Consider the backward heat equation ∂tu + uxx = 0 with u(0, x) = g(x) in...

Consider the backward heat equation ∂tu + uxx = 0 with u(0, x) = g(x) in the periodic domain x ∈ T (Peridoc) . Use separation of variables to solve this equation and prove that as long as g is not a constant, the solution does not decay whent → ∞.

Expert Solution

Colby Messinger answered 1 month ago

Consider the equation utt = uxx x ∈ (0, pi) ux(0,t) = u(pi,t) = 0 Write...

Consider the equation utt = uxx x ∈ (0, pi) ux(0,t) = u(pi,t) = 0 Write the series expansion for a solution u(x,t)

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < 1,...

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < 1, t > 0, u(0,t) = 0 = u(1,t), u(x,0) = x/2

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < pi,...

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < pi, t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinx - sin3x

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < 1,...

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < 1, t > 0, ux(0,t) = 0 = ux(1,t), u(x,0) = 1 - x2

Find the solution to the heat equation on 0 < x < l, with u(0, t)...

Find the solution to the heat equation on 0 < x < l, with u(0, t) = 0, ux(l, t) = 0, and u(x, 0) = phi(x). This is sometimes called a "mixed" boundary condition.

Use separation of variables to solve uxx+2uy+uyy=0, u(x, 0) = f(x), u(x, 1) = 0, u(0,...

Use separation of variables to solve uxx+2uy+uyy=0, u(x, 0) = f(x), u(x, 1) = 0, u(0, y) = 0, u(1, y) = 0.

Solve the nonhomogeneous heat equation: ut-kuxx=sinx, 0<x<pi, t>0 u(0,t)=u(pi,t)=0, t>0 u(x,0)=0, 0<x<pi

Please solve the following: ut=uxx, 0<x<1, t>0 u(0,t)=0, u(1,t)=A, t>0 u(x,0)=cosx, 0<x<1

Consider the equation uux + uy = 0 with the initial condition u(x, 0) = h(x)...

Consider the equation uux + uy = 0 with the initial condition u(x, 0) = h(x) = ⇢ 0 for x > 0 uo for x < 0, with uo< 0. Show that there is a second weak solution with a shock along the line x = uo y / 2 The solution in both mathematical and graphical presentation before and after the shock.

Consider the nonhomogenous heat equation with time dependent sources ut = uxx + xt, an initial...

Consider the nonhomogenous heat equation with time dependent sources ut = uxx + xt, an initial condition u(x, 0) = x2 and inhomogenous boundary conditions ux(0, t) = 2t and ux(1, t) = 4. Use eigen function expansion to find the solution u(x, t).

Question

Consider the backward heat equation ∂tu + uxx = 0 with u(0, x) = g(x) in...

Solutions

Expert Solution

Related Solutions

Consider the equation utt = uxx x ∈ (0, pi) ux(0,t) = u(pi,t) = 0 Write...

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < 1,...

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < pi,...

Solve the following heat equation using Fourier Series uxx = ut, 0 < x < 1,...

Find the solution to the heat equation on 0 < x < l, with u(0, t)...

Use separation of variables to solve uxx+2uy+uyy=0, u(x, 0) = f(x), u(x, 1) = 0, u(0,...

Solve the nonhomogeneous heat equation: ut-kuxx=sinx, 0<x<pi, t>0 u(0,t)=u(pi,t)=0, t>0 u(x,0)=0, 0<x<pi

Please solve the following: ut=uxx, 0<x<1, t>0 u(0,t)=0, u(1,t)=A, t>0 u(x,0)=cosx, 0<x<1

Consider the equation uux + uy = 0 with the initial condition u(x, 0) = h(x)...

Consider the nonhomogenous heat equation with time dependent sources ut = uxx + xt, an initial...