Similarity Method
M.M. Yovanovich
SIMMETHOD.MWS
> restart:
Similarity method applied to diffusion equation.
The similarity method is applied to the one-dimensional diffusion equation (heat equation) defined in the half-space
. The PDE
is transformed into the ODE
by means of the similarity parameter defined as
or as
. Both definitions lead to the identical solution of the PDE.
The second definition will be used in this Maple worksheet.
Partial differential equation.
> PDE:= diff(T(x,t),x$2) - diff(T(x,t),t)/alpha = 0;
Similarity parameter.
> eta(x,t):= x/(2*sqrt(alpha*t));
Partial derivatives of the similarity parameter.
> diff(eta(x,t),x); diff(eta(x,t),t);
Conversion of the partial derivatives of the PDE.
> Tx:= diff(T(eta),eta)*diff(eta(x,t),x);
> Txx:= diff(Tx,eta)*diff(eta(x,t),x);
> Tt:= diff(T(eta),eta)*diff(eta(x,t),t);
Transformed ODE.
> ODE:= Txx-Tt/alpha = 0;
> ODE2:= subs(x = 2*sqrt(alpha*t)*eta, ODE);
The transformed ODE.
> ODE3:= expand(4*alpha*t*ODE2);
Solution of the transformed ODE.
>
unassign(eta):
sol:= subs(_C1=C1, _C2=C2, dsolve(ODE3, T(eta)));
Alternative form of the solution.
> sol2:= C3*erfc(eta) + C4;
Both forms appear in solutions of the one-dimensional diffusion equation in the half-space
.