tailieunhanh - Handbook of mathematics for engineers and scienteists part 192

Handbook of mathematics for engineers and scienteists part 192. 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. | . Parabolic Equations 1305 4. 2W o wm bw -m. dt dx dx J Functional separable solution x A 2 m bm2 11 m X A B F t -m 2 - --------F t F t 1 4a m 1 J where A B and C are arbitrary constants. w x t F t - C - J t 5. 2W. aA w bwi-n. dt dx dx J Generalized traveling-wave solution x Ci hn2 A n w x t - ------ C2 - kt _ C2 - kt 3a n 1 where C1 and C2 are arbitrary constants. . 2a n 1 k --------- n 6. 1 . dw d Xw dw dt dx dx Solutions 2 x A w x t ln A WB - 2at J W x t 1 ln A Bx - A where A B C and D are arbitrary constants. 2 . There are solutions of the following forms w x t F z w x t G w x t H n 2kt n xe kXt w x t U Z - A-1 ln t Z x k ln t where k and fi are arbitrary constants. z kx fit e xt-1 2 D 2aCt traveling-wave solution self-similar solution dw d I dw 7. dt dx f w d J This equation occurs in nonlinear problems of heat and mass transfer and flows in porous media. 1 . Traveling-wave solution in implicit form 2 f f w dw 7 k2 ------ - kx At C2 J Aw C1 where C1 C2 k and A are arbitrary constants. To A 0 there corresponds a stationary solution. 2 . Self-similar solution w w z z xt 1 2 where the function w z is determined by the ordinary differential equation f w w z Z 2 zw z 0. 1306 Nonlinear Mathematical Physics Equations 8. dw d dt dx x dw f w dx g w . This equation occurs in nonlinear problems of heat and mass transfer with volume reaction. 1 . Traveling-wave solutions w w z z x Xt where the function w z is determined by the autonomous ordinary differential equation f w w z Z - Xw z g w 0. 2 . Let the function f f w be arbitrary and let g g w be defined by z s A g w B f w where A and B are some numbers. In this case there is a functional separable solution that is defined implicitly by y f w dw At - 2Bx2 C1x C2 where C1 and C2 are arbitrary constants. 3 . Now let g g w be arbitrary and let f f w be defined by . A1A2w B A2A3 f w -----TX----- T Z dw 1 g w g w J Z - - 2 where A1 A2 and A3 are some numbers. Then there are generalized traveling-wave solutions of the form