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Tham khảo tài liệu 'heat conduction basic research part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 114 Heat Conduction - Basic Research and it follows from 2.7 and 2.8 that m W t V i 1 m ư - Z iai i 1 Ằ 2i 1 i Ằ2 2jU 2.10 ựẰ 2i -1 2 i -1 and so on. Here the prime denotes the derivative with respective to r. To determine u explicitly we take the following four steps Step 1. Determine the integer m by substituting Eq. 2.7 along with Eq. 2.8 into Eq. 2.5 or 2.6 and balancing the highest-order nonlinear term s and the highest-order partial derivative. Step 2. Substitute Eq. 2.7 with the value of m determined in Step 1 along with Eq. 2.8 into Eq. 2.5 or 2.6 and collect all terms with the same order of together the left-hand side of Eq. 2.5 or 2.6 is converted into a polynomial in . Then set each coefficient of this polynomial to zero to derive a set of algebraic equations for k c a0 and ai for i 1 2 . m. . Step 3. Solve the system of algebraic equations obtained in Step 2 for k c a0 and V for Ỉ 1 2 . m by use of Maple. Step 4. Use the results obtained in the above steps to derive a series of fundamental . . G __ solutions u r of Eq. 2.5 or 2.6 depending on I G I since the solutions of Eq. 2.8 have been well known for us we can obtain exact solutions of Eqs. 2.1 and 2.2 . 2.2 The Exp-function method According to the classic Exp-function method it is assumed that the solution of ODEs 2.5 or 2.6 can be written as g 2 an exp nr u r ---------- bm mr m -p af exp f r a- g exp -gr bp exp pr b-q exp -qr 2.11 where f g p and q are positive integers which are unknown to be further determined and an and bm are unknown constants. 115 Exact Travelling Wave Solutions for Generalized Forms of the Nonlinear Heat Conduction Equation 3. A generalized form of the nonlinear heat conduction equation 3.1 Application of the G ZG -expansion method Introducing a complex variable rj defined as Eq. 2.3 Eq. 1.1 becomes an ordinary differential equation which can be written as -kcU - ak2 Un - U Un 0 a 0 3.1 or equivalently -kcU - ak2n n - 1 Un-2U 2 - ak2nUn-1U - U Un 0 3.2 To get a closed-form