Уравнения математической физики. Сборник задач. Даишев Р.А - 51 стр.

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∂U(ξ,t)
∂t
= a
2
∂
2
U(ξ,t)
∂ξ
2
q
2
π
sin λξ. ξ 0 ∞
U(x, t)
d
¯
U
(s)
(λ, t)
dt
+ a
2
λ
2
¯
U
(s)
(λ, t) = 0,
¯
U
(s)
(λ, t) U(x, t)
¯
U
(s)
(λ, t) =
s
2
π
∞
Z
0
U(ξ, t) sin λξdξ
¯
U
(s)
(λ, t)
t = 0
¯
U
(s)
(λ, 0) =
s
2
π
∞
Z
0
U(ξ, 0) sin λξdξ =
=
s
2
π
∞
Z
0
f(ξ) sin λξdξ =
¯
f
(s)
(λ).
¯
U
(s)
(λ, t) =
¯
f
(s)
(λ)e
−a
2
λ
2
t
U(x, t) =
s
2
π
∞
Z
0
¯
U
(s)
(λ, t) sin λxdλ =
=
2
π
∞
Z
0
f(ξ)dξ
∞
Z
0
e
−a
2
λ
2
t
sin λξ sin λxdλ =
                                                                         ∂U (ξ,t)            2 U (ξ,t)
   Ð å ø å í è å. Óìíîæèì îáå ÷àñòè óðàâíåíèÿ                              ∂t
                                                                                    = a2 ∂    ∂ξ 2
   q
       2
íà    sin λξ. Èíòåãðèðóÿ ïî ξ îò 0 äî ∞, ïîëó÷èì äëÿ ñèíóñ-
       π
îáðàçà Ôóðüå ôóíêöèè U (x, t) äèôôåðåíöèàëüíîå óðàâíåíèå

                     dŪ (s) (λ, t)
                                    + a2 λ2 Ū (s) (λ, t) = 0,
                           dt
ãäå Ū (s) (λ, t)-ôóðüå îáðàç ôóíêöèè U (x, t), îïðåäåëÿåìûé êàê
                                           s     ∞
                      (s)                      2Z
                 Ū         (λ, t) =               U (ξ, t) sin λξdξ
                                               π
                                                    0


Íà÷àëüíîå óñëîâèå äëÿ Ū (s) (λ, t) ïîëó÷àåì èç îïðåäåëåíèÿ ïðè
t = 0:                   s ∞
               (s)           2Z
             Ū (λ, 0) =           U (ξ, 0) sin λξdξ =
                             π
                                                0
                         s
                                 Z∞
                             2
                     =                f (ξ) sin λξdξ = f¯(s) (λ).
                             π
                                 0

Ðåøåíèå äèôôåðåíöèàëüíîãî óðàâíåíèÿ äëÿ Ôóðüå-îáðàçà ïðè
ýòèõ íà÷àëüíûõ óñëîâèÿõ èìååò âèä:

                            Ū (s) (λ, t) = f¯(s) (λ)e−a λ t
                                                        2 2




Ïðèìåíÿÿ ê íåìó îáðàòíîå ñèíóñ-ïðåîáðàçîâàíèå Ôóðüå, ïîëó-
÷èì:                   s ∞
                        2 Z (s)
            U (x, t) =     Ū (λ, t) sin λxdλ =
                        π
                                           0
                     Z∞                Z∞
                 2                                  2 λ2 t
             =            f (ξ)dξ           e−a              sin λξ sin λxdλ =
                 π
                     0                 0



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