Numerical techniques software packages

A large number of chemical/biological processes will be presented, modeled and efficient numerical techniques will be developed and programmed using MATLAB 2. This is a sophisticated numerical software package. MATLAB is powerful numerically through its built-in functions and it allows us to easily develop and evaluate complicated numerical codes that fulfill very specialized tasks. Our solution techniques will be developed and discussed from both the chemical/biological point of view and the numerical point of view. 

Warner [176] has given a comprehensive discussion of the principal approaches to the solution of stiff differential equations, including a hundred references among the most pertinent books, papers and application packages directed at simulating kinetic models. Emphasis has been put not only on numerical and software problems such as robustness, improving the linear equation solvers, using sparse matrix techniques, etc., but also on the availability of a chemical compiler, i.e. a powerful interface between kineticist and computer. 

Graphical techniques have lost favor because of the availability of computers and software packages that make numerical procedures much less tedious than graphical techniques. Furthermore, numerical procedures are less subjective and are usually more accurate than graphical procedures. The method of least squares is a numerical procedure for finding a continuous function to represent a set of data points. Our data points are represented by ordered pairs of numbers, (xi, yi), (x2, y2), (x3, ys), etc. where x is the independent variable. We assume that there is some function... 

Even though it is possible to convince software packages that second-order ODEs can be solved using techniques for first-order ODEs, all numerical methods require that both boundary condition for a second-order ODE must be known at the starting point. In other words, both boundary conditions must be known at the same value of the spatial coordinate. Split boundary value problems do not conform to this requirement. The mass balance for diffusion and chemical reaction is typically classified as a split boundary value problem. 

The state-space model in Eq. 6-75 contains both dependent variables, the elements of x, and independent variables, the elements of u and d. But why is x referred to as the state vector This term is used because x(t) uniquely determines the state of the system at any time, t. Suppose that at time t, the initial value x 0) is specified and u(t) and d t) are known over the time period [0, t]. Then x(t) is unique and can be determined from the analytical solution or by numerical integration. Analytical techniques are described in control engineering textbooks (e.g., Franklin et al., 2005 Ogata, 2008), while numerical solutions can be readily obtained using software packages such as MA TLAB or Mathematica. 

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