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u : R2 Q (t, x) u (t, x) R (t, x) R2 2u 2u - a2 = f (t, x), (t, x) Q, t2 x2 Q = (0, ) (0, l) (0, l) (0, ) = {t| 0 < t < } Q u|x=0 = (1) (t), u|x=l = (2) (t), t (0, ).

u|t=0 = (x), x (0, l), kl + t = a u = (x), x (0, l), t kl+ t= a k (0, 1, 2,...) [0, l] k k = 0 = 0 k = 0 [0, l] kl + u t = a 2u 2u kl (k + 1)l - a2 = 0, t,, x [0, l], t2 x2 a a kl (k + 1)l u|x=0 = (1)(t), u|x=l = (2)(t), t, a a u|t= = (k)(x), x [0, l], kl a u = (k)(x), x [0, l], t kl t= a u = (x), x [0, l], t kl+ t= a (k) (j) (j = 1, 2) (k) (j) (j = 1, 2) (k) (k) (k)(x + ) - (k)( - x) =a (k) ( - x) - (k) (x + ) kl + - x -2(1) + 2(x), x [0, ], a (k)(x + ) + (k)(x - ) =a (k) (x - ) - (k) (x + ) + +2(x), x [, l - ], (k)(2l - x - ) - (k)(x - ) =a (k) (2l - x - ) - (k) (x - ) + kl + x - l + +2(2) + 2(x), x [l -, l], a [0, l/2] (k)(x + ) - (k)( - x) =a (k) ( - x) - (k) (x + ) kl + - x -2(1) + 2(x), x [0, l - ], a (k)(2l - x - ) + (k)( - x) = a (k) (2l - x - ) - (k) ( - x) + kl + - x kl + + x - l +2(1) + 2(2) - 2(x), x [l -, ], a a (k)(2l - x - ) - (k)(x - ) =a (k) (2l - x - ) - (k) (x - ) + kl + x - l + +2(2) - 2(x), x [, l], a d(k)( - x) [l/2, l] (k) ( - x) = d( - x) kl + - x kl + - x kl + - x (1) = d(1) d a a a (k) [0, l/2] x kl + x - (k)(3 - x) - (k)(x - ) = 2(2 - x) - 2(1) a -a(k) (3 - x) +a(k) (x - ), x (, 2), kl + l - x - (k)(2l - 3 - x) - (k)(x + ) =2(2l - 2 - x) - 2(2) a -a(k) (x + ) + a(k) (2l - 3 - x), x [l - 2, l - ].

(k) [0, l/2] = 0 = 0 (k)(x) = (x) x [0, l] 2 = l - = l/3 l 2l kl x l (k)(l - x) - (k) x - =2 - x - 2(1) + - 3 3 a a 3a l -a(k) (l - x) + a(k) x -, 3 l l l l (k) x + + (k) x - = 2(x) + a(k) x - - a(k) x +, 3 3 3 3 l 4l kl 2l x (k)(l - x) - (k) x + =2 - x - 2(2) + - 3 3 a 3a a l -a(k) x + + a(k) (l - x) x [l/3, 2l/3] (k) 2l 4l kl x l (k)(l - x) =(x) + - x + - x - (1) + - 3 3 a a 3a kl 2l x l l -(2) + - + a(k) x - - a(k) x +, a 3a a 3 l 2l 4l kl x l (k) x - = (x) - - x + - x + (1) + - 3 3 3 a a 3a kl 2l x l -(2) + - + a(k) (l - x) - a(k) x +, a 3a a l 2l 4l kl x l (k) x + = (x) + - x - - x - (1) + - + 3 3 3 a a 3a kl 2l x l +(2) + - + a(k) x - - a(k) (l - x), a 3a a x [l/3, 2l/3] (k) l l kl + x x + - - x + (l - x) + (1) 3 3 a kl l x 2l 2l -(2) + - + a(k) - x - a(k) + x, x [0, l/3), a 3a a 3 l l kl 2l x (l - x) + x - + + x - (1) + - 3 3 a 3a a (k)(x) = kl l x 2l 4l -(2) - + + a(k) - x - a(k) - x, x (l/2, 2l/3), a 3a a 3 l 5l kl x 2l x - + (l - x) - - x - (1) + - + 3 3 a a 3a kl + l - x 2l 4l +(2) + a(k) x - - a(k) - x, x (2l/3, l).

a 3 (k) = l/ = 2l/2l 2l 2l 2l (k) x + - (k) - x =a (k) - x - (k) x + 3 3 3 kl 2l x l -2(1) + - + 2(x), x 0,, a 3a a 4l 2l 4l 2l (k) - x + (k) - x = a (k) - x - (k) - x + 3 3 3 kl 2l x kl l x +2(1) + - + 2(2) - + - 2(x), x [l/3, 2l/3], a 3a a a 3a a 4l 2l 4l 2l (k) - x - (k) x - = a (k) - x - (k) x - + 3 3 3 kl l x +2(2) - + - 2(x), x [2l/3, l].

a 3a a [l/3, 2l/3] l l (k) x + - (k)(l - x) = a (k) (l - x) - (k) x + 3 kl l x l -2(1) + - + 2 x -, x [l/3, 2l/3], a a a l l l l (k) x + +(k) x - = a (k) + x - (k) x - + 3 3 3 kl l x kl 2l x +2(1) - + + 2(2) + - - 2(l - x), x [l/3, 2l/3], a 3a a a 3a a l l (k)(l - x)-(k) x - = a (k) (l - x) - (k) x - + 3 kl x l +2(2) + - 2 x +, x [l/3, 2l/3].

a a l l kl l x (k) x + = a(k) (l - x) - a(k) x - - (1) + - + 3 3 a a a kl l x kl 2l x kl x +(1) - + +(2) + - + (2) +, a 3a a a 3a a a a l l kl l x (k)(l - x) = a(k) x + - a(k) x - + (1) - + + 3 3 a 3a a kl 2l x kl x kl l x l +(2) + - + (2) + + (1) + - - 2 x -, a 3a a a a a a a l l kl l x (k) x - = a(k) + x - a(k) (l - x) + (1) - + + 3 3 a 3a a kl 2l x kl x kl l x +(2) + - - (2) + + (1) + - - 2(l - x).

a 3a a a a a a a 2l 2l kl x a(k) + x - a(k) - x + (1) + + 3 3 a a kl l x kl l x +(2) + - - (2) + + + a 3a a a 3a a kl l x 2l +(1) - - - 2 - x, x [0, l/3), a 3a a 4l 2l kl 2l x a(k) - x - a(k) - x + (1) + - + 3 3 a 3a a kl l x kl l x (k)(x) = +(2) - + + (2) + - + a 3a a a a a kl x 2l (1) + - 2 - x, x (l/3, 2l/3), a a 2l 2l kl 4l x a(k) - x - a(k) x - - (1) + - + 3 3 a 3a a kl 2l x kl l x +(1) - + + (2) + - + a 3a a a a a kl l x +(2) + -, x (2l/3, l) a a a = 2l/ = l kl + l - x kl + x (k)(l - x) = (1) + (2) - (x), x [0, l], a a kl + x kl + l - x (k)(x) = (1) + (2) - (l - x), x [0, l].

a a (k) C1[0, l] kl kl (k)(0) = (1), (k)(l) = (2).

a a = (k) = l/3 = 2l/ = l (k) (k+1)l (k) C2[0, l] C1[0, l] (j) C2 kl, j = 1, a a kl kl l kl (k)(0) = (1), (0) = (1) +, a2(k) (0) = (1), a a 3a a kl kl l kl (k)(l) = (2), (l) = (2) +, a2(k) (l) = (2), a a 3a a u C2 Q(k) kl (k + 1)l = l/3 Q(k) =, (0, l) Q(k) Q(k) C2 Q(k) a a Q(k) (k) (0) (k) (k) C1[0, l] (k) (k) (j) (j = 1, 2) [0, l/3] [l/3, 2l/3] [2l/3, l] (k) (k) C1[0, l] (k) x = l/x = 2l/3 x = l/l 2l kl l kl l (k) = 2 - (0) + (1) + - (2) + a(k) - a(k) (l) = 3 3 a 3a a 2l kl l kl l =2 + (0) - (1) + - (2) + a(k) - a(k) (l).

3 a 3a a (k) x = l/ (k) (k) x = 2l/ (k) C1[0, l] (k) (k) C2 Q(k) (k) (k) (j) (j = 1, 2) (k) (j) (j = 1, 2) u C2 Q(k) = 2l/3 = l kl (k + 1)l (k) C2[0, l] C1[0, l] (j) C2, j = 1, a a kl kl 2l kl (k)(0) = (1), (0) = (1) +, a2(k) (0) = (1), a a 3a a kl kl 2l kl (k)(l) = (2), (l) = (2) +, a2(k) (l) = (2), a a 3a a C2 Qk = 2l/ (k) C1[0, l] (k+1)l (k) C2[0, l] C1[0, l] (j) C2 kl, j = 1, a a kl (k + 1) l kl (k)(0) = (1), (0) = (1), a2(k) (0) = (1), a a a kl (k + 1)l kl (k)(l) = (2), (l) = (2), a2(k) (l) = (2), a a a C2 Qk = l (k) C1[0, l] = l k = Q(0) k (0) (k) (k-1) C2 Q(k-1) Q(k-1) = l C2 Q(m) m = 0, 1,..., k = l m = 0 (0)(x) = (x) (0) (k) (j) (j = 1, 2) (j) (j = 1, 2) ( = l/3) ( = 2l/3) ( = l) k-[m + s(m)]l + (-1)mx (0)(x) = (-1)m(1) + a m=k [m - 1 + s(m)]l + (-1)mx + (-1)m-1(2) + (k) s(k)l + (-1)kx, a m= s(m) s m s(m) = 0 m (k) (j) (j = 1, 2) (k) = l/3 = 2l/3 = l (0) C1[0, l] (0)(0) = (1) (0), (0)(l) = (2) (0).

2u 2u - a2 = 0, (t, x) Q, t2 xu|x=0 = (1)(t), u|x=l = (2)(t), t (0, ), u u|t=0 = (x), = (0)(x), x (0, l).

t t= C2[0, l] (0) C1[0, l] (j) C2[0, ) (j = 1, 2) (0) = (1)(0), a2 (0) = (1) (0), (l) = (2)(0), a2 (l) = (2) (0), C2 Q u (0) (k) (k) Q Q(m) Q(m) Q(m) Q(m) m = 0, 1, 2,...

1 2 3 C2[0, l] C1[0, l] (j) C2[0, ] j = 1, kl kl (0) = (1) +, (l) = (2) +.

a a a a C2 Q k = 0, 1, 2,... = 0, l/3, 2l/3, l u x+at-ml (m)(x - at + ml) + (m)(x + at - ml) u(t, x) = + (m)()d, t, x Q(m), 2 2a x-at+ml (m)(x + at - ml) + (m)(at - x - ml) x u(t, x) = + (1) t - + 2 a at+x-ml + (m)()d, t, x Q(m), 2a at-x-ml (m)(x - at + ml) - (m)(-x - at + (m + 1)l) x - l u(t, x) = +(2) t + + 2 a l-x-at+(m+1)l + (m)()d, t, x Q(m), 2a x-at+ml (m)(l - x - at + (m + 1)l) + (m)(at - x - ml) x u(t, x) = + (1) t - + 2 a l-x-at-(m+1)l x - l + (2) t + + (m)()d, t, x Q(m), a 2a at-x-ml Q Q(m) Q Q(m) r = 1, 2, 3, r r (0) (k) (0) (1) (2) (1) (2) (1) (2) Q m [j - s(m + j)]l + (-1)m+j-1x (m)(x) = (-1)m (s(m)l + (-1)mx) + (-1)m+j (1) + a j=[j - s(m + j - 1)]l + (-1)m+jx + (2), m = 1, 2, 3,..., a (m)(x) = (-1)k-m(k)(s(k - m)l + (-1)k-mx)+ k-[j - s(k - j + 1)]l - (-1)k-jx +(-1)k-m (-1)k-j (1) + a j=m [j - s(k - j)]l + (-1)k-jx +(2), m = 0, 1,..., k - 1.

a (m)(x) = (-1)m-k(k)(s(m - k)l + (-1)m-kx)+ m [j - s(m - j)]l + (-1)m-j+1x + (-1)m-j (1) + a j=k+[j - s(m - j + 1)]l + (-1)m-jx + (2), m = k + 1, k+2, k + 3,....

a u Q (t, x) u(t, x) R v Q (t, x) v(t, x) R v|x=0 = (1)(t), v|x=l = (2)(t), t [0, ), v v|t=0 = 0, = p(x), x [0, l].

t kl+ t= a f Q (t, x) f(t, x) R v w(t,, x) 2w 2w - a2 = 0, (t, x) Q, t2 x [0, ) t t+l w|x=0 = f(, y)dy = g(1)(t, ), w|x=l = f(, y)dy = g(2)(t, ), t [0, ), 0 w w|t=0 = 0, = f(, x), x [0, l], t t= f [0, l] x f [0, ) f : [0, ) [0, ) (, x) f(, x) R f(, x) = f(, x) (, x) Q f C0,1 Q f C0,1 R2 C0,++ R2 = [0, )[0, ) = {(, x)| 0, x < } ++ g(j) C0,2 R2 C0,2 R++ ++ [0, ) g(1)(t, ) g(2)(t, ) t [0, ) = f(, 0) = f(, l) t t t=0 t= w(t,, x) w C2,0,2 R2 [0, l] ++ w v t v(t, x) = w(t -,, x)d.

v C2 Q kl+ a v w(t -,, x) v|t=0 = 0, = d = p(x), x [0, l], t kl+ t kl+ t= t= a a v p t = (kl + )/a t t t (1)(t) = w(t -,, 0)d, (2)(t) = w(t -,, l)d.

0 u u = + v v 2 - a2 = 0, (t, x) Q, t2 x|x=0 = (1)(t) - (1)(t), |x=l = (2)(t) - (2)(t), t [0, ), |t=0 = (x), = (x) - p(x), x [0, l].

t kl+ t= a C2 Q C2[0, l] ( - p) C1[0, l] (j) - (j) C2[0, ) (j = 1, 2) (0) = (1)(0) - (1)(0), a2 (0) = (1) (0) - (1) (0), (l) = (2)(0) - (2)(0), a2 (l) = (2) (0) - (2) (0), kl + kl + (0) - p(0) = (1) - (1), a a kl + kl + (l) - p(l) = (2) - (2) a a = 0, l/3, 2l/3, l k = 0, 1, 2,...

f C0,1(Q) p C1[0, l] (j) C2[0, ) (j = 1, 2) kl + kl + (j)(0) = 0, j = 1, 2, p(0) = (1), p(l) = (2), a a (1) (0) = f(0, 0), (2) (0) = f(0, l).

(0) = (1)(0), a2 (0) = (1) (0) - f(0, 0), (l) = (2)(0), a2 (l) = (2) (0) - f(0, l), kl + kl + (0) = (1), (l) = (2).

a a C2[0, l] C1[0, l] (j) C2[0, ) j = 1, f C 0,1 Q C2 Q k = 0, 1, 2,... = 0, l/3, 2l/3, l u = + v v f C2 Q -p (j) -(j) j = 1,











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