2. We are lo solve y" -ky -) (O < x < L) subject to the boundary conditions y(0)y(L)0. a) Find Gr...
8. (a) Determine the Fourier sine series for the function { f(x) L 2 0 (b) Using your answer to part (a), solve the diffusion equation at for (a,t):0 < L, t>0} subject to the boundary conditions (0, t) (L, t) (x,0) f(x) 8. (a) Determine the Fourier sine series for the function { f(x) L 2 0 (b) Using your answer to part (a), solve the diffusion equation at for (a,t):0 0} subject to the boundary conditions (0, t)...
please solve #2 Solve the following problems subject to the given boundary conditions. Show the formulas for any arbitrary constants (Ao, An, Bn), but you do not need to actually calculate them tu a(0. t)=0. u(1, t) = 5 u(z,0-82-1 2 0< x<2, t0 u(0, t) = 0, u(2. t) = 0 a(x, 0) 0, tr(r,0) = 0 3 ー+-=-10, 0
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it. 6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
Solve equation (4) in Section 5.2 FIdywx) dxA (4) subject to the appropriate boundary conditions. The beam is of length L, and wo is a constant. (a) The beam is embedded at its left end and simply supported at its right end, and w(x) wo, 0 < x < L. усх) (b) Use a graphing utility to graph the deflection curve when wo 48EI and L = 1. = y y 0.2 0.4 0.6 0.8 1,0 0.2 0.4 0.6 0.8...
please help Problem 1: Solve the following differential equations subject to the specified boundary conditions: a. 40, subject to y(o) 30 and y(1) 10 br,subject to T(O2) 0 c. da-a29-0, (a is a constant), subject to θ(0) 50 and θ(x-.00) dx2 d20 dx? 10
Solve the heat equation Ut = Uxx + Uyy on a square 0 <= x <= 2, 0<= y<= 2 with the following boundary and initial conditions 2. Solve the heat equation boundary conditions uvw on a square O S r s 2, 0 S vS 2 with the (note the mix of u and tu) and with initial condition 0 otherwise Present your answer as a double trigonometric sum. 2. Solve the heat equation boundary conditions uvw on a...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0) = sin(x), 0<x<a
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
Solve equation (4) in Section 5.2 E = w(x) (4) subject to the appropriate boundary conditions. The beam is of length L, and wo is a constant. (a) The beam is embedded at its left end and simply supported at its right end, and w(x) = wg. 0<x<L. y(x) = (b) Use a graphing utility to graph the deflection curve when wo = 48E1 and L = 1. y + 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1...