Derivatives and Differential Equations

Some computational problems may involve calculating or obtaining derivatives of functions. Depending on the complexity of the function, it may not be possible to obtain an explicit analytical expression for the derivative. Similarly, some of the problems may involve obtaining the derivative from observed data. For example, an experiment conducted for determination of the kinetics of a reaction will yield concentration-time data. An alternative method of determining the rate constant for the reaction involves regressing the rate of the reaction as a function of concentration. The rate of the reaction is defined as ; thus, the problem involves estimating the derivative from the concentration-time data. One of the numerical techniques for obtaining the derivative is represented by equation 4.17.

The subscripts refer to the time period. Thus, C_Ai is the concentration at time t_i, and so on. The derivative is approximated by the ratio of differences in the quantities. This formula is termed the forward difference formula, as the derivative at t_i is calculated using values at t_i and t_i+1. Similarly, there are backward and central difference formulas that are also applied for the calculation of the derivative [4, 10]. The comparative advantages and disadvantages of the different formulas are beyond the scope of this book and are not discussed further.

Similarly, the numerical techniques for integration of ordinary and partial differential equations are beyond the scope of this book. Interested readers may find a convenient starting point in reference [4] for further knowledge of such techniques.