This section is concerned with the analytical (mathematical) solution of dynamic models. Analytical solution is the 'clever' way of solving the models and produces equations that allow generalisations to be made about system responses. Numerical solution (where the equations are solved by a computer) is a less subtle way of solving the models. Numerical solution only allows a solution for a particular set of circumstances to be produced - to produce general information it is necessary to perform several simulations under different conditions.
Although analytical solution might appear, at first glance, to be the best way of handling process models it does suffer from one very severe problem: there are no general solutions for most non-linear differential equations. Since virtually all equations that you'll generate in modelling a process system will be non-linear, this is a serious problem! The method used to get around this is to produce 'linearised' equations which approximate the behaviour of the non-linear equations in a limited region around a specified operating point. Although these equations can be solved, they are approximate solutions and this limits the extent to which they can provide general information about a process response.
This section starts with a discussion of what a non-linear differential equation is. It then goes on to describe the technique of using deviation variables. This is simple a way of transforming an equation to remove the steady-state components, which will be the same at the start and the end of the response. Using deviation variables considerably simplifies the maths. The next topic covered is linearisation, where you'll discover how to approximate non-linear differential equations with a 'linearised' equivalent. The last section presents standard solutions to several different types of linear differential equations.