Complex Mechanisms that Require Numerical Integration

Complex Mechanisms that Require Numerical Integration... 

For reaction mechanisms that have explicit solutions to the set of differential equations, it is always also possible to define the derivatives dC /dp explicitly. In such cases the Jacobian can be calculated in explicit equations and time consuming numerical differentiations are not required. The equations are rather complex, although implementation in Matlab is straightforward. The calculation of numerical derivatives is always possible and for mechanisms that require numerical integration, it is the only option. 

There are many analytical chemistry textbooks that deal with the chemical equilibrium in fairly extensive ways and demonstrate how to resolve the above system explicitly. However, more complex equilibrium systems do not have explicit solutions. They need to be resolved iteratively. In kinetics, there are only a few reaction mechanisms that result in systems of differential equations with explicit solutions they tend to be listed in physical chemistry textbooks. All other rate laws require numerical integration. 

In two later sections, we will deal with numerical integration, which is required to solve the differential equations for complex mechanisms. Before that, we will describe nonlinear fitting algorithms that are significantly more powerful and faster than the direct-search simplex algorithm used by the MATLAB function fminsearch. Of course, the principle of separating linear (A) and nonlinear parameters (k) will still be applied. 

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