet. Actual applications of molecular mechanics are much better carried out on a software package designed for that purpose, however. 

Those interested in doing mathematics on a computer have an increasingly wide selection of software packages to choose from as noted above. At Saint John s University, Mathematica and Mathcad 3.0 have been used in the... 

Some computer exercises will be discussed briefly in this section. They serve as laboratory exercises for the second semester of physical chemistry sequence when quantum chemistry is taught. These computer exercises are also used in the lecture demonstration format. Sometimes they are given as extended problem assignments. Four software packages are used to various degrees in these exercises BASIC programs, spreadsheet templates, Mathematica, and Mathcad. 

Here i is a complex frequency in form of 8 -i- to /. For example, a differential equation describing the behavior of a capacitor = Av(f)IAt C, is transformed to [/(Ol = I s) where I s) = C s-Vis) - 5 v(0). I would like to emphasize here that due to advances in symbolic computing both direct and inverse Laplace transform are no longer the laborious table-searching procedures they used to be. Analytical software such as Maple or Mathematica can yield the Laplace transform (if it exists) of a function of almost any complexity in a matter of seconds, which makes this techifique very convenient to use for any researcher. So, 1 encourage the reader not to be intimidated by Eq. (2), get hold of some appropriate software, and think Laplace transform as something easily done. 

We anticipate that the students will come with a good background in math and in computations. Those who feel a little bit behind in terms of computational skills are strongly encouraged to download tutorials available on the World Wide Web for using EXCEL, MATHCAD, MATLAB , or MATHEMATICA that will come in very handy in solving complex problems. We leave the student and the instructor free to choose any or none of the above-listed available software. 

In this equation the initial concentrations [X]o and [Y]q appear explicitly and (0) = 0 for all initial concentrations. The solutions of such equations, (t), can be used to obtain the entropy production, as will be shown explicitly in section 9.5. Differential equations such as these, and more complicated systems, can be numerically solved on a computer, using software such as Mathematica or Maple. Sample Mathematica codes are provided in Appendix 9.1. 

Linear stability analysis does not provide a means of determining how the system will evolve when a state becomes unstable. To understand the system s behavior fully, the full nonlinear equation has to be considered. Often we encounter nonlinear equations for which solutions cannot be obtained analytically. However, with the availability of powerfiil desktop computers and software, numerical solutions can be obtained without much difficulty. To obtain numerical solutions to nonlinear equations considered in the following chapter, Mathematica codes are provided at the end of Chapter 19. 

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