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Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 37.9 Hints Hint 37.1 Hint 37.2 Hint 37.3 Hint 37.4 Hint 37.5 Hint 37.6 Hint 37.7 Impose the boundary conditions u 0 t u 2n t Uỹ 0 t Uỹ 2n t . Hint 37.8 Apply the separation of variables u x y X x Y y . Solve an eigenvalue problem for X x . Hint 37.9 Hint 37.10 1734 Hint 37.11 Hint 37.12 There are two ways to solve the problem. For the first method expand the solution in a series of the form u x t an t sin ÍIEỊL . X L n 1 Because of the inhomogeneous boundary conditions the convergence of the series will not be uniform. You can differentiate the series with respect to t but not with respect to x. Multiply the partial differential equation by the eigenfunction sin nnx L and integrate from x 0 to x L. Use integration by parts to move derivatives in x from u to the eigenfunctions. This process will yield a first order ordinary differential equation for each of the an s. For the second method Make the change of variables v x t u x t x where x is the equilibrium temperature distribution to obtain a problem with homogeneous boundary conditions. Hint 37.13 Hint 37.14 Hint 37.15 Hint 37.16 Hint 37.17 .
