Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

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Московский государственный индустриальный университет

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dτ = a√2tdS

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f (x)

1 ∞

l −∞

|f (x)|dx

f (x) f (x)

∞

f (x) = (A(α) cos αx + B(α) sin αx)dα,

∞

f (x − 0) + f (x + 0) = (A(α) cos αx + B(α) sin αx)dα. 2

−∞

A(α) B(α)
A(α) =	1	∞ f (τ ) cos ατ dτ ;
A(α) =	π	∞ f (τ ) cos ατ dτ ;
	π	−∞
	1
B(α) =	1	∞ f (τ ) sin ατ dτ.
B(α) =		∞ f (τ ) sin ατ dτ.
	π
		−∞

πl , 2lπ , . . . , kπl , . . .

α [0, +∞) .

f (x) A(α) B(α)

		∞
2		0
A(α) =			f (τ ) cos ατ dτ, B(α) = 0,
A(α) =	π		f (τ ) cos ατ dτ, B(α) = 0,
			∞
f (x) = 0				A(α) cos αxdx.
				f (x)
						∞
				2		0
A(α) = 0,			b(α) =				f (τ ) sin ατ dτ,
A(α) = 0,			b(α) =		π		f (τ ) sin ατ dτ,

∞

f (x) = B(α) sin αxdx.

∞		∞
f (x) =	1		f (τ )(cos ατ cos αx + sin ατ sin αx)dτ dα.

	π

0−∞

cos α(τ − x) = cos ατ cos αx + sin ατ sin αx

		∞	∞
f (x) =	1		dα	f (τ ) cos α(τ − x)dx.

	π
		0	−∞

l → ∞

u(x, t) = X(x)T (t)

uλ(x, t) = (A(λ) cos λx + B(λ) sin λx)e−λ2a2t.

λ λk

∞

u(x, t) = (A(λ) cos λx + B(λ) sin λx)e−λ2a2tdλ.

A(λ) B(λ) t = 0

∞

u|t=0 = ϕ(x) (A(λ) cos λx + B(λ) sin λx)dλ = ϕ(x).

		ϕ(x)
			A(λ) B(λ)
A(λ) =	1	∞ ϕ(τ ) cos λτ dτ ;
A(λ) =	π	∞ ϕ(τ ) cos λτ dτ ;
	π	−∞
	1
B(λ) =	1	∞	ϕ(τ ) sin λτ dτ.
B(λ) =	π		ϕ(τ ) sin λτ dτ.
	π
		−∞

		∞	∞
	1		dλ	2	2
u(x, t) =				ϕ(τ ) cos λ(τ − x)e−λ a		tdτ.
u(x, t) =	π			ϕ(τ ) cos λ(τ − x)e−λ a		tdτ.
		0	−∞

∞

cos λ(τ − x)e−λ a

tdλ =

e−(τ −x)

/(4a

t).

∞

tdλ dτ =

u(x, t) =

ϕ(τ )

cos λ(τ − x)e−λ a

−∞

∞

2a√

ϕ(τ )e−(τ −x)

/(4a

t)dτ.

πt

−∞

∞

		1				2	2
u(x, t) =	2a√					ϕ(τ )e−(τ −x)	/(4a	t)dτ.
			πt
					−∞
				0,		x < x
u(x, t) t=0	=	u0,				x1 1x x2
\|				0,		x > x2.

t > 0

	u			2	2
u(x, t) =	2a√0			e−(τ −x)	/(4a	t)dτ.
u(x, t) =	2a√0	πt		e−(τ −x)		t)dτ.
			x1

(z)

			x
1			0	2
(z) =	√			e−S	/2dS,
		2π

(−z) = −(z) (−∞) = −0, 5, (∞) = 0, 5.

			x2	−x
			a√		2t
	u			2
							/2a√2tdS =
u(x, t) =	2a√0					e−S
		πt

x1−x

a√

−x

e−S2/2dS =

a√

e−S2/2dS +

√2π

x x

1−

a√

−x

x1−x

a√

e−S2/2dS

= u0

e−S2/2dS

√2π

−

√2π

a√−2t

u(x, t) = u0 a√−2t

−

x2√− x

x1√− x

x < x1

x > x2

a 2t

x1 < x < x2

(z)

(∞) = 0, 5

= 0

x < x1

x > x2

x1 < x < x2

a x1

a = 1 x1 = 1 x2 = 3

√2t

−

√2t

u(x, t) = u

− x

1 − x

u(x, t)

x = 0 x = 0 u(0, t) = 0

(x) x < 0

t=0

								(x)
	1		∞	f (τ ) e−	(τ −x)2		(τ +x)2	dτ.
u(x, t) =	2a√		0			− e−
					4a2t		4a2t
		πt			4a2t		4a2t

∂u ∂x x=0

	1	∞		e−	(τ − x)2	+ e−	(τ + x)2		dτ.
u(x, t) =	1		f (τ )		4a2t			4a2t
u(x, t) =			f (τ )		4a2t			4a2t
	0
	2a√πt
	2a√πt

=			−
				u x=0				= u x=L	= 0,
		x L/2,			x			(0, L/2),
		0,			x / (0, L/2).
					∂u
	L		−
	L		−
=	100	(L x),			∂x	x=0, u x=L			= 0,
=		(L x),					x [0, L]

= 0

(x)

				∂u		∂u	= 0,
u	=
u	=
				∂x	x=0 =	∂x x=L
		100, x		(0, L/2],
		100, x		(0, L/2],
		20, x	(L/2, L].
t=0		20, x	(L/2, L].

x(−∞, +∞)

			100(1 − x/10),	x	[0, 10],
u		=	100(1 + x/10),	\|	\|
u	t=0	=	100(1 + x/10),	x	[−10, 0),
			0,	x > 10.
			0,	x > 10.

x (0, +∞)

							U	,
							U	,
u x=0 = 0,				u t=0 =			0,0
	∂u						U0,
			= 0,	u		=	0,
	∂x
	∂x	x=0			t=0

x[2, 10],

x(0, 2) (10, ∞).

0 < x < L, x ≥ L.

∂u − a2 ∂2u = 0. ∂t ∂x2

(n+1,m) (n+1,m-1) (n+1,m) (n+1,m+1)

		h	h
	τ		τ
h		h
(n,m-1)	(n,m)	(n,m+1)	(n,m)

0 x l 0 t T

u(0, x) = ϕ0(x),

				∂u
					∂u
0			1			2
α0u		x=0	+ α1		∂x x=0 = γ1(t);


β	u		+ β			= γ (t).

				∂x
		x=l

(n, m)

∂2u	n	umn +1		2umn + un
	m =		−		m−1	+ O(h2).
∂x2	m =			h2	m−1	+ O(h2).

∂2u n

∂x2 m

umn+1

−

umn

umn +1

−

2umn + umn

m = 1, 2, 3, . . . , M 1,

− a2

−

= 0,

n = 0, 1, 2, . . . , N −−1

um = ϕ0(xm),

m = 0, 1, 2, . . . , M,

n+1

α0u

+ α1

−

= γ1(t

n = 0, 1, . . . , N

−

h un+1

−

un+1

−

+ β1

−

= γ2(tn+1),

n = 1, 2, . . . , N

β0uM

n+1

(M + 1)

(N +

r = τ a /h

um0 = ϕ0(xm),

m = 0, 1, 2, . . . , M,

(a)

un+1 = (1 2r)un + r(un

+ un

m = 1, 2, . . . , M

(b)

−

m−1

m+1

n = 1, 2, . . . , N

−1,

−

α1/h

n+1

)

−

n+1

γ1(t

−

n = 0, 1, 2, ..., N

−

(c)

α /h

−

− n+1

uMn+1 =

β1/h

uMn+11 + γ2(t

) ,

n = 0, 1, 2, ..., N

(d)

β + β /h

−

+ β /h

n = 0

t = 0

(m = 1, 2, . . . , M − 1) n + 1 t = (n + 1)τ

t = (n + 1)τ n = n + 1

n < N

							(n + 1, m)
n + 1
	umn+1 − umn			n+1	− 2umn+1	n+1		m = 1, 2, . . .
	umn+1 − umn	−	a2 um+1		− 2umn+1	+ um−1 = 0,		m = 1, 2, . . .
	τ	−			h2			n = 1, 2, . . .

u n + 1

								n + 1
u1n+1, u2n+1, . . . , un+1					u		n
u1n+1, u2n+1, . . . , un+1
M−1
am · umn+1−1 + bm · umn+1 + cm · umn+1+1 = fm, m = 1, 2, . . . , M − 1,
am = cm = −r bm = 1 + 2r fm = umn									u	n + 1
									u	n + 1
							ym = umn+1
amym−1 + bmym + cmym+1 = fm,							m = 1, 2, . . . , M			−	1;
y0 = L0y1 + K0;										−
	−
yM = L˜M yM				1 + K˜M ,
L0 =				−α1/h	; K0 =		γ1(tn+1)	;
L0 =		α0 − α1/h			; K0 =			;
		α0 − α1/h				α0 − α1/h
˜				β1/h	˜		γ2(tn+1)
LM =					; KM =				.
LM =			β0 + β1/h		; KM =		β0 + β1/h		.