Example 3.2.8 Diffusion of a Substrate in an Enzyme Catalyzed Reaction - BVPs with Removeable Singularity

Boundary value problems in cylindrical and spherical coordinates have an inherent singularity at x = 0. These problems can be tackled using Maple's inbuilt midpoint methods. For example, when obtaining the diffusion of a substrate in an enzyme catalyzed reaction,[6] the governing equation for the dimensionless concentration is

where f is a dimensionless function which describes the change of diffusion coefficient as a function of concentration:

f=1+

The boundary conditions are:

and

y

This boundary value problem is solved in Maple below for different values of λ:

> | restart:

> | with(plots):

> | f(y):=1+lambda/(y(x)+0.01)^2;

| (1)

> | Eq:=1/x^2*diff((x^2*f(y)*diff(y(x),x)),x)=10*y(x)/(y(x)+0.1);

| (2)

> | eq:=expand(subs(lambda=0,Eq));

| (3)

> | BCs:=(D(y)(0),y(1)-1);

| (4)

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

Error, (in dsolve/numeric/bvp) system is singular at left endpoint, use midpoint method instead |

Maple identifies the singularity at x = 0 and suggests the midpoint method:

> | sol:=dsolve({eq,BCs},{y(x)},type=numeric,method=bvp[midrich]);

| (5)

> | odeplot(sol,[x,y(x)],0..1,thickness=4,title="Figure Exp. 3.2.15.",axes=boxed);

> | sol(0);

| (6)

> | L:=[0,1e-2,1e-1,1,10];

| (7)

> | MM:=nops(L);

| (8)

> | clr:=[red,green,gold,blue,magenta];

| (9)

> | for i to MM do

> | eq:=expand(subs(lambda=L[i],Eq));

> | sol:=dsolve({eq,BCs},{y(x)},type=numeric,method=bvp[midrich]);

> | p[i]:=odeplot(sol,[x,y(x)],0..1,thickness=4,color=clr[i],title="Figure Exp. 3.2.16.",axes=boxed);

> | od:

> | display({seq(p[i],i=1..MM)});

> |