Soluzioni di Equazioni di Fourier 

[Graphics:../Images/index_gr_1.gif]

Caso in cui l'equazione di Fourier non è omogenea per la presenza di un termine di sorgente      [Graphics:../Images/index_gr_2.gif] = [Graphics:../Images/index_gr_3.gif] - q(x) 

[Graphics:../Images/index_gr_4.gif]
[Graphics:../Images/index_gr_5.gif]
[Graphics:../Images/index_gr_6.gif]
[Graphics:../Images/index_gr_7.gif]
[Graphics:../Images/index_gr_8.gif]
[Graphics:../Images/index_gr_9.gif]

[Graphics:../Images/index_gr_10.gif]

[Graphics:../Images/index_gr_11.gif]
[Graphics:../Images/index_gr_12.gif]
[Graphics:../Images/index_gr_13.gif]

[Graphics:../Images/index_gr_14.gif]

[Graphics:../Images/index_gr_15.gif]

Cerchiamo la soluzione nello stato stazionario  t->∞ 

[Graphics:../Images/index_gr_16.gif]
[Graphics:../Images/index_gr_17.gif]
[Graphics:../Images/index_gr_18.gif]
[Graphics:../Images/index_gr_19.gif]
[Graphics:../Images/index_gr_20.gif]
[Graphics:../Images/index_gr_21.gif]
[Graphics:../Images/index_gr_22.gif]
[Graphics:../Images/index_gr_23.gif]
[Graphics:../Images/index_gr_24.gif]
[Graphics:../Images/index_gr_25.gif]

[Graphics:../Images/index_gr_26.gif]

[Graphics:../Images/index_gr_27.gif]
[Graphics:../Images/index_gr_28.gif]
[Graphics:../Images/index_gr_29.gif]

[Graphics:../Images/index_gr_30.gif]

I coefficienti  si ricavano attraverso la formula data da Fourier 

[Graphics:../Images/index_gr_31.gif]
[Graphics:../Images/index_gr_32.gif]
[Graphics:../Images/index_gr_33.gif]
[Graphics:../Images/index_gr_34.gif]
[Graphics:../Images/index_gr_35.gif]
[Graphics:../Images/index_gr_36.gif]

[Graphics:../Images/index_gr_37.gif]

La soluzione prodotto ν(x,t) è data da una  sommatoria contenente il parametro n 

[Graphics:../Images/index_gr_38.gif]
[Graphics:../Images/index_gr_39.gif]

[Graphics:../Images/index_gr_40.gif]

[Graphics:../Images/index_gr_41.gif]

[Graphics:../Images/index_gr_42.gif]

[Graphics:../Images/index_gr_43.gif]
[Graphics:../Images/index_gr_44.gif]

[Graphics:../Images/index_gr_45.gif]

[Graphics:../Images/index_gr_46.gif]
[Graphics:../Images/index_gr_47.gif]

[Graphics:../Images/index_gr_48.gif]

[Graphics:../Images/index_gr_49.gif]
[Graphics:../Images/index_gr_50.gif]

[Graphics:../Images/index_gr_51.gif]

[Graphics:../Images/index_gr_52.gif]

Possiamo concludere che per t->∞ la ν(x,t)  ->0

La vista in 3D lo conferma 

[Graphics:../Images/index_gr_53.gif]

[Graphics:../Images/index_gr_54.gif]

[Graphics:../Images/index_gr_55.gif]

Ma qui si vede meglio 

[Graphics:../Images/index_gr_56.gif]

[Graphics:../Images/index_gr_57.gif]

[Graphics:../Images/index_gr_58.gif]

Tuttavia, per t=0,  la ν(x,t) non  "coincide"  con la νic(x,0).  Teniamo quindi
  conto di altri termini dello sviluppo in serie; poniamo n=9. 

[Graphics:../Images/index_gr_59.gif]

[Graphics:../Images/index_gr_60.gif]

[Graphics:../Images/index_gr_61.gif]

Ovviamente   si nota la discontinuità per x=1. 

[Graphics:../Images/index_gr_62.gif]

[Graphics:../Images/index_gr_63.gif]

[Graphics:../Images/index_gr_64.gif]
[Graphics:../Images/index_gr_65.gif]

[Graphics:../Images/index_gr_66.gif]

[Graphics:../Images/index_gr_67.gif]
[Graphics:../Images/index_gr_68.gif]

[Graphics:../Images/index_gr_69.gif]

[Graphics:../Images/index_gr_70.gif]
[Graphics:../Images/index_gr_71.gif]

[Graphics:../Images/index_gr_72.gif]

[Graphics:../Images/index_gr_73.gif]
[Graphics:../Images/index_gr_74.gif]

[Graphics:../Images/index_gr_75.gif]

[Graphics:../Images/index_gr_76.gif]

La soluzione somma si presenta come segue 

[Graphics:../Images/index_gr_77.gif]

[Graphics:../Images/index_gr_78.gif]

[Graphics:../Images/index_gr_79.gif]
[Graphics:../Images/index_gr_80.gif]

[Graphics:../Images/index_gr_81.gif]

[Graphics:../Images/index_gr_82.gif]

Converted by Mathematica      May 26, 2003