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2. Find the solution to the following non-homogeneous heat equation with given initial and boundary conditions on [0, π]. UtUxx = e sin (3x), u(0, t) = u(n, t) = 0, u(x,0) = sinx. Recall that for solving this boundary value problem, we solve for time-dependent Fourier coefficients bn (t) for the solution u(x, t) = bn(t) sin(nx). n=1

2. Find the solution to the following non-homogeneous heat equation with given initial and boundary conditions on [0, π]. UtUxx = e sin (3x), u(0, t) = u(n, t) = 0, u(x,0) = sinx. Recall that for solving this boundary value problem, we solve for time-dependent Fourier coefficients bn (t) for the solution u(x, t) = bn(t) sin(nx). n=1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Find the solution to the following non-homogeneous heat equation with given initial and boundary conditions on [0, π]. -2t Ut Uxx = e sin(3x), u(0, t) = u(t, t) = 0, u(x, 0) sin x. = Recall that for solving this boundary value problem, we solve for time-dependent Fourier coefficients bn (t) for the solution u(x, t): ∞ =Σbn(t) sin(nx). n=1 =
Transcribed Image Text:2. Find the solution to the following non-homogeneous heat equation with given initial and boundary conditions on [0, π]. -2t Ut Uxx = e sin(3x), u(0, t) = u(t, t) = 0, u(x, 0) sin x. = Recall that for solving this boundary value problem, we solve for time-dependent Fourier coefficients bn (t) for the solution u(x, t): ∞ =Σbn(t) sin(nx). n=1 =
2 Each bn (t) satisfies the first-order ODE bn(t) + n²bn(t) = ¶n (t), wherein qn(t) = Q(s, t)ds. Here, Q(x, t) e-t sin(3x). Also, note that the initial condition for each first order ODE comes from the initial condition of the heat equation. = As for the ODE, recall that for an equation of the form y' (t) + ky(t) = g(t), the homogeneous solution is of the form y(t) = Ce-kt, and the particular solution can be found using the Cauchy formula: Yp(t) = e •t - kt f *e²³9Ddi. ekt But in special cases, such as when g(t) is an exponential or polynomial function, one could assume an exponential or polynomial solution of unknown coefficients and find the coefficients by plugging that presumed solution into the equation.
Transcribed Image Text:2 Each bn (t) satisfies the first-order ODE bn(t) + n²bn(t) = ¶n (t), wherein qn(t) = Q(s, t)ds. Here, Q(x, t) e-t sin(3x). Also, note that the initial condition for each first order ODE comes from the initial condition of the heat equation. = As for the ODE, recall that for an equation of the form y' (t) + ky(t) = g(t), the homogeneous solution is of the form y(t) = Ce-kt, and the particular solution can be found using the Cauchy formula: Yp(t) = e •t - kt f *e²³9Ddi. ekt But in special cases, such as when g(t) is an exponential or polynomial function, one could assume an exponential or polynomial solution of unknown coefficients and find the coefficients by plugging that presumed solution into the equation.
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