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6.4. Ecuat¸ii stiff 141<br />
1.4<br />
1.2<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
1.0001<br />
0.9999<br />
ode23s<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />
x 10 4<br />
0<br />
1<br />
1 1.02 1.04 1.06 1.08 1.1 1.12<br />
x 10 4<br />
Figura 6.6: Propagarea unei flăcări, δ = 0.0001, eroarea relativă 1e-4 (sus) s¸i un<br />
zoom pe solut¸ie (jos), obt¸inute cu ode23s<br />
412 function evaluations<br />
99 partial derivatives<br />
106 LU decompositions<br />
318 solutions of linear systems<br />
A se compara cu cea generată de ode45:<br />
3040 successful steps<br />
323 failed attempts<br />
20179 function evaluations<br />
0 partial derivatives<br />
0 LU decompositions<br />
0 solutions of linear systems<br />
Se poate obt¸ine solut¸ia exactă a problemei (6.4.1). Ecuat¸ia este cu variabile separabile.<br />
Integrând se obt¸ine o ecuat¸ie implicită care-l dă pe y:<br />
1<br />
y +ln<br />
<br />
1<br />
y −1<br />
<br />
= 1<br />
δ +ln<br />
<br />
1<br />
δ −1<br />
<br />
−t.<br />
Expresia solut¸iei exacte este<br />
y(t) =<br />
1<br />
W(ae a−t )+1
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