MATH 412: Theory of PartialDifferential EquationsLecturer: Prof. Wolfgang BangerthBlocker Bldg., Ro om 507D(979) 845 [email protected]://www.math.tamu.edu/~bangerthHomework assignment 10 – due Thursday 11/16/2006Problem 1 (Solutions of the heat equation with inhomogenous bound-ary conditions). In class, we have seen how one can solve the heat equationwith inhomogenous boundary conditions by a) writing the solution u(x, t) asu(x, t) = uE(x) + ˜u(x, t), b) solving for the equilibrium temperature uE(x), andc) solving for the remainder ˜u, which has to satisfy a homogenous PDE withhomogenous boundary conditions.Use this technique to derive the solution of heat equation on [0, 1]:∂u(x, t)∂t− k∂2u(x, t)∂t2= 0,u(0, t) = 1,u(1, t) = 2,u(x, 0) = sin(πx).(5 points)Problem 2 (Eigenfunction expansion). Solve problem 8.3.2 in the book,i.e. derive a solution of the formu(x, t) =∞Xn=1an(t) sinnπxL,with time-dependent coefficients an(t). State the differential equations the an(t)have to satisfy and derive that the solution must converge to a steady state underthe conditions stated in the problem. (4 points)Problem 3 (Eigenfunction expansion). Use the method of eigenfunctionexpansions to solve the following problem:∂u(x, t)∂t− k∂2u(x, t)∂t2= 1,u(0, t) = 0,u(1, t) = 0,u(x, 0) = 0.You may use the formulas from Problem 2, but this time need to solve for theexplicit form of the coefficients an(t). (4 points)1Problem 4 (Eigenfunction expansion of a different equation). Con-sider the following variant of the heat equation (note the additional term in thePDE):∂u(x, t)∂t− k∂2u(x, t)∂x2+ αu(x, t) = q(x),u(0, t) = 0,u(1, t) = 0,u(x, 0) = f (x).As for the heat equation, the solution can be written asu(x, t) =∞Xn=1an(t) sinnπxL.However, the coefficients an(t) now have to satisfy a different ordinary differen-tial equation. Go back to your notes to see how the ODE for an(t) was derivedfor the heat equation and adjust this process to the present equation. Statewhich ODE an(t) has to satisfy here. (3
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