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:
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f(y):=1+lambda/(y(x)+0.01)^2; |
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(1) |
| > |
Eq:=1/x^2*diff((x^2*f(y)*diff(y(x),x)),x)=10*y(x)/(y(x)+0.1); |
| > |
eq:=expand(subs(lambda=0,Eq)); |
 |
(3) |
 |
(4) |
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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]); |
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(5) |
| > |
odeplot(sol,[x,y(x)],0..1,thickness=4,title="Figure Exp. 3.2.15.",axes=boxed); |
![[x = 0., y(x) = 0.227913456960032040e-1, diff(y(x), x) = 0.]](images/Example3.2.8 Rev 1_17.gif) |
(6) |
![[0, 0.1e-1, .1, 1, 10]](images/Example3.2.8 Rev 1_18.gif) |
(7) |
 |
(8) |
| > |
clr:=[red,green,gold,blue,magenta]; |
![[red, green, gold, blue, magenta]](images/Example3.2.8 Rev 1_20.gif) |
(9) |
| > |
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); |
| > |
display({seq(p[i],i=1..MM)}); |