Question

7. (10 points extra credit, graded in OFFICE) solve the following ini- tial boundary value problem using separation of variables. Graph the fortieth partial sum of the solution for some values of t. Ytt =9y... + x20 <<4 t>O (15) g(0, 1) = 4(4,1) = 0 t>0 (16) y(2,0) = 0 yt(1,0) = 0 0 <<4 (17)

For the graph use software like Matlab or something similar.

Class: Partial Differential Equations I

Answer 1

/ 1 Date Page equation & + = quax + x² y (ot) = glut) 2.0 g(x,0) = Ye (x, 0) 20 0<x<4, to +70° OLX34 (x, t) a v(x, t) + w (x) be the solution then get a Vet you a Vaxt Wax I Vtta qe (v xx + Wax) + A D Alca) Condition! y(x, o 20 vero) wa) Condition 2 (x,o) a 0 OVE (0,0) 20 G (o,t) 20 Condition 3 Cladition 4 4 (4,t) zo v-14, t) - will Now let wia) be solution of coaxt Axao wco) 20 C ²W (o) 2 AX3 7 on to w 1420 - C20 64 AT 4 B 20 & 2-BX3 of ut + = c²Vz. vlagt) is sol in 711,0)2 - Vtixo) o Vlo,t) 2 0 2 v(uit)

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* Wolfram Mathematica 7.0 - [Untitled-29) File Edit Insert Format Cell Graphics Evaluation EXPECTEQQ.com Untitled-2 Palettes Window Help eqnid = (@.y[x, t]-90. Y[*, t] =2,[x, 0] = 0, Derivative[0, 1] [Y][X, 0] = 0, y[0, t] = 0, y[4, t] = 0) (2x, t] - Sy2.) (x, t) = r°, yix, 01 = 0, y 0,2)(x, 01 = 0, y[0, 1] = 0, y14, t] = 0) 3= sollb = y(x, t) !. NdSolve [egnib, y(x, t), (1, 0, 1), (0, 0, 4), PrecisionGoal - 3] [[1]] 3 Interpolatingiunction (10., 4.1, (0., 4.13, ][x, t) = Plot3D (2011), (1, 0, 1), (0, 0, 4, Arestabel - ("x", "T }, Ticis - [[0, 1, 2, 3), (0, 1, 2), (-3, 0]] WINX 1