> restart;

Now that we have the differential equations describing the system, we want to transform them so that they are in the dimensionless variables presented in the text.

First, input the two differential equations in their current forms.

> de1:=diff(Ts(r,t),t)=alpha/r^2*diff(r^2*diff(Ts(r,t),r),r);

> de2:=diff(Tf(t),t)=-3*alpha/B/R*diff(Ts(r,t),r);

> with(PDEtools,dchange):

Now to define the transformations we wish Maple to carry out.

> trans1:={Ts=T1-Thetas*(T1-T0),r=epsilon*R,t=tau*R^2/alpha};

Use dchange to carry out the transformations

> newde1:=dchange(trans1,de1,[Thetas(epsilon,tau),epsilon,tau]);

Maple seems to have not cancelled any terms, so we will do some simplification.

> simplify(newde1/(-1)/alpha*R^2/(T1-T0));

This is equivalent to equation 11.1-36 presented in the text.

Now for the second transformation.

> trans2:={Ts=T1-Thetas*(T1-T0),Tf=T1-Thetaf*(T1-T0),r=epsilon*R,t=tau*R^2/alpha};

> newde2:=dchange(trans2,de2,[Thetas(epsilon,tau),Thetaf(tau),epsilon,tau]);

> simplify(newde2/(-1)/alpha*R^2/(T1-T0));

This is equivalent to equation 11.1-40 in the text. We have therefore verified that the problem stated in dimensionless variables is indeed as presented in the text.