偏微分方程总结.pdfVIP

1 Ù  ‚ 5 ‡ ©§ © a †I O. §2.1 Ä SN J‡ ˜ . ü‡ gCþ  ‚ 5 ©§ © aÚ zI O. 1. § © a  Äü‡ gC þx, y  ‚5© a u + 2a u + a u + a u + b u + cu = f (x, y), (x, y) ∈ Ω. (2.1.1) 11 xx 12 xy 22 yy 1 x 1 y Xa11(x, y), a12(x, y), a22 (x, y) 3Ωþ Ø Ǒ" . O∆ = a2 − a11a22. 12 (1) e3 : (x , y ) ? , O∆ > 0, K  (2.1.1)3 : (x , y ) ǑV . ; 0 0 0 0 (2) e3 : (x , y )? , O∆ = 0,K  (2.1.1)3 : (x , y ) ǑÆ . ; 0 0 0 0 (3) e3 : (x , y )? , O∆ < 0, K  (2.1.1)3 : (x , y )Ǒý . . 0 0 0 0 , , X J  (2.1.1)3 Ω z : ǑV . , o  (2.1.1)3  Ω V . .Ó ,X J (2.1.1)3 Ω z : ǑÆ . (ý .),  o  (2.1.1)3 Ω Æ . (ý .). 2. A Ɛ§ © a11(dy)2 − 2a12dxdy + a22 (dx)2 = 0, (2.1.2) Ǒ© (2.1.1)A Ɛ§ , A Ɛ (2.1.2)È© ‚ (qÏ È© ½ö Ï ) Ǒ (2.1.1)A Æ ‚ ½A Ƃ .X Ja11 = 0, oŒ dy a12 + √∆ dy a12 − √∆ = , = (2.1.3) dx a11 dx a11  | (2.1.3)Ǒ Ǒ© (2.1.1)A Ɛ§| . 3. zI O. V . : ∆ > 0. d  | (2.1.3)küx Ø ÓÈ© ‚ µ φ (x, y) = c , φ (x, y) = c . 1 1 2 2 C † ξ = φ (x, y), η = φ (x, y). 1 2  (2.1.1)Œ z Ǒ1 IO . u = A u + B u