Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

a

umn+1 − umn

+ b

umn +1 − umn

= C; m = 0, ±1, ±2, . . . ; n = 0, 1, 2, . . .

τ

h

m

± ±

(mh),

m = 0, 1, 2, . . .

u0 = ϕ0

a

un+1

n

un

un

m

− um

+ b

m

−h m−1

= C; m = 0, ±1, . . . ; n = 0, 1, . . . .

0

τ

(n+1,m)

(n+1,m)

(n+1,m)

(n,m)

(n,m+1)

(n,m-1)

(n,m)

(n,m-1)

(n,m)

(n,m+1)

(∂u/∂x)mn

(umn +1 − umn −1)/ (2h)

a

umn+1

umn

umn +1 − umn −1

−

+ b

= C; m = 0, 1, . . . ; n = 0, 1, . . . .

0

τ

2h

um

= ϕ0(mh); m = 0, 1, 2 . . .

0

τ−

h

a

umn+1

umn

+ b

umn+1+1 − umn+1

= C; m = 0, 1, . . . ; n = 0, 1, . . .

n

um = ϕ0(mh); m = 0, 1, . . .

u0 = f0(nτ ),

n = 0, 1, 2, . . .

(n+1,m)

(n+1,m+1)

(n+1,m-1)

(n+1,m)

(n,m)

(n,m)

0

τ−

h

umn+1

umn

n+1

a

+ b

umn+1 − um−1

= C; m = 0, 1, . . . ; n = 0, 1, 2, . . .

n

um = ϕ0(mh),

m = 0, 1, 2, . . .

u0 = f0(nτ ),

n = 0, 1, 2, . . .

umn+1+1

umn+1+1 = umn+1

−

ha

(umn+1 − umn ) +

Ch

, n = 0, 1, 2, . . . , m = 0, 1, 2, . . .

bτ

b

um1 +1 = um1

−

ha

(um1 − um0 ) +

Ch

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

bτ

b

u

z = y ln(x2 − y2);

1 ∂z

+

1 ∂z

=

z

x ∂x

y

∂y

y2

y

∂2u

∂2u

u = arctg

;

+

= 0

x

∂x2

∂y2

u = cos(x − y) + ln(x + y);

∂2u

=

∂2u

∂y2

∂x2

u = arcsin(x + ay);

∂u

= a

∂u

.

∂y

∂x

u = et sin x;

∂u

+

∂2u

= 0.

∂t

∂x2

u = cos(xy) + y2;

uxx + uy + y2u = y4.

z = z(x, y)

z2x − yz + x2 = 0;

zxy(2xz − y) + 2xzxzy + 2zzy = zx.

z

−

x

zxy

− zx + zy = 1.

= 1;

y

z

∂2u

=

x2

.

∂x∂y

1 + y2

∂2u

y

=

.

∂x∂y

1 + y2

∂2z

=

√

y

.

∂x∂y

1 − x

2

∂3V

= xzexy.

∂x∂y∂z

Lu = f,

L

u f

u

L

f

a

∂u

+ b

∂u

, 0 x ∞, 0

t ∞,

Lu =

∂t

∂x

u(0, x),

0 x < ∞,

0 t < ∞,

u(t, 0),

δf (h) → 0

h → 0, τ → 0,

u(h) → [u]h

h → 0 τ → 0