The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 181
to do this using Newon’s method we must compute the Jacobian matrix is
⎛
⎞
∂F 1 /∂A n ∂F 1 /∂B n ∂F 1 /∂X n ∂F 1 /∂E n
J = ⎜∂F 2 /∂A n ∂F 2 /∂B n ∂F 2 /∂X n ∂F 2 /∂E n
⎟
⎝∂F 3 /∂A n ∂F 3 /∂B n ∂F 3 /∂X n ∂F 3 /∂E n
⎠
∂F 4 /∂A n ∂F 4 /∂B n ∂F 4 /∂X n ∂F 4 /∂E n
⎛
⎞
1/h + αE n 0 −β αA n
(9.162)
= ⎜ 0 1/h −γ 0
⎟
⎝ −αE n 0 1/h + (γ + β) −αA n
⎠ (9.163)
0 0 1 1
We can implement the Jacobian calculation in Mathematica as follows
jac[f , vars ] := Module[dF,
If[Length[f] ≠ Length[vars], Return[$Failed]];
dF[func ] := Map[D[func, #] &, vars];
Return[Map[dF, f]]
]
and then we can calculate the Jacobian with
F={ (A n -A n−1 )/h + (α*A n *e n -β*X n ),
(B n -B n−1 )/h - γ*X n ,
(X n -X n−1 )/h - (α*A n *e n - (β + γ)*X n ),
e n +X n -1 };
jac[F, {A n , B n , X n , e n }]
Note that because the symbol E is reserved in Mathematica for the exponential e,
we have replaced the symbol for the enzyme with a lower case e.
At each time step we need to apply the iteration formula and invert it using
Newton’s formula, e.g.,
nextStep[h , {ALast , BLast , XLast , eLast }] :=
Module[F,J,A,X,e,B,∆,IJ,IJF,ɛ=10 −12 ,A n ,X n ,e n ,B n ,
F={ (A n -ALast)/h + (α*A n *e n -β*X n ),
(B n -BLast)/h - γ*X n ,
(X n -XLast)/h - (α*A n *e n - (β + γ)*X n ),
e n +X n -1 };
J= jac[F, {A n , B n , X n , e n }];
IJ = Inverse[J];
IJF = IJ.F;
{A, B, X,e} = ALast, BLast, XLast, eLast;
For [i = 0, i 5, i++,
∆ = -IJF /. {A n -> A, B n -> B, X n -> X, e n ->e};
{A, B, X, e} += ∆;
If[Abs[∆] < ɛ, Break[]];
];
Return[A, B, X, e]
]
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge