tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 4

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | We calculate the derivatives of Ơ and T. ơx 0 ơy 2y Tx 2x Ty 0 Then we calculate the derivatives of u. ux 2xuT uy 2yuơ uxx 4x2uTT 2uT uyy 4y uơơ 2uơ Finally we transform the equation to canonical form. y2 uxx x2uyy 0 ơ 4tutt 2uT T 4auơơ 2uơ 0 uơơ uTT n uơ n uT 2ơ 2t Equilibrium Solutions Example Consider the equilibrium solution for the following problem. ut uxx u x 0 x ux 0 t ux 1 t 0 Setting ut 0 we have an ordinary differential equation. d2 u x 0 dx2 This equation has the solution u ax b. 1694 Applying the boundary conditions we see that u b. To determine the constant we note that the heat energy in the rod is constant in time. u x t dx Jo b dx Jo u x 0 dx Jo x dx Jo Thus the equilibrium solution is u x 2. .

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