W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)

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1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4) X (x) αx(x) (5) X(), X() (6) (3) T () f(x), T () (7) (6), (7) (4) (5) (5) X(x) Ae αx + Be αx (α > ) X(x) Ax + B (α ) A cos αx + B sin αx (α < ) A, B (6) X(x) X(x) B n sin nπx (α n2 π 2 ) n 1, 2,, n B n T (t) T (t) C n cos nπat u(x, t) B n C n sin nπx 2 (C n : ) cos nπat (1) u 3 (1) u (1) u(x, t) A n sin nπx cos nπat (1) A n : B n C n 4 (7) f(x) A n sin nπx A n 2 f(x) sin nπx dx u(x, t) 2 f(y) sin nπy dy sin nπx cos nπat (2)-(3) (1)

2 xy t P z(x, y, t) 2 z t 2 c2 ( 2 ) z x z y 2, c >. (1) z(r, t) R(r)T (t) (5) (2) T (t) c 2 kt (t) R (r) + 1 r R (r) kr(r) (6) D : a x(r, θ) r cos θ, z(x, y, t) z(r, θ, t) ( r a, θ 2π) y(r, θ) r sin θ (1) 2 ( z 2 t 2 z c2 r z r r ) z r 2 θ 2 (2) 1 (1) (2) (2) z(r, θ, ) φ(r), z(r, θ, ) t (3) J ( ) α n J (α n ) k 2 (5) (2) (6) (6) (3) (4) T (t) c 1 cos cα n a t R(r) J ( αn a r ) α 1 < α 2 < < α n < A n z(r, t) A n J ( αn a r ) cos cα n a t z (a, θ, t) (4) (3)-(4) (2) 2 θ z(r, θ, t) z(r, t) A n (3) φ(r) A n J ( αn a r ) A n 2 {J 1 (α n )} 2 1 sφ (αs) J (αs) ds

3 E H ϵ E t rot H J, ϵ div E ρ µ H t rot E J m, ρ J ϵ, µ 1 ϵ µ µ div H ρ m ρ m J m (1) c m/s ρ m, J m (2) P M ρ ρ div P J J + P t + rotm H H + M D B (3) D ϵ E + P, B µ (H + M) (4) (1) D t rot H J, div D ρ B t rot E, µ div B 1 (2)-(4) (1) (5) (5) 2 V A B rot A, E grad V A t (6) div A + ϵ µ V t (7) (1) 2 V t 2 1 V + ρ ϵ µ ϵ 2 µ 2 A t 2 1 A + 1 J ϵ µ ϵ (8) : ϵ µ 2 t 2 V ρ ϵ, A µ J 2 (6)-(7) (1) (8) (8) E (E 1, E 2, E 3 ) ( E3 rot E y E 2 z, E 1 z E 3 x, E 2 x E ) 1 y div E E 1 x + E 2 y + E 3 z V V (x, y, z) ( V grad V x, V y, V ) z V div (grad V ) V A (A 1, A 2, A 3 ) grad (div A) rot (rot A) A

4 u(x, y, z, t) 3 2 u t 2 c2 2 ( 2 u x u y u z 2 ), c >. (1) u r x 2 + y 2 + z 2 t u u (r, t) 2 u x 2 1 u r r x2 u r 3 r + x2 2 u r 2 r 2 (2) 2 u y 2 1 u r r y2 u r 3 r + y2 2 u r 2 r 2 (3) 2 u z 2 1 u r r z2 u r 3 r + z2 2 u r 2 r 2 (4) r 2 x 2 + y 2 + z u x u y u z (ru) r r 2 (5) (2)-(4) (5) (1) 2 (ru) t 2 c 2 2 (ru) r 2 (6) (6) 1 g 1, g 2 u (r, t) 1 r g 1 (r ct) + 1 r g 2 (r + ct) (7) c 2 (7) u(r, t) 1 (6) 3 (, m, n) c (1) u (x, y, z, t) f (x + my + nz ct) A, k u (x, y, z, t) A sin k (x + my + nz ct) (8) (1) A k : (k, km, kn) k (x + my + nz ct) r (x, y, z) (8) 3 u A sin (k r kct) A, k u (x, y, z, t) A cos k (x + my + nz ct) A cos (k r kct) (1) e ix cos x + i sin x u A exp (i (k r kct))

5 u u(x, y) u xx + u yy ( u ) (1) u U { (x, y) x 2 + y 2 1 } x 2 + y 2 1 u(x, y) F (x, y) (2) (2) (1)-(2) x r cos θ, y r sin θ u u(r, θ) u xx + u yy u rr + 1 r u r + 1 r 2 u θθ 4 (1)-(2) r 2 u rr + ru r + u θθ, r < 1 u(1, θ) f(θ), θ 2π u(r, θ) u(r, θ + 2π) x 2 + y 2 r 2 1 F (x, y) F (r, θ) F (1, θ) f(θ) u(r, θ) R(r)Θ(θ) (3) (3) Θ(θ) Θ(θ + 2π) Θ(θ) Θ(θ) A n cos nθ + B n sin nθ, n 1, 2, α n 2 R(r) R(r) C n r n a n A n C n, b n B n C n u(r, θ) r n (a n cos nθ + b n sin nθ), n 1, 2, u(r, θ) a 2 + r n (a n cos nθ + b n sin nθ) u(1, θ) f(θ) f(θ) a 2 + (a n cos nθ + b n sin nθ) a n 1 π b n 1 π f(φ) cos nφdφ f(φ) sin nφdφ u(r, θ) 1 f(φ)dφ 2π + 1 r (cos n nθ f(φ) cos nφdφ π ) + sin nθ f(φ) sin nφdφ (4) (3) Θ (θ) + αθ r 2 R (r) + rr (r) αr(r) α : f(θ) sin 2 θ (4)

6 f(θ) sin 2 θ (3) (3) f(φ)dφ sin 2 φdφ 1 cos 2φ dφ 2 [ φ sin 2φ 2 4 ] 2π π f(φ) cos nφdφ 1 cos 2φ cos nφdφ 2 { cos nφ cos(n + 2)φ + cos(n 2)φ 2 { π 2, n 2,, n 2. cos(n 2)φdφ } dφ f(φ) sin nφdφ 1 cos 2φ 2 sin nφdφ. (3) u(r, θ) 1 2π π + 1 ( π r2 cos 2θ π ) 2 1 ( 1 r 2 cos 2θ ). 2

7 δ(x)dx 1, δ(x) (x ) f(x) f(x)δ(x ξ)dx f(ξ) s 2 f s (x) 1 ) exp ( x2 2πs 2s 2 (1) s δ(x) 1 1. f s (x) δ(x) 1 (1) 2 u u(x, y) f s (x)dx 1 u xx + u yy ρ(x, y) (2) ρ(x, y) ρ(x, y) (2) (2) 3 (2) G xx + G yy δ(x)δ(y) (3) G G (x, y) (2) u(x, y) G (x ξ, y η) ρ (ξ, η) dξdη (4) 2 (4) u (2) (2) r u(r) ρ(r) (5) (3) (4) G(r) δ(r) u(r) G ( r r ) ρ(r )dr r (x, y) G(r) 1 og r (6) 2π r (x, y, z) G(r) 1 4π r (7) 3 (7) r G(r) (6) r G(r)

8 (6) r G(r) r (x, y) (6) G(r) G(x, y) 1 2π og x 2 + y 2 G x (x, y) 1 2π 2x x 2 + y 2 2 x 2 + y 2 x 2π(x 2 + y 2 ) G y (x, y) 1 2π 2y x 2 + y 2 2 x 2 + y 2 y 2π(x 2 + y 2 ) G xx (x, y) x2 + y 2 x 2x 2π(x 2 + y 2 ) 2 x 2 y 2 2π(x 2 + y 2 ) 2 G yy (x, y) x2 + y 2 y 2y 2π(x 2 + y 2 ) 2 y 2 x 2 2π(x 2 + y 2 ) 2. G(r) G xx + G yy (r ).

9 [, a] f(x) b n 2 a a f(x) b n sin nπ a x f(x) sin nπ a x dx, n 1, 2, 1 f(x) cos x ( x π) 2 D {(x, y) : x a, y b} (1) D u u xx + u yy D u(x, ), u(x, b), x a u(, y) φ(y), u(a, y), y b (2) u u(x, y) u(x, y) A n sinh nπ b (a x) sin nπ b y x (2) φ(y) A n sinh nπ b a sin nπ b y 2 A n 3 D u u xx + u yy ρ(x, y) (3) D u(x, ), u(x, b), x a u(, y), u(a, y), y b (4) u u(x, y) u(x, y) m1 A mn sin mπ a x sin nπ b y (4) (3) A mn ρ(x, y) x ρ(x, y) ρ m (y) 2 a a m1 ρ m (y) sin mπ a x ρ(ξ, y) sin mπ ξ dξ, m 1, 2, a ρ m (y) y (3) B mn A mn ( mπ ) 2 ( nπ + a b B mn 4 ab a b ) 2 ρ(ξ, η) sin mπ a ξ sin nπ b η dηdξ ρ(x, y) 1 xy (4) 4 (3) u

10 ρ(x, y) 1 xy (4) 4 (3) u ρ(x, y) 1 xy 4 B mn 1 ( a ξ sin mπ ) ( b ab a ξdξ η sin nπ ) b ηdη a ξ sin mπ a ξdξ [ a mπ ] a ξ cos mπ a ξ + a mπ a2 cos mπ + a mπ mπ a a2 mπ ( 1)m a2 mπ ( 1)m+1 cos mπ a ξdξ [ a mπ ] a sin mπ a ξ b η sin nπ b ηdη b2 nπ ( 1)n+1 B mn 1 a 2 b 2 ab mπ ( 1)m+1 nπ ( 1)n+1 ab mnπ 2 ( 1)m+n A mn ( 1) m+n ( mπ ) 2 ( nπ + a b ) 2 ab mnπ 2 u(x, y) m1 ( mπ a ( 1) m+n ) 2 ( nπ + b sin mπ a x sin nπ b y ) 2 ab mnπ 2

11 u u xx + u yy + u zz (1) u(x, y, z) X(x)Y (y)z(z) X (x) k1 2X(x), Y (x) k2 2Y (y) Z (x) k3 2Z(z) (2) k 1, k 2, k 3 k k k 2 3 (3) (2) X(x) c 1 e k 1x + c 1e k 1x Y (y) c 2 e k 2y + c 2e k 2y Z(z) c 3 e k 3z + c 3e k 3z c 1, c 1, c 2, c 2, c 3, c 3 (1) u(x, y, z) k 1,k 2,k 3 C k1 k 2 k 3 e k1x+k2y+k3z C k1 k 2 k 3 k 1 k 2 k 3 (3) k 1, k 2, k 3 2 (ρ, φ, z) x ρ cos φ, y ρ sin φ, z z (4) (1) u ρρ + 1 ρ u ρ + 1 ρ 2 u φφ + u zz (5) 1 (1) (4) (5) u (ρ, φ, z) R(ρ)Φ(φ)Z(z) (5) ) R (ρ) + R (ρ) + (α 2 m2 ρ ρ 2 R(ρ) (6) Φ (φ) m 2 Φ(φ), Z (z) α 2 Z(z) α, m Φ(φ), Z(z) Φ(φ) A cos mφ + B sin mφ Z(z) Ce αz + De αz (A, B, C, D : ) R(ρ) (6) α 1 m 3 (r, θ, φ) x r sin θ cos φ, z r cos θ y r sin θ sin φ (1) u rr + 2 r u r+ 1 r 2 u θθ+ 1 r 2 tan θ u 1 θ+ r 2 sin 2 θ u φφ u (r, θ, φ) R(r)Θ(θ)Φ(φ) Φ(φ) A sin mφ + B cos mφ (A, B : ) R (r) + 2 R (r) n (n + 1) R(r) r r 2 (7) { d (1 µ 2 ) dθ } } + {n(n + 1) m2 dµ dµ 1 µ 2 Θ µ cos θ (7) R(r) r λ λ

12 (7) R(r) r λ λ R(r) r λ λ (λ 1) r λ 2 + 2λr λ 2 n (n + 1) r λ 2 λ (λ 1) + 2λ n(n + 1) (λ n) (λ + n + 1) λ n, n 1 R(r) r n, r n 1

13 u t ku xx, k > (1) u u(x, t) t x k x [, ] (1) u (, t) u (, t) (2) u (x, ) φ(x) (3) φ(x) φ() φ() 1 (2)-(3) (1) u (x, t) 2 exp ( k n2 π 2 φ(y) sin nπ y dy 2 ) t sin nπ x 1) u(x, t) X(x)T (t) (1) X(x) T (t) 2) (2) X(x) T (t) u(x, t) 3) t φ(x) 2 x (, ) (1) (2) 1 u(x, t) (a cos λx + b sin λx) e kλ2t. a, b λ 2 a, b λ a(λ), b(λ) u λ (x, t) {a(λ) cos λx + b(λ) sin λx} e kλ2 t 1 (1) u(x, t) u λ (x, t) dλ {a(λ) cos λx + b(λ) sin λx} e kλ2t dλ (4) (3) φ(x) {a(λ) cos λx + b(λ) sin λx} dλ a(λ) 1 2π b(λ) 1 2π φ(ξ) cos λξdξ (5) φ(ξ) sin λξdξ (6) (4) u(x, t) 1 2π dλ φ(ξ) cos λ (x ξ) e kλ2t dξ (5), (6) (4) (7) (7)

14 (5), (6) (4) (7) u(x, t) [{ 1 2π { 1 + 2π 1 dλ 2π e kλ2t dξ 1 dλ 2π } φ(ξ) cos λξdξ } φ(ξ) sin λξdξ cos λx ] sin λx e kλ2t dλ φ(ξ) (cos λξ cos λx + sin λξ sin λx) φ(ξ) cos λ (ξ x) e kλ2t dξ

15 x [, ] u t ku xx, k > (1) u u(x, t) x t x, x ū, ū h, h K u x h (u x ū ) x K u (2) x h (u x ū ) x K ū ū (2) u(x, t) w(x, t) + α + βx (3) w w(x, t) α, β (3) K w x h w x + h α Kβ h ū x K w x h w x + h α x + (h + K) β h ū (4) α, β K w x h w x x K w (5) x h w x x 1 (3) (2) (4) 2 (3) w w t kw xx (6) u u(x, ) φ(x) w w(x, ) φ(x) α βx : φ 1 (x) w(x, t) (a cos λx + b sin λx) e λ2 kt (7) a, b λ (5) Kλb h a Kλa sin λ Kλb cos λ h a cos λ + h b sin λ (8) tan λ K (h + h ) λ K 2 λ 2 h h h, h a tan λ λ nπ λ nπ (7) w(x, t) ( 2 sin nπx ) dy φ 1 (y) sin nπy ( exp n2 π 2 kt 2 ) (9) λ nπ (7) (9) a 9 15

16 λ nπ (7) (9) a λ nπ (7) a w(x, t) b sin nπx e w(x, t) n 2 π 2 2 kt b n sin nπx n 2 π 2 e 2 kt (1) t φ 1 (x) b n sin nπx b n 2 φ 1 (y) sin nπy dy (1) w(x, t) ( 2 sin nπx (9) ) dy φ 1 (y) sin nπy ( exp n2 π 2 kt 2 )

17 x [, ] u t ku xx, k > u x (, t) u x (, t) (1) t [, T ] J, N x : J, t : T N 2 u(x, ) φ(x) (1) ( 2 u(x, t) φ(y) cos nπ ) y dy exp ( k n2 π 2 ) t cos nπ x (3) 3 2 x j : j x, j, 1, 2,, J t n : n t, n, 1, 2,, N u n j : u (x j, t n ) (a) (b) (1) u t un+1 j u n j t (1) u n 1 un x, u xx un j+1 2un j + un j 1 ( x) 2, u n u n 1, u n J un J 1 x u n J u n J 1 (1) σu n j+1 + (1 2σ)un j + σun j 1 u n un 1, u n+1 j un J un J 1 σ : k t ( x) 2 (2) 1 (1) (2) MATLAB k 1/2; 1; T 1; J 1; N 1; dx /J; dt T/N; s k dt/(dx 2 ); for x 1 : 1 : J u(x, 1) cos(pi x/j); end for t 1 : 1 : N 1 u(1, t + 1) s u(2, t) + (1 s) u(1, t); for x 2 : 1 : J 1 u(x, t + 1) s u(x + 1, t)... +(1 2 s) u(x, t) + s u(x 1, t); end u(j, t + 1) (1 s) u(j, t) + s u(j 1, t); end f(x, t) u t ku xx + f(x, t), k > fj n : f(x j, t n ) 9 15

18 f(x, t) u t ku xx + f(x, t), k > f n j : f(x j, t n ) u n+1 j u n j t k un j+1 2un j + un j 1 ( x) 2 + f n j u n+1 j u n+1 j u n j + σ(u n j+1 2u n j + u n j 1) + t fj n σu n j+1 + (1 2σ)u n j + σu n j 1 + t fj n σ : k t ( x) 2

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( 5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx

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1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

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