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3 x a c = a x a 2 1 c = a x 2 c R = e x = a x x c R f(x) =e x f n (x) c(x) f(x) =1+ x 1! + x2 2! + x3 +, <x< 3! f n (x) =1+ x 1! + x2 2! + x3 xn + + 3! n! c(x) f n (x) f(x) = xn+1 (n +1)! xn+2 (n +2)! f(x) x a f(x) f(a) 3

4 2.2 a = ,b = x, y a b =.35 = a, b /1.234 < a b e R e R = (x y) (a b) a b (x a) (y b) a b 1 5 = a,b a = ,b = ,c = a + b = a + c = = a a b c n +1 x,x 1,,x n f(x) f(x ),f(x 1 ),,f(x n ) n g(x) g(x j )=f j = f(x j ) (j =, 1,,n) (3.1) g(x) f(x) g(x) g(x i )=c + c 1 x i + c 2 x i + + c n x n i = f(x i ), g(x i ) x j (j =, 1,,n) x f(x) g(x) f(x) g(x) 4

5 1 n L n,k (x),k =, 1,,n g(x) L n,k (x) L n,k (x) = g(x) = n c k L n,k (x) (3.2) k= {, x = xj,j k 1, x = x k (3.3) g(x i )=c i i =, 1,,n c i = f(x i ),i=, 1,,n g(x) L i,k (x) = n i=,i k x x i x k k i (3.4) (3.3) (3.2) x = x k 1 d y f(x k 1 ) x k 1 h x k, (h = x k x k 1 ) f(x k )=f(x k 1 )+(x k x k 1 )f (x k 1 )+ (x k x k 1 ) 2 f (x k x k 1 )+ (3.5) 2 (3.5) 3 f(x k )=f(x k 1 )+(x k x k 1 )f (x k 1 ) (3.6) f(x) = x 3.6 x k (3.6) f(x k )= x k = x k 1 f(x k 1) f (x k 1 ) (3.7) x k x k 1 x f(x) [a, b] b a f(x)dx (3.8) 5

6 [a, b] f(x) a b n x i (i =, 1, 2,,n) f(x i ) h (3.9) b a f(x)dx h(f(x 1 )+f(x 2 )+ + f(x n )), h(f(x )+f(x 1 )+ + f(x n 1 )), (h =(b a)/n) (3.9) [a, b] m x j [x j 1,x j ] (3.1) (3.1) b a b a f(x)dx = m j=1 xj x j 1 f(x)dx m j=1 h 2 (f j 1 + f j ) (3.1) f(x)dx h 2 (f + f m )+h(f 1 + f f m 1 ), (h =(b a)/m) (3.11) (3.12) dy dx = f(x, y), y(x )=y (3.12) y(x) =y y(x) Y j y(x) y(x j )=y(x j 1 )+hφ(x j 1,y(x j 1 ); h) (3.13) φ F (3.13) Y j = Y j 1 + hf (x j 1,Y j 1 ; h) (3.14) Y j (3.14) Y j F (x, y(x); h) f(x, y) φ y = f(x, y) y(x) h y(x + h) = y(x)+hy (x)+ h2 2! y (x)+ + hp p! y(p) (x)+ hp+1 (p +1)! y(p+1) (x + θh) = y(x)+hf(x, y(x)) + h2 d hp d p 1 f(x, y(x)) + + f(x, y(x)) 2! dx (p)! dxp 1 + hp+1 d p f(x + θh, y(x + θh)), <θ<1 (3.15) (p +1)! dxp 6

7 (3.15) φ(x, y(x); h) = f(x, y(x)) + h d 2! dx F + hp d p F (x, y(x); h) = f(x, y(x)) + h d 2! dx (3.16) (3.17) f(x, y(x)) + + hp 1 p! d p 1 f(x, y(x)) dxp 1 f(x + θh, y(x + θh)) (3.16) (p +1)! dxp f(x, y(x)) + + hp 1 p! d p 1 f(x, y(x)) (3.17) dxp 1 F (x, y(x); h) φ(x, y(x); h) = O(h p ) (3.18) (3.18) (3.14) p p =1 (3.14) p =1 (3.17) F (x, y(x); h) =f(x, y(x)) (3.19) (3.14) (3.19) Y = y Y j = Y j 1 + hf(x j 1,Y j 1 ) x j = x j 1 + h (j =1, 2,,n) 3.5 p (3.14) p =2 p =2 (3.19) F (x, y(x); h) =f(x, y(x)) + h d f(x, y(x)) (3.2) 2! dx y(x) (3.12) f(x, y(x)) x = f x (x, y(x)) + f y (x, y(x))y (x) = f x (x, y(x)) + f(x, y(x))f y (x, y(x)) (3.21) f x (x, y(x)) = f x f y (x, y(x)) = f 7

8 (3.2) (3.21) F (x, y(x); h) =f(x, y(x)) + h(f x(x, y(x)) + f(x, y(x))f y (x, y(x))) 2 (3.23) (3.22) F (x, y(x); h) =αf(x, y(x)) + βf(x + γh,y + γhf(x, y(x))) (3.23) (3.22) 2 (3.24) f(x + γh,y + γhf(x, y(x))) = f(x, y(x)) + γhf x (x, y(x)) + γhf(x, y(x))f y (x, y(x)) + O(h 2 ) (3.24) (3.23) F (x, y(x); h) = αf(x, y(x)) + β{f(x, y(x)) + γhf x (x, y(x)) + γhf(x, y(x))f y (x, y(x)) + O(h 2 )} = (α + β)f(x, y(x)) + βγh{f x (x, y(x)) + f(x, y(x))f y (x, y(x))} + O(h 2 ) (3.25) F = F (x, y(x); h), f = f(x, y(x)), f x = f x (x, y(x)), f y = f y (x, y(x)) (3.22) (3.25) F = f + h(f x + ff y )/2 (3.26) F = (α + β)f + βγh(f x + ff y )+O(h 2 ) (3.27) α + β =1,βγ =1/2 β λ (3.23) F (x, y(x); h) =(1 λ)f(x, y(x)) + λf(x + h 2λ,y+ h f(x, y(x))) (3.28) 2λ (3.17) (3.18) (3.19) (3.27) (3.28) F (x, y(x); h) φ(x, y(x); h) = O(h 2 ) (3.28) F (3.14) Y j = Y j 1 + hf (x j 1,Y j 1 ; h) 2 λ =1/2 (3.29) F (x, y(x); h) = 1 (f(x, y(x)) + f(x + h, y + hf(x, y(x))) (3.29) 2 Y j = y(x j ) (3.14) (3.25) Y j = Y j 1 + h(f(x j 1,Y j 1 )+f(x j 1 + h, Y j 1 + hf(x j 1,Y j 1 ))/2 (3.3) y j (j =1, 2,,n) Y = y k 1 = hf(x j 1,Y j 1 ) k 2 = hf(x j 1 + h, Y j 1 + k 1 ) Y j = Y j (k 1 + k 2 ) x j = x j 1 + h 8

9 3.6 p p =4 k 1 = f(x, Y ) (3.31) k 2 = f(x + αh, Y + βk 1 ) (3.32) k 3 = f(x + α 1 h, Y +(β 1 k 1 + γ 1 k 2 )h) (3.33) k 4 = f(x + α 2 h, Y +(β 2 k 1 + γ 2 k 2 + δ 2 k 3 )h) (3.34) (3.35) (3.15) y = f(x, y(x)) y,y y = df dx = ( y = d2 f dx 2 = x + f ( x + f ) f = f x + ff y (3.36) ) (f x + ff y )=f xx + f x f y + ff xy + ff yx + ff 2 y + f 2 f yy (3.37) D = x + f f = f(x,y ) [ ] x = x,y = y y,y,y y = f y = [f x + ff y ] =[Df] y = [f xx +2ff xy + f 2 f yy + f y (f x + ff y )] =[D 2 f + f y Df] (3.16) x = x φ(x,y(x ); h) = [f + h h2 Df + 2! 3! (D2 + f y D)f ] + h3 4! (D3 + f y D 2 + fy 2 D +3D2 f y )f +O(h 5 ) (3.38) D l = l k= ( ) l k f l k l x k l k (3.16) p =4 f(x + u, y + v) = l= ( 1 u l! x + v ) l f(x, y) D l 1 = ( ) l l α k (βf ) l k l k x k l k k= 9

10 D 1 = α x + βf hd 1 = αh x + βhf D 11 k 1 = f(x,y )=f (3.39) k 2 = f(x + αh, y + βf h) [ = f + D 11 f + D2 11 f + D3 11 f + D4 11 f ] + 2! 3! 4! [ ] = f + hd 1 f + h2 2! D2 1 f + h3 3! D3 1 f + h4 4! D4 1 f + (3.4) D l 2 = ( ) l l α k k 1{(β 1 + γ 1 )f } l k l x k l k k= D 2 = α 1 x +(β 1 + γ 1 )f hd 2 = α 1 h x +(β 1 + γ 1 )hf α 1 h x +(β 1k 1 + γ 1 k 2 )h = hd 2 +(β 1 k 1 + γ 1 k 2 β 1 k 1 γ 1 k 1 )h = hd 2 +(k 2 f )γ 1 h [ = hd 2 + γ 1 h 2 D 1 f + h ] 2! D2 1 f + h2 3! D3 1 f + D 21 k 3 = = = f[x + α 1 h, y +(β 1 k 1 + γ 1 k 2 )h] [ f + D 21 f + D2 21 f + D3 21 f + D4 21 f ] + 2! 3! 4! [ [ ] f + {hd 2 + γ 1 h 2 D 1 f + h2 2! D2 1f + h2 3! D3 1f [ {h 2 D 22 +2hD 2 γ 1 h 2 D 1 f + h ] 2! 2! D2 1f + } f 1

11 = + γ1h [D f + h ] 2 2 } 2! D2 1f {h 3 D 32 3! +3h2 D 22 γ 1h [D 2 1 f + h2 ] 2! D2 1 f + [f + hd 2 f + h2 2! D2 2f + h3 3! D3 2f + + γ 1 h 2 + { f y D 1 f + h 2! f yd1f 2 + hd 1 fd 2 f y + h2 3! f yd1 3 f + h2 2! D2 1 fd 2f y + h2 2! γ 1f yy (D 1 f) 2 }] + h2 2! D 1fD2 2 f y + } ] f + (3.41) D l 3 = ( ) l l α k k 2{(β 2 + γ 2 + δ 2 )f } l k l x k l k k= D 3 = α 2 x +(β 2 + γ 2 + δ 2 )f hd 3 = α 2 h x +(β 2 + γ 2 + δ 2 )hf α 2 h x +(β 2k 1 + γ 2 k 2 + δ 2 k 3 )h = hd 3 +(β 2 k 1 + γ 2 k 2 + δ 2 k 3 β 2 f γ 2 f δ 2 f )h = hd 3 +[γ 2 (k 2 f )+δ 2 (k 3 f )]h [ { = hd 3 + h 2 γ 2 D 1 f + h } 2! D2 1f + h2 3! D3 1f + +2δ 2 γ 1 hf y D 1 f)+ h2 3! D3 1 f + h2 2! γ 1f y D 2 1 f + h2 γ 1 D 1 fd 2 f y + }] D 31 k 4 = = [ f + D 31 f + D2 31 f + D3 31 f + D4 31 f ] + 2! 3! 4! [f + hd 3 f + h 2 f y {γ 2 (D 1 f + h ) 2 D21f + h2 3! D3 1f + ( + δ 2 D 2 f + hγ 1 f y D 1 f + h 2 D2 2 f + h2 )} 3! D3 2 ff y + 2 γ 1f y D1 2 f + h2 γ 1 D1 2 fd 2f y + h2 + 1 ) {h 2 D 23 2! f h2 +2h3 h2 D 3 f y (γ 2 (D 1 f + D21 f + 3! D3 2 f + ( + δ 2 D 2 f + hγ 1 f y D 1 f + h 2 D2 2 f + ))+h 4 f yy (γ2 2 (D 1f) 2 +2γ 2 δ 2 D 1 fd 2 f 11

12 = } + δ2(d 2 2 f) 2 + ) + 1 3! {h3 D3f 3 +3h 4 D3f 2 y (γ 2 D 1 f + δ 2 D 2 f + )+ } + 1 4! {h4 D3 4 f + }+ ] [f + hd 3 f + h2 2! D2 3f + h3 3! D3 3f + + h 2 (γ 2 D 2 f)f y + h 3 (γ 2 D 1 f + δ 2 D 2 f y )D 3 f y + h3 2 (γ 2D 2 1 f + δ 2D 2 2 f +2γ 1δ 2 f y D 1 f)f y ] (3.42) (3.14) 1 Y j = Y j 1 + hf (x j 1,Y j 1 ; h) = Y j 1 + h(c 1 k 1 + c 2 k 2 + c 3 k 3 + c 4 k 4 ) (3.43) (3.39) (3.4) (3.41) (3.42) (3.38) hf : c 1 + c 2 + c 3 + c 4 =1 h 2 Df : c 2 D 1 f + C 3 D 2 f + c 4 D 3 f = Df 2! h 3 D 2 f : c 2 D1 2 f + c 3D2 2 f + c 4D3 2 f = 2! 3! D2 f h 3 f y Df : c 3 γ 1 D 1 f + c 4 (γ 2 D 1 f + δ 2 D 2 f)= 1 3! Df h 4 D 3 f : (3.44) c 2 D1 3 f + c 3D2 3 f + c 4D3 3 f = 3! 4! D3 f h 4 f y D 2 f : c 3 γ 1 D1 2 f + c 4(γ 2 D1 2 f + δ 2D2 2 f)= 2 4! D2 f h 4 DfDf y : c 3 γ 1 D 1 fd 2 f y + c 4 (γ 2 D 1 f + δ 2 D 2 f)d 3 f y = 3 4! DfDf y h 4 fy 2 Df : c 4 γ 1 δ 2 D 1 f = 1 4! Df D, D s (s =1, 2, 3) f D s/d l l f(s, l =1, 2, 3) D 1 f = α f f +β x D 1f = f x +f f D 2 f Df D 3 f Df α = β (3.45) { α1 = β 1 + γ 1 α 2 = β 2 + γ 2 + δ 2 (3.46) 12

13 D 1 f = αdf D 2 f = α 1 Df D 3 f = α 2 Df (3.47) (3.47) (3.44) 1 8 c 1 + c 2 + c 3 + c 4 = 1 (a) c 2 α + c 3 α 1 + c 4 α 2 = 1/2 (b) c 2 α 2 + c 3 α c 4 α 2 2 = 1/3 (c) c 2 α 3 + c 3 α c 4 α 3 2 = 1/4 (d) c 3 αγ 1 + c 4 (αγ 2 +(α 1 δ 2 ) = 1/6 (e) c 3 α 2 γ 1 + c 4 (α 2 γ 2 +(α 2 1 δ 2) = 1/12 (f) c 3 αα 1 γ 1 + c 4 (αγ 2 +(α 1 δ 2 )α 2 = 1/8 (g) c 4 αγ 1 δ 2 = 1/24 (h) (e) (f) c 3 (h) c 4 C 4 (α 1 α)α 1 δ 2 = 1 12 α 6 (α 1 α)α 1 δ 2 224αγ 1 δ 2 = 1 12 α 6 2αγ 1(2α 1) = α 1 (α α 1 ) (i) (3.48) (e) (f) c 4 (i) γ c 3 αγ 1 (α 2 α 1 )= α c 3 (α 2 α 1 ) α 1(α α 1 ) 2(α 1) = α ( α2 c 3 α 1 (α 2 α 1 )(α α 1 )=(2α 1) 3 1 ) 4 (b) αα 2 (α + α 1 ) (c)+(d) c 3 α 1 (α 2 α 1 )(α α 1 )= 1 2 αα (α + α 2)+ 1 4 (j) (k) (j) (k) (h) α ( α2 (2α 1) 3 1 ) = αα (α + α 2)+ 1 4 α(α 2 1) =, α 2 = 13

14 (a) (b) (c) c 1,c 2,c 3,c 4 c 1 = 6αα 1 2(α + α 1 )+1 12αα 2α 1 1 c 2 = 12α(α 1 α)(1 α) 1 2α c 3 = 12α 1 (α 1 α)(1 α 1 ) c 4 = 6αα 1 4(α + α 1 )+3 12(1 α)(1 α 1 ) (a) (b) (c) (d) γ 1 γ 2 δ 1 α, α 1 (3.45) (3.46) β = α, β 1 = α 1 γ 1 (α, α 1 )β 2 = α 2 γ 2 (α, α 1 ) δ 2 (α, α 1 ) α = α 1 = 1 (a) (b) (c) 2 (d) /2 1/2 1 1/2 1/4 1/4 1 1/3 1/8 1/8 1 1/4 c 1,c 2,c 4 c 3 1 1/ /3 1 1/6 C 1 = 1 6 C 2 = 2 3 C 3 C 3 = 1 6 (3.49) (3.45) (3.46) (3.49) (e) (f) (g) γ 1 = 1 6c 3 γ 2 =1 3c 3 δ 2 =3c 3 β = α = 1 2 β 1 = α 1 γ 1 = c 3 β 2 = α 2 γ 2 γ 2 = (3.5) c 3 =1/3 c 1 =1/6,c 2 =1/3,c 3 =1/3,c 4 =1/6,γ 1 =1/2,γ 2 =,δ 2 =1,β 1 =,β 2 = Y = y j (j =1, 2,,n) k 1 = hf(x j 1,Y j 1 ) k 2 = hf(x j 1 + h/2,y j 1 + k 1 /2) 14

15 k 3 = hf(x j 1 + h/2,y j 1 + k 2 /2) k 4 = hf(x j 1 + h, Y j 1 + k 3 ) Y j = Y j (k 1 +2k 2 +2k 3 + k 4 ) x j = x j 1 + h m y 1 (x),...,y m (x) y = dy j dx = f j(x, y 1,...,y m ), y j (x )=y j, (j =1, 2,,m) y 1. y m, y = y 1. y m f 1 (x, y 1,,y m ), f(x, y) =. f m (x, y 1,,y m ) dy dx = f(x, y) y(x )=y (3.51) 2 j(j =1, 2,,n) dy dx = f 1(x, y, z) (3.52) dz dx = f 2(x, y, z) (3.53) y(x )=y, z(x )=z Y = y,z = z, k 1 = hf 1 (x j 1,Y j 1,Z j 1 ) m 1 = hf 2 (x j 1,Y j 1,Z j 1 ) k 2 = hf 1 (x j 1 + h/2,y j 1 + k 1 /2,Z j 1 + m 1 /2) m 2 = hf 2 (x j 1 + h/2,y j 1 + k 1 /2,Z j 1 + m 1 /2) k 3 = hf 1 (x j 1 + h/2,y j 1 + k 2 /2,Z j 1 + m 2 /2) 15

16 m 3 = hf 2 (x j 1 + h/2,y j 1 + k 2 /2,Z j 1 + m 2 /2) k 4 = hf 1 (x j 1 + h, Y j 1 + k 3,Z j 1 + m 3 ) m 4 = hf 2 (x j 1 + h, Y j 1 + k 3,Z j 1 + m 3 ) Y j = Y j (k 1 +2k 2 +2k 3 + k 4 ) Z j = Z j (m 1 +2m 2 +2m 3 + m 4 ) x j = x j 1 + h x 1,x 2,,x n u u 1 F (x 1,x 2,,x n,u, u,, 2 u 2 u,, )= x 1 x 1 x x, y u(x, y) x 2 1 a(x, y) 2 u x 2 +2b(x, y) 2 u x + c(x, u y) 2 + d(x, y) u 2 x + e(x, y) u + f(x, y)u = g(x, y) a, b, c, d, e, f, g b 2 ac > b 2 ac < b 2 ac = x x i 1 <x i <x i+1 (x i+1 x i = x i x i 1 = x) y = y(x) 3 (x i 1,y i 1 ), (x i,y i ), (x i+1,y i+1 ) (x i,y i ) tan θ x θ tan θ = dy dx = lim y x x = lim y i+1 y i x x x tan θ =(y i+1 y i )/ x dy/dx dy dx = y i+1 y i x 16

17 dy dx = y i y i 1 x tan θ =(y i+1 y i 1 )/(2 x) dy dx = y i+1 y i 1 2 x u = u(x, y) x a, y b x i = m y j = n (x, y) (i, j) u u(x, y) u ij u i 1,j,u i,j 1 x i, y j u/ x, u/ u x = u i+1,j u i,j x i (4.1) u = u i,j+1 u i,j y j (4.2) u x = u i,j u i 1,j x i 1 (4.3) u = u i,j u i,j 1 y j 1 (4.4) u x = u i+1,j u i 1,j x i 1 + x i (4.5) u = u i,j+1 u i,j 1 y i 1 + y j (4.6) 2 2 u/ x 2 u x = x ( ) u = ( u/) i+1,j ( u/) i 1,j x i 1 + x i ( u/) i+1,j i +1 u ( u/) i 1,j i 1 ( ) u = u i 1,j+1 u i 1,j 1 i 1,j y j 1 + y j ( ) u = u i+1,j+1 u i+1,j 1 i+1,j y j 1 + y j 17

18 2 u x = u i+1,j+1 u i+1,j 1 u i 1,j+1 + u i 1,j 1 ( x i 1 + x i )( y i 1 + y i ) (4.7) 2 u/ x 2 u x = u i+1,j+1 u i+1,j 1 u i 1,j+1 + u i 1,j 1 ( x i 1 + x i )( y i 1 + y i ) (4.8) 2 u x = 2 u x (4.9) 2 u/ x 2 2 u/ 2 u/ x u/ 2 u x 2 = x ( ) u = ( u/ x) i+1,j ( u/ x) i,j x x i ( u/ x) i+1,j ( u/ x) i,j ( ) u = u ( ) i+1,j u i,j u, = u i,j u i 1,j x i+1,j x i x i,j x i 1 2 ( u x 2 = ui+1,j u i,j x i u )/ i,j u i 1,j x i (4.1) x i 1 y 2 u 2 = ( ui,j+1 u i,j y j u i,j u i,j 1 y j 1 ) / y j (4.11) x i 1 = x i = h y j 1 = j i = k h k 2 u x = 1 2h (u i+1,j u i 1,j ) (4.12) u = 1 2k (u i,j+1 u i,j 1 ) (4.13) 2 u x = 1 4hk (u i+1,j+1 u i+1,j 1 u i 1,j+1 + u i 1,j 1 ) (4.14) 2 u x 2 = 1 h 2 (u i+1,j 2u i,j u i 1,j ) (4.15) 2 u 2 = 1 k 2 (u i,j+1 2u i,j u i,j 1 ) (4.16) 18

19 2 u(x + h, y) u(x h, y) (4.17) ( u/ x) u(x + h, y) = u(x, y)+h u x + h2 2 u 2! u(x h, y) = u(x, y) h u x + h2 2! u u(x + h, y) u(x, y) = x h u(x + h, y) =u i+1,j,u(x, y) =u i,j u x = u i+1,j u i,j h x 2 + h3 3 u + 3! x3 (4.17) 2 u x 2 h3 3 u + 3! x3 (4.18) h 2 u 2! x 2 h2 3 u 3! x 3 + h (4.18) u x = u i,j u i 1,j h O(h) (4.19) O(h) (4.2) O(h) (4.17) (4.18) ( u/ x) u x u(x + h, y) u(x h, y) = 2h = u i+1,j u i 1,j 2h h2 3 u 3! x 3 O(h 2 ) (4.21) h 2 (4.17) (4.18) ( u/ x) 2 u x 2 = u(x + h, y) 2u(x, y)+u(x h, y) h 2 h2 4 u 12 x 4 = u i+1,j 2u i,j + u i 1,j h 2 O(h 2 ) (4.22) 2 u/ x 2 h u t = c2 2 u x 2 (4.23) u(x, ) = f(x), x L u(,t)=p(t),u(l, t) =q(t), t 19

20 c (4.23) (4.13) (4.15) u(x, t + k) u(x, t) k 2 u(x + h, t) 2u(x, t)+u(x h, t) = c h 2 λ = kc 2 /h 2 u(x, t + k) =λu(x + h, t)+(1 2λ)u(x, t)+λu(x h, t) (4.24) u(x + h, t),u(x, t),u(x h, t) 3 u(x, t + k) (4.13) (4.15) {u t (x, t) c 2 u xx (x, t)} {c 2 /(λh 2 )}{u(x, t + k) λu(x + h, t) (1 2λ)u(x, t) λu(x h, t)} = O(h 2 ) u (4.23) k = λh 2 /c 2 u(x, t + k) =λu(x + h, t)+(1 2λ)u(x, t)+λu(x h, t)+o(h 4 ) O(h 4 ) [,L] N h = L/N, k = λh 2 /c 2 x m = mh (m =, 1,,N),t n = nk (n =, 1, ) { x L, t } (x m,t n ) u(x m,t n ) U m,n (4.23) (4.24) U m,n+1 = λu m 1,n +(1 2λ)U m,n + λu m+1,n (n =, 1, ; m =1, 2,,N 1) (4.25) { Um, = u(x m, ) = f(x m ) (m =, 1,,N) U,n = u(,t n )=p(t n ), U n,m = u(l, t n )=q(t n ) (n =, 1, ) (4.26) U m, = f(x m ) (m =, 1,,N) U,n = p(t n ),U N,n = q(t n ) (n =, 1, ) n (n =, 1, 2, ) U m,n+1 = λu m 1,n +(1 2λ)U m,n + λu m+1,n (m =1, 2,,N 1) λ λ λ 4.1 λ = kc 2 /h 2 λ 1/2 U m,n h u(x m,t n ) x 6 2

21 u x u =; x a, y b (4.27) 2 u(, y)=p(y) u(x, ) = v(x), u(a, y) =q(y),u(x, b) =w(x) (4.28) p, q, v, w x a, y b h, k i =, 1,,m, j =, 1,,n f(x, y) =f i,j (4.15) (4.16) (4.27) u i,j 1 h 2 (u i+1,j 2u i,j + u i 1,j )+ 1 k 2 (u i,j+1 2u i,j + u i,j 1 )= u i,j = 1 2(h 2 + k 2 ) {k2 (u i+1,j + u i 1,j )+h 2 (u i,j+1 + u i,j 1 )} (4.29) u p(y) q(y) v(x) w(x) u i,j i =1, 2,,m 1; j =1, 2,,n 1 u i,j = (4.29) u u (1) i,j u(1) i,j (4.29) u(2) i,j k k +1 ε (u i,j (k+1) u (k) (k+1) i,j )/u i,j ε; (4.3) i =1, 2,,m 1; j =1, 2,,n 1 u (k+1) i,j u i,j m 1 i=1 n 1 j=1 u i,j (k+1) (k) u i,j / m 1 n 1 i=1 j=1 u i,j (k+1) ε (4.31) (4.31) (4.29) u (4.29) { } (k+1) 1 u i,j = 2(h 2 + k 2 k 2 (u (k) i+1,j + u (k+1) i 1,j )+h 2 (u (k) i,j+1 + u (k+1) i,j 1 ) (4.32) ) (4.32) - (4.32) u (k) i,j u (k) i,j [ { } ] u (k+1) i,j = u (k) 1 i,j + 2(h 2 + k 2 k 2 (u (k) i+1,j + u (k+1) i 1,j )+h 2 (u (k) i,j+1 + u (k+1) (k) i,j 1 ) u i,j (4.33) ) (4.33) [ ] u i,j (k) 1 u i,j (k+1) u i,j (k) u i,j (k+1) u i,j (k) =[ ] [ ] u i,j (k+1) u i,j (k) 21

22 [ ] [ ] ω [ u (k+1) i,j = u (k) i,j + ω 1 2(h 2 + k 2 ) { } ] k 2 (u (k) i+1,j + u (k+1) i 1,j )+h 2 (u (k) i,j+1 + u (k+1) (k) i,j 1 ) u i,j (4.34) ω (4.34) SOR ω ω i =, 1, 2,,m; j =, 1, 2,,n h = k ω = 2, µ =cos π 1+ 1 (µ/2) 2 m +cosπ n (4.35) 7 22

24 [1] 21 [2] 1998 [3] 1999 [4] 21 [5] 1993 [6] PAD,PASCAL,C 1999 [7] [8] 1997 [9] 1999 [1] C 1989 [11] Samuel P.Harbison,Guy L.Steele Jr., C 1992 [12] C 1993 [13] W.H.Press,B.P.Flannery,S.A.Teukolsky,W.T.Vetterling Numerical Recipes in C[ ] 1993 [14] 1997 [15] 1993 [16] - C 1992 [17] C 1998 [18] 199 [19] - C/C

25 7 =1= =2= =3= =4= =5= - =6= =7=

1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =

1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f = 1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i