MATLAB Runge-Kutta Routines

Both versions of ode45 use variable step Runge-Kutta algorithms. Six slopes or derivatives are used to calculate an approximation of order five, and then another of order four. These are compared to develop an estimate of the error at the current step. It is required that the estimated error be less than a predetermined amount. Otherwise, the step is reduced. In the old version of ode45 and ode23, the error checking amount is computed as 

The new version of ode45 and ode23 takes a more sophisticated approach to error checking. The solution is computed at step k and each component of the solution is required to satisfy its own error condition by satisfying one of the following inequalities 

The syntax for the old version of ode45 shown using help ode45 

0DE45 Solve differential equations, higher order method. 

String containing name of user-supplied problem description. 

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The kinetic profiles displayed in Figure 3-32 have been integrated numerically with Matlab s stiff solver odel5s using the rate constants /ci=1000 M 1s 1, /c2=100 s 1 for the initial concentrations [A]o=l M, [Cat]o=10 4 M and [B]o=[C]o=0 M. For this model the standard Runge-Kutta routine is far too slow and thus useless. 

Numerical integration of sets of differential equations is a well developed field of numerical analysis, e.g. most engineering problems involve differential equations. Here, we only give a very brief introduction to numerical integration. We start with the Euler method, proceed to the much more useful Runge-Kutta algorithm and finally demonstrate the use of the routines that are part of the Matlab package. 

The fourth order Runge-Kutta method is the workhorse for the numerical integration of ODEs. Elaborate routines with automatic step-size control are available in Matlab. 

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