tailieunhanh - Handbook of mathematics for engineers and scienteists part 202

Handbook of mathematics for engineers and scienteists part 202. 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. | . Systems OF General Form 1375 1 . Solution u t c exp y f t bx - G0 dt d x t w t b exp y f t b Y - G0 dt d x t where ự ự t and f f t are determined by the system of ordinary differential equations ự t ựf t bự - cf g t bự - cf ft ff t bự - cf h t bự - cf and the function ỡ ỡ x1 . xn t satisfies linear equation 7 Ị. Remark 1. The coefficients of the linear differential operator L can be dependent on t. 2 . Let us multiply the first equation by b and the second one by -c and add the results together to obtain L Z Zf t z bg t z - ch t z z bu - cw. 1 dt This equation will be considered in conjunction with the first equation of the original system du _ . . . 7J7 L u uf t z g t z . 2 dt Equation 1 can be treated separately. Given a solution of this equation z z x1 . xn t the function u u x1 . xn t can be determined by solving the linear equation 2 and the function w w x1 . xn t is found as w bu - z c. Remark 2. Let L be a constant-coefficient differential operator with respect to the independent variable X X1 and let the condition d cf t z bg t Z - ch t z 0 dt L J hold true for example it is valid if the functions f g h are not implicitly dependent on t . Then equation 1 admits an exact traveling-wave solution z Z z where z kx - xt with arbitrary constants k and A. 2- Li u uf W LzW w fj U . ot w J ot w J Here L1 and L2 are arbitrary constant-coefficient linear differential operators of any order with respect to x. 1 . Solution u ekx-Xty w ekx-Xtz x. - Yt where k A 3 and y are arbitrary constants and the functions y y and z z are determined by the system of ordinary differential equations Mi y Ay yf y z 0 Mztz Az zg y z 0 Mi y e kx Li ekxy M2 z e kx L2 ekx z . To the special case k A 0 there corresponds a traveling-wave solution. 2 . If the operators L1 and L2 contain only even derivatives there are solutions of the form u C1 sin kx C2 cos kx t w C1 sin kx C2 cos kx t u C1 exp kx C2 exp -kx t w C1 exp kx C2 exp -kx t u C1x C2 t w C1x C2 t where C1 C2 and k are .