Петухова Н.Ю. Методические указания по подготовке к экзамену по курсу уравнения математической физики
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°
°
R
2
R
1
x
0
∈
∈ R
n
(x, t) ∈ R
n+1
∂
2
u
∂t
2
= a
2
∆u
∆u =
n
P
i=1
∂
2
u
∂x
2
i
(x
0
, t
0
) ∈ R
n+1
∂u
∂t
= a
2
∆u (x
0
, t
0
) ∈
∈ R
n+1
∆u =
∂u
∂x
1
+ x
2
∂u
∂x
4
+ e
x
3
x
0
∈ R
n
(x, y, z) ∈ R
3
R
3
u
xx
+ u
yy
+ (x
2
− 1)u
zz
= u
z
u
xx
+ xyu
yy
+ e
x
u
zz
= 0
sin xu
xx
+ u
yy
= u
z
(x
2
+ y
2
+ z
2
− 1)u
xx
+ (2 + cos y)u
yy
+ 3u
zz
= xu
x
(x, y)
(x
2
+ y
2
− 1)u
xx
+ xyu
yy
− u
x
= 0
xu
xx
+ u
xy
− yu
yy
= u
x
xu
xx
− (x + y)u
xy
+ yu
yy
− u
y
= 0
x
2
u
xx
+ xyu
xy
+ u
yy
= 2u
y
u
xx
+ 2e
x
u
xy
+ u
yy
+ e
x
u
y
= 0
e
y
u
xx
+ u
xy
− u
yy
= u
x
u
xx
− 3x
2
u
xy
+ 2x
4
u
yy
+ 2u
y
= 0
16u
xx
+ 8u
xy
+ u
yy
− 16u
x
+ 4u
y
= 0
9u
xx
+ 6u
xy
+ u
yy
− 12u
x
− 4u
y
= 0
25u
xx
+ 10u
xy
+ u
yy
+ 20u
x
+ 4u
y
= 0
u(x, y) (x, y) ∈ R
2
4xu
xy
− 3u
y
= 0 2yu
xy
− u
y
= 0
u
xy
− 2xu
y
= 0
u(x, y)
u
xx
− 3u
xy
+ 2u
yy
= 0 u
xx
− u
xy
− 2u
yy
= 0
49u
xx
+ 28u
xy
+ 3u
yy
= 0 u
xx
+ 12u
xy
+ 27u
yy
= 0
48u
xx
+ 16u
xy
+ u
yy
= 0 75u
xx
+ 20u
xy
+ u
yy
= 0
(x, y) ∈ R
2
u
xx
− 4u
xy
+ 3u
yy
= 0 2u
xx
+ u
xy
− u
yy
= 0
u
xx
− 3x
2
u
xy
+ 2x
4
u
yy
− 2u
x
= 0
3x
2
u
xx
+ 2xyu
xy
− y
2
u
yy
= yu
x
xu
xx
− xu
yy
+ yu
x
= 0 y
2
u
xx
− x
2
u
yy
− u = 0
R
2
(x, y) ∈ R
2
u
xy
= 0,
u(x, 0) = x
2
+ 1,
u(0, y) = sin y + 1;
u
xy
= 0,
u(x, 0) = sin x + 1,
u(0, y) = e
y
+ y
2
;
½
u
xy
= 0,
u|
x=y
= 2y, u
x
|
x=y
= 1;
½
u
xy
= 0,
u|
x=y
= 2y, u
x
|
x=y
= 2y.
x ∈
∈ R
1
t > 0
½
9u
tt
= u
xx
,
u|
t=0
= e
3x
, u
t
|
t=0
= 2;
½
4u
tt
= u
xx
,
u|
t=0
= sin 2x, u
t
|
t=0
= e
−x
;
½
u
tt
= 4u
xx
,
u|
t=0
= x
2
+ 1, u
t
|
t=0
= 3x
2
;
½
u
tt
= 16u
xx
,
u|
t=0
= cos
2
x, u
t
|
t=0
= cos
2
x;
½
u
tt
= 9u
xx
,
u|
t=0
= x + e
−x
, u
t
|
t=0
= e
−x
;
½
5u
tt
= u
xx
,
u|
t=0
= x sin x, u
t
|
t=0
= x cos x
2
.
x ∈ R
1
t > 0
½
u
tt
= 4u
xx
+
x
t + 1
,
u|
t=0
= u
t
|
t=0
= 0;
½
u
tt
= a
2
u
xx
+
xt
t + 2
,
u|
t=0
= u
t
|
t=0
= 0.
R
2
R
3
R
2
t > 0 (x, y) ∈ R
2
½
9u
tt
= ∆u,
u|
t=0
= x
2
, u
t
|
t=0
= y;
½
u
tt
= a
2
∆u,
u|
t=0
= sin xe
y
, u
t
|
t=0
= 2;
½
u
tt
= a
2
∆u,
u|
t=0
= 0, u
t
|
t=0
= (2x + y)
2
;
(
u
tt
=
1
4
∆u,
u|
t=0
= 0, u
t
|
t=0
= (2x − 3y)
2
;
½
u
tt
= 4∆u,
u|
t=0
= 4x
2
+ 5y
2
, u
t
|
t=0
= 0;
½
u
tt
= 25∆u,
u|
t=0
= 5x
2
− 6y
2
, u
t
|
t=0
= 0.
R
3
t > 0 (x, y, z) ∈ R
3
½
u
tt
= 4∆u,
u|
t=0
= xy, u
t
|
t=0
= z;
½
u
tt
= a
2
∆u,
u|
t=0
= x + y, u
t
|
t=0
= sin z;
(
u
tt
=
1
4
∆u,
u|
t=0
= x
2
+ y
2
+ z
2
, u
t
|
t=0
= 0;
½
u
tt
= ∆u,
u|
t=0
= sin(x
2
+ y
2
+ z
2
), u
t
|
t=0
= 0;
½
u
tt
= 2∆u,
u|
t=0
= 0, u
t
|
t=0
= (x − 2y + z)
2
;
½
u
tt
= 16∆u,
u|
t=0
= x
2
+ y
2
− z
2
, u
t
|
t=0
= 0;
½
4u
tt
= ∆u,
u|
t=0
= 2x
2
+ y
2
− 2z
2
, u
t
|
t=0
= 0;
½
u
tt
= 9∆u,
u|
t=0
= 0, u
t
|
t=0
= (3x + 4y + 2z)
2
.
R
1
t > 0 x > 0
½
u
tt
= a
2
u
xx
,
u|
t=0
= u
t
|
t=0
= 0, u|
x=0
= sin t;
½
u
tt
= 4u
xx
,
u|
t=0
= u
t
|
t=0
= 0, u
x
|
x=0
= sin t;
½
9u
tt
= u
xx
,
u|
t=0
= u
t
|
t=0
= 0, u
x
|
x=0
= t;
½
u
tt
= 2u
xx
,
u|
t=0
= u
t
|
t=0
= 0, u
x
|
x=0
= t
2
.
(x, t) ∈ R
2
(x, t) ∈ R
n
n > 1
(x, y) ∈ R
2
t > 0
½
9u
t
= ∆u,
u|
t=0
= (xy)
2
;
½
u
t
= 4∆u,
u|
t=0
= sin x + sin y;
½
u
t
= 4∆u,
u|
t=0
= sin x · sin y;
½
4u
t
= ∆u,
u|
t=0
= e
x
y
2
.
R
3
t > 0 (x, y, z) ∈ R
3
½
9u
t
= ∆u,
u|
t=0
= x
2
+ y
2
+ z
2
;
½
u
t
= 2∆u,
u|
t=0
= sin x + sin y + cos z;
½
u
t
= 4∆u,
u|
t=0
= sin x · sin y · cos z;
½
4u
t
= ∆u,
u|
t=0
= xy + z
2
.
x
t
f(x)
g(x) F (x, t)
x
t
½
u
tt
= a
2
u
xx
, t > 0, 0 < x < π,
u|
t=0
= f(x), u
t
|
t=0
= g(x), u|
x=0
= 0 = u
x
|
x=π
;
½
u
tt
= 4u
xx
+ 4u + cos t, t > 0, 0 < x < 1,
u|
t=0
= f(x), u
t
|
t=0
= g(x), u
x
|
x=0
= 0 = u
x
|
x=1
;
½
4u
tt
= u
xx
+ F (x, t), t > 0, 0 < x < 2,
u|
t=0
= f(x), u
t
|
t=0
= g(x), u
x
|
x=0
= 0 = u
x
|
x=2
;
½
u
tt
= 9u
xx
+ F (x, t), t > 0, 0 < x < 2,
u|
t=0
= f(x), u
t
|
t=0
= g(x), u|
x=0
= 0 = u|
x=2
;
½
u
t
= a
2
u
xx
− 3u + x
2
e
t
, t > 0, 0 < x < 1,
u|
t=0
= f(x), u
x
|
x=0
= 0 = u|
x=1
;
½
9u
t
= u
xx
+ F (x, t), t > 0, 0 < x < 3,
u|
t=0
= f(x), u
x
|
x=0
= u|
x=3
= 0.
x
y t
x
y t
u
tt
= a
2
∆u, t > 0, 0 < x < 1, 0 < y < 1,
u|
t=0
= 2xy, u
t
|
t=0
= 0,
u|
x=0
= u|
x=1
= 0, u|
y=0
= u|
y=1
= 0;
u
tt
= 4∆u + xyt, t > 0, 0 < x < 1, 0 < y < π,
u|
t=0
= 0, u
t
|
t=0
= y,
u
x
|
x=0
= u
x
|
x=1
= 0, u|
y=0
= u|
y=π
= 0;
½
9u
t
= ∆u, t > 0, 0 < x < 1, 0 < y < 2,
u|
t=0
= x + y, u|
x=0
= u
x
|
x=1
= 0, u|
y=0
= u|
y=2
= 0;
½
u
t
= 25∆u, t > 0, 0 < x < π, 0 < y < 2π,
u|
t=0
= xy, u
x
|
x=0
= u|
x=π
= 0, u|
y=0
= u
y
|
x=2π
= 0.
R
3
R
3