tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 9

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Solutions Solution We write the derivatives of u x t in terms of derivatives of F n . ut axta-1F á F t Ux taF 1 u . t2a F F uxx x2 We substitite these expressions into the heat equation. a - F t x2 F ax 1F - t n We can write this equation in terms of F and n only if a -1 2. We make this substitution and solve the ordinary differential equation for F n . F n F - 2 iog F - .2 c 4- Í n2 A F 1 c exp I .1 4-7 n2 F C1 exp I - 4- I dn C2 We can write F in terms of the error function. F C1 erf C2 2V 1894 We write this solution in terms of x and t. u x t C1 erf C2 This is not the general solution of the heat equation. There are many other solutions. Note that since x and t do not explicitly appear in the heat equation A_ n ti x -x0 I . u x t Cl erf c2 2Vv t - t0 J is a solution. Solution We write the derivatives of X in terms of f. X lAf xaf t- f dtdC xf u ộx f axa-1tf r dxdự J Xxx f -i- ax -1t axa 1taxa 1t-df ch ỚẸ Xxx a2x2a-2t2 f a a - 1 xa-2tf x 2 a2 2f a a - l f We substitute these expressions into the diffusion equation. Tf x 2t a2 2f 11 a a - l f 9 In order for this equation to depend only on the variable we must have a -2. For this choice we obtain an ordinary .