Метод спуска

u1(x,t)

1

( y)dy

2

2

2 a t

(at)

| x

|y x| at

y |

u2 (x,t)

1

( y)dy

(at)

2

| x

2

2 a

|y x| at

y |

u3 (x,t)

1

t

d

f ( y, )dy

2

2

2

2 a

a

(t

)

0

|y x| a(t )

| x y |

R

y ( y1, y2 )

F( y1 )dy

x1 R

F( y1 )dy1

R

2

| x

2

|y x| R

y |

x1 R

y

2

R2 ( y x )2

R

1

1

( y1 x1 )2 y22 R2

x1 R x1

x1 R

y1

y

2

R2 ( y x )2

1 1

y

2

R2 ( y x )2

1

1

F( y1 )dy

x1 R

R2 ( y1 x1 )2

dy2

F( y1 )dy1

R

2

| x

2

R

2

( y1 x1)

2

2

|y x| R

y |

x1 R

R2 ( y1 x1 )2

y2

R2 ( y x )2

dy2

1

1

2

2

R2 ( y x )2

y2

R

( y1 x1 )

1

1

2

R2 ( y x )2

y2

1

1

arcsin

R2 ( y x )2

2

2

1

1

R ( y1 x1 )

F( y1 )dy

x1 R

F( y1 )dy1

R

2

| x

2

|y x| R

y |

x1 R

1

( y1)dy

1

x1 at

u1(x1,t)

( y1 )dy1

2

2

2 a t

|y x| at

(at) | x y |

2 a t

x at

1

1

(x1

at) a (x1

at) ( a)

(x1 at) (x1 at)

2a

2

u2 (x1,t)

1

( y1)dy

2

2

2 a |y x| at

(at) | x y |

1

x1 at

1

x1 at

( y )dy

(ξ)dξ

2a

2 a

1 1

x1 at

x1 at

1

t

f ( y , )dy

u3(x1,t)

2

2

2

d

1

2 a 0

|y x| a(t )

a (t ) | x y |

1

1

1

t

x a(t )

1

t

x a(t )

d

f ( y , )dy

d

f (ξ, )dξ

2a

2 a

1

1

0

x1 a(t )

0

x1 a(t )

(x at) (x at)

1

x at

1 t

x a(t )

u(x,t)

2

(ξ)dξ

d

f (ξ, )dξ

2a

2a

x at

0

x a(t )

2

u2

a2

u f (x,t), x R3 , t 0

t

3

u(x,0) (x), x R

u

(x,0)

(x), x R

3

t

x R3

x R3

T sup | (x) |

T 2

sup | f ( x,t) |

2

x R3

x R3 ,0 t T

u (x,t)

1

1

( y)dS

4 a2

1

t t

y

|y x| at

u (x,t)

1

2

( y)dS

y

2

4 a t |y x| at

,t

| x

y |

1

f y

a

u (x,t)

dy

2

3

4 a |y x| at

| x y |