3. Find the solution to the two-dimensional heat equation, ut = u -oo < ! < oo,-oo < y < oo) with...
Find a formula for the solution of the initial value problem for for t>0, -oc < x < oo ut = uzz-u a(1:0) = g(z) -x < 1 < x where g is continuous and bounded.( Hint: use v(x, t) = et u(z. t).)
Find a formula for the solution of the initial value problem for for t>0, -oc
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients
3. Consider the...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1.
Let...
The two-dimensional heat equation reduces to Laplace's equation to = 0 if the temperature u is steady-state. u(x, y) is defined in 0<x<2 and 0 Sys2 and satisfy u(x,0) = u(x, 2) = u(0, y) = 0 and u(2, y) = 80 sin my. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables. (2) Find u(x, y) satisfying the boundary condition. (3) Obtain the value of u(1,5).
Write out the solution
please
Find the steady-state solution of the heat conduction equation α2uxx-ut that satisfies the given set of boundary conditions. ux(0, t)-u(0, t) = 0, u(L, t)-T v(x) =
Find the steady-state solution of the heat conduction equation α2uxx-ut that satisfies the given set of boundary conditions. ux(0, t)-u(0, t) = 0, u(L, t)-T v(x) =
3. (5 points) Find the solution u(x,t) of the equation ut = uxx, subject to the boundary conditions u(0,t) = 1, u(2,t) = 3, and the initial condition u(x,0) = 3x + 1.
1. Complete the example started in class: u(r, t) is the solution of the 3 dimensional wave equation with radial symmetry, tze(r +2ur/r), with initial data u(r,0) -0, and u(r,0)-9(r) with -1 935 0, r>a. Using the formula developed in class, find u(r, t).
1. Complete the example started in class: u(r, t) is the solution of the 3 dimensional wave equation with radial symmetry, tze(r +2ur/r), with initial data u(r,0) -0, and u(r,0)-9(r) with -1 935 0, r>a. Using...
(40 marks) Find the solution of the two-dimensional Laplace equation$$ u_{x x}+u_{y y}=0 \quad 0<x<1,0<y<1 $$with the boundary conditions$$ u(x, 1)=x, u(x, 0)=u(0, y)=0, u(1, y)=y $$
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr)
Let u be the solution to the initial boundary value problem for the...
6. Find the solution of the 1-dimensional heat equation on the interval 0, : Uxx, Ur (t, 0) U1(t, T) = 0, u(0, x) = 100 cos 2x
6. Find the solution of the 1-dimensional heat equation on the interval 0, : Uxx, Ur (t, 0) U1(t, T) = 0, u(0, x) = 100 cos 2x