TC2411 Tutorial 7 1. For each of the following PDEs, determine if the PDE, boundary conditions or initial conditions are linear or nonlinear, and, if linear, whether they are homogeneous or nonhomogeneous. Also, determine the order of the PDE. (a) u xx + u xy = 2u , u x (0, y ) = 0 (b) u xx + xu xy = 2 , u ( x,0) = 0 , u ( x,1) = 0 (c) u xx ut = f ( x, t ) , ut ( x,0) = 2 (d) u xx = ut , u ( x,0) = 1 , u (1, t ) = 0 (e) u xut + u xt = 2u , u (0, t ) + u x (0, t ) = 0 (f) u xx + e utt = u cos x , t u ( x,0) + u ( x,1) = 0 2. Solve ut = u xx for 0 x , given u (0, t ) = 0 , u ( , t ) = 0 , u ( x,0) = 10 sin x . 3. Solve ut = u xx for 0 x 1 , given u (0, t ) = 0 , u (1, t ) = 2 , u ( x,0) = 3 . 4. Solve ut = u xx for 0 x 1 , given u (0, t ) = 2 , u (1, t ) = 2 , u ( x,0) = e x . 5. Consider 1D transient heat conduction along a thin rod ( 0 x 2 ). T 2T =3 2 t x The boundary conditions are T = 0 at x = 0, 2 . The initial condition is 0 x 1 x T = 1 x 2 2 x Determine T. 6. Solve ut = u xx for 0 x 1 , given u x (0, t ) = 0 , u x (1, t ) = 0 , u ( x,0) = cos x . 7. Solve ut = u xx for 0 x 1 , given u (0, t ) = 0 , u x (1, t ) = 0 , u ( x,0) = 1 . 8. Solve u xx + u yy = 0 for 0 x 1 , 0 y 2 , given u ( x,0) = 0 , u ( x,2) = x , u (0, y ) = 0 , u (1, y ) = 0 . 9. Solve u xx + u yy = 0 for 0 x 1 , 0 y 1 , given u ( x,0) = 1 x , u ( x,1) = x , u (0, y ) = 0 , u (1, y ) = 0 . 10. Solve u xx + u yy = 0 for 0 x 1 , 0 y 1 , given u ( x,0) = 0 , u y ( x,1) = 0 , u (0, y ) = 0 , u (1, y ) = 1 . Answer: 1. (a) 2nd order linear homogeneous PDE; linear homogeneous BC (b) 2nd order linear nonhomogeneous PDE; linear homogeneous BCs (c) 2nd order linear nonhomogeneous PDE; linear nonhomogeneous IC (d) 2nd order linear homogeneous PDE; linear homogeneous BC; linear nonhomogeneous IC (e) 2nd order nonlinear homogeneous PDE; linear homogeneous BC (f) 2nd order linear homogeneous PDE; linear homogeneous IC 2. u ( x, t ) = 10e t sin x 2 6 2 2 (1) n e n t sin nx 3. u ( x, t ) = 2 x + n n =1 n 4 2 n 2 2 (1 (1) n e 1 ) (1 (1) n ) e n t sin nx 4. u ( x, t ) = 2 + 2 2 n n =1 1 + n 8 5. T ( x, t ) = n 2 n =1 n 4 n 2 2 t nx e sin 2 2 3 sin 2 6. u ( x, t ) = e t cos x 2 1 n + 2 e 2 1 n + 2 7. u ( x, t ) = n=0 8. u ( x, y ) = n =1 2 2t 1 sin n + x 2 2(1) n +1 sinh ny sin nx n sinh 2n 2 sinh ny 2 9. u ( x, y ) = cosh ny [(1) n + cosh n ] sin nx n sinh n n =1 n 10. u ( x, y ) = n=0 1 1 sinh n + x sin n + y 1 1 2 2 n + sinh n + 2 2 2