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An example. Exponential decay

A single phase exponential decay is defined by this equation.


Simple calculus shows that this has the property that the derivative of Y is proportional to the value of Y.

 dY/dX = -K*Y

How to fit a differential equation with Prism.

Prism can fit a model defined by a differential equation.

Choose the differential equation type at the top of the Equation dialog, and define Y' ( the derivative of Y with respect to X) as a function of X and parameters. For the example, enter:

 Y' = -K*Y

That's it. Putting an apostrophe after the Y on the left side of the equation tells Prism that  you are defining the derivative of Y with respect to X.  


Prism does not understand the other nomenclature for differential equations. Don't try to define an equation that starts with "dY/dX = ".

Note that X doesn't actually appear in the equation. That's ok. It is there in spirit, since Y' defines the derivative of Y with respect to X.

When you look at that equation, there appears to be only one parameter, K. In fact, the equation has two parameters. Prism generates a parameter Y[0], which is the value of Y at X=X0.

When you go to add constraints and initial values, Y[0] appears just like the other parameter K.

You can set X0 to any constant value you want, but it is usually set to 0.0. If you want to choose a different value, set it on the bottom of the initial values tab of the dialog that defines the equation. Note this is the dialog that defines the equation, not the dialog used for each fit.

Fitting a differential equation requires more calculations, so it takes noticeably more time that fitting the usual kind of equation.

It is only possible to define Y'. It is not possible to use differential equations to define intermediate variables. This would be useful for fitting compartmental models, but Prism cannot (yet) fit this kind of model.

With models defined as differential equations, Prism 6 did not allow you do define different models for different data sets using the <A> ..<B>.. notation. Prism 7 does allow this. But note that in every case, you must define Y'. It is not possible to define Y' for some data sets and define Y for others.

Prism 6 always fit the Y[0] value and shared its value for all data sets. Prism 7 lets you set it to a constant value, and lets you share it among data sets or not.


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