Метод спуска

u (x,t)

1

1

( y)dS

t

(x atξ)dS

4 a2

4

1

t t

y

t

ξ

|y x| at

|ξ| 1

1

(x atξ)dSξ

at

( ,ξ)dSξ

4

4

|ξ| 1

|ξ| 1

| u ( x,t) |

1

sup | | dS

at

sup | | dS

1

3

ξ

3

ξ

4

4

|ξ| 1 x R

|ξ| 1 x R

sup |

|

at sup | |

x R3

dS

x R3

dS

sup | |

aT

sup | |

ξ

ξ

4

4

3

3

|ξ| 1

|ξ| 1

x R

x R

2

t

u (x,t)

1

( y)dS

(x atξ)dS

y

ξ

2

4 a t |y x| at

4

|ξ| 1

t

t sup | |

x R3

| u2 (x,t) |

sup | | dSξ

dSξ T sup | |

4

3

4

x R

3

|ξ| 1

x R

|ξ| 1

y,t

| x

y |

1

f

a

u (x,t)

dy,

M sup | f ( x,t) |

2

3

3

4 a |y x| at

| x y |

x R ,0 t T

| u (x,t) |

1

M

dy

M

dy

2

2

3

4 a |y x| at | x

y |

4 a |y x| at | x y |

M

2

at r2dr

M

(at)2

Mt2

d

sin d

2 2

4 a

2

r

4 a

2

2

2

0

0

0

MT 2

T 2

sup

| f (x,t) |

2

2

x R3 ,0 t T

x R3

x R3

T sup | (x) |

T 2

sup | f (x, t) |

2

x R3

x R3 ,0 t T