Example 3.2.6 Diffusion with Second Order Reaction

Example 3.2.1 is solved here using Maple's 'dsolve' command. The boundary condition at the surface, x = 1 is taken as:

Example 3.2.1 is solved below in Maple with a modified boundary condition:

> | restart:

> | with(plots):

Enter the governing equation:

> | Eq:=diff(c(x),x$2)=Phi^2*c(x)^2;

| (1)

The value of the parameter is substituted here:

> | eq:=subs(Phi=1,Eq);

| (2)

The boundary conditions are entered here:

> | BCs:=(D(c)(0),D(c)(1)=100*(1-c(1)));

| (3)

The numerical solution is obtained here:

> | sol:=dsolve({eq,BCs},{c(x)},numeric);

| (4)

The concentration profile obtained is plotted here:

> | odeplot(sol,[x,c(x)],0..1,thickness=4,title="Figure Exp. 3.2.10.",axes=boxed,color=gold);

Next, the problem is solved for a higher value of Φ:

> | eq:=subs(Phi=10,Eq);

| (5)

> | BCs:=(D(c)(0),D(c)(1)=100*(1-c(1)));

| (6)

> | sol:=dsolve({eq,BCs},{c(x)},numeric);

| (7)

> | odeplot(sol,[x,c(x)],0..1,thickness=4,title="Figure Exp. 3.2.11.",axes=boxed,color=brown);

We observe that as Φ increases, the profile becomes steeper and the time taken to solve the problem also increases.

> |