Handbook of mathematics for engineers and scienteists part 189. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 1284 Linear Equations and Problems of Mathematical Physics . Equations of the Form 92w k v_ _ n2d2w h v_ cw . dt2 dt a dx2 dx CW PU d2w dw 2 d2w dt2 dt dx2 For k 0 and b 0 it is the telegraph equation. The substitution w x t exp -2kt u x t leads to the equation J b L k2 U dt2 dx2 4 which is discussed in Subsection . d2w dw 2 d2w dw dt2 dt a dx2 dx cw x t The substitution w x t exp -2a 2bx - 2kt u x t leads to the equation u _ a2 u i c _1 k2 _ 2b2 u pvi N-a 2bx kt b x t a yc 4 4 a u j u exp 2 a ux 2 t - x t tx which is discussed in Subsection . . Elliptic Equations . Laplace Equation Aw 0 The Laplace equation is often encountered in heat and mass transfer theory fluid mechanics elasticity electrostatics and other areas of mechanics and physics. The two-dimensional Laplace equation has the following form 1 d. r dr d2w d2w 0 dx2 dy2 dw 1 d2w _ o dr r2 dp2 in the Cartesian coordinate system in the polar coordinate system where x _ r cos p y _ r sin p and r _ y x2 y2. . Particular solutions and methods for their construction. 1 . Particular solutions in the Cartesian coordinate system w x y Ax By C w x y A x2 - y2 Bxy w x y A x3 - 3xy2 B 3x2y - y3 Ax By w x y x2 y2 C w x y exp px A cos py B sin py w x y A cos px B sin px exp py w x y A sinh px B cosh px C cos py D sin py w x y A cos px B sin px C sinh py D cosh py where A B C D and p are arbitrary constants. . Elliptic Equations 1285 2 . Particular solutions in the polar coordinate system w r A In r B w r Arm B rm C cos mp D sin mp where A B C and D are arbitrary constants and m 1 2 . 3 . If w x y is a solution of the Laplace equation then the functions W3 w1 Aw Xx C1 Xy C2 B w2 Aw x cos 3 y sin 3 -x sin 3 y cos 3 x y x2 y2 x2 y2 are also solutions everywhere they are defined A B C1 C2 3 and X are arbitrary constants. The signs at X s in the formula for w1 are taken arbitrarily. 4 . A fairly general method for constructing particular solutions involves the following. Let f z u
