tailieunhanh - Acoustic Waves From Microdevices to Helioseismology Part 3

Tham khảo tài liệu 'acoustic waves from microdevices to helioseismology part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 68 Acoustic Waves - From Microdevices to Helioseismology to the ODE G u u u . 0 u du. OQ In general the left hand side of Eq. is a polynomial in u and its various derivatives. Step 2. The F-expansion method gives the solution of in the form N u x f u Q E a Fi Q aN 0 i 0 where ai i 0 1 2 . N are constants to be determined and F Q satisfies the first order nonlinear ODE in the form F Q 2 q0 q2 F2 Q q4 F4 Q where q0 q2 and q4 are constants and N in Eq. is a positive integer that can be determined by balancing the nonlinear term s and the highest order derivatives in Eq. . Step 3. Substituting the F-expansion into and using setting each coefficient of the polynomial to zero yields a system of algebraic equations involving a0 1 .aN k and w. Step 4. Solving these equations probably with the aid of Mathematica or Maple then a0 1 .aN k and w can be expressed by q0 q2 q4. Step 5. Substituting these results into F-expansion then a general form of traveling wave solution of the NLPDE can be obtained. Many solutions of equation have been reported in 13 14 . Substituting the values of q0 q2 q4 and the corresponding JEF solution F Q into the general form of solution we may get several classes of exact solutions of equations involving JEFs. Also we give a brief description of the mapping method to seek the traveling wave solutions of in the form u x f u n n kx w f 70 70 is an arbitrary constant. Thus Eq. reduces to Eq. whose solution can be express in the form n u n E Aif n i 0 where n is a balancing number Ai are constants to be determined and f n satisfies the DE 2 . 2 f f2 Here p q and r are constants. After substituting Eq. into the ODE and using Eq. the constants Ai k and w may be determined. By using the solutions of auxiliary nonlinear equation many JEF solutions of NLEEs have been obtained 19 20 . The JEFs sn Q sn Q m cn Q cn Q m and dn Q dn Q m are double periodic and have the .