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III condition tl (0,2)-Sill utO, (2e) (6) Write down the solutions to the following initial-bound...
4.2.3. Write down the solutions to the following initial-boundary value problems for the wave equ...
I
need help with this PDE problem, part A and D
4.2.3. Write down the solutions to the following initial-boundary value problems for the wave equation in the form of a Fourier series: a) utt = uzz , u(t, 0) = u(t, π) = 0, u(0,x) = 1, ut(0,z) = 0; (b) utt = 2uxx, a(t, 0) = u(t, π) = 0, a(0,x) = 0, ut(0,x) = 1; c) utt = 3uzz , ti(t, 0)=u(t, π)=0, u(0,x) = sin3 x,...
42.3) Write down the solutions to the follig inboundary value problems for the wave equation in t...
#4.2.3 (e, f, g only) & #4.3.11 (all of it)
42.3) Write down the solutions to the follig inboundary value problems for the wave equation in the form of a Fourier series: (a) utt = uzz , u(t, 0) = u(t, π) = 0, a(0,2) 1, ut(0,x) = 0; (d) ut4u (e) ut.-uzz , u(t, 0)u1) 0, u(0,), u, (0,x; u(t, 0)=ux(t, 1)=0, a(0,2)=1, ut(0,2)=0; (g) utt = uzx , ux(t, 0)-u, (t, 1) = 0, u(0,x)-x(1-x), ut(0,2 )-0. Explain...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
Apply method of images and Fourier transformation to solve the following boundary value problem f...
Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0
Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0
Solve the initial-boundary value problem. Theorem 11.3.5 (b) will simplify the computa- tion of the coefficients...
Solve the initial-boundary value problem. Theorem 11.3.5 (b) will simplify the computa- tion of the coefficients in the Fourier sine series. 64uz 0<x<3, t>0, u(0,t) = 0, u(3, t) = 0, t > 0, u(x,0) = (22 – 9), ut(2,0) = 0, 0<x<3. Utt
Problem 1: Solve the initial value / Dirichlet problem on the half-line and find the value...
Problem 1: Solve the initial value / Dirichlet problem on the half-line and find the value u(1, 2): (8 points) Utt(t, 2) – Uzz(t, x) = x+t, (t, x) ER [0, +co), u(0,x) = = cos(2), ut(0, 2) = e", u(t,0) = 1+t.
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), usi...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
2. (P5, page 105, 10pts) Solve the following initial-boundary value problem: utt = 90 , tu(0,1)...
2. (P5, page 105, 10pts) Solve the following initial-boundary value problem: utt = 90 , tu(0,1) = u(2,1) = 0, u(3,0) = 0, ut(2,0) = 4(x) = e-, for 0<x<2,t> 0, for t>0, for 0<x<2, for 0 < x < 2.
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
2. Let u(z,t) be a differentiable function on R x [0, 0o). a) Show that the directional derivativ...
2. Let u(z,t) be a differentiable function on R x [0, 0o). a) Show that the directional derivative of u at (x, t) = (zo, to) along v is Dvu(x, t) = ▽u(ro, to) , v b) Solve the following homogeneous linear transport equation ul + uz = 0, u(x,0) =-2 cosx c) Solve the following non-homogeneous equation ut-2uz--2 cos (x-t), u(x, 0) = sin x d) Solve the following second-order homogeneous linear euqation u(z,0) = sin x, ut (z,...
I
need help with this PDE problem, part A and D
4.2.3. Write down the solutions to the following initial-boundary value problems for the wave equation in the form of a Fourier series: a) utt = uzz , u(t, 0) = u(t, π) = 0, u(0,x) = 1, ut(0,z) = 0; (b) utt = 2uxx, a(t, 0) = u(t, π) = 0, a(0,x) = 0, ut(0,x) = 1; c) utt = 3uzz , ti(t, 0)=u(t, π)=0, u(0,x) = sin3 x,...
#4.2.3 (e, f, g only) & #4.3.11 (all of it)
42.3) Write down the solutions to the follig inboundary value problems for the wave equation in the form of a Fourier series: (a) utt = uzz , u(t, 0) = u(t, π) = 0, a(0,2) 1, ut(0,x) = 0; (d) ut4u (e) ut.-uzz , u(t, 0)u1) 0, u(0,), u, (0,x; u(t, 0)=ux(t, 1)=0, a(0,2)=1, ut(0,2)=0; (g) utt = uzx , ux(t, 0)-u, (t, 1) = 0, u(0,x)-x(1-x), ut(0,2 )-0. Explain...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0
Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0
Solve the initial-boundary value problem. Theorem 11.3.5 (b) will simplify the computa- tion of the coefficients in the Fourier sine series. 64uz 0<x<3, t>0, u(0,t) = 0, u(3, t) = 0, t > 0, u(x,0) = (22 – 9), ut(2,0) = 0, 0<x<3. Utt
Problem 1: Solve the initial value / Dirichlet problem on the half-line and find the value u(1, 2): (8 points) Utt(t, 2) – Uzz(t, x) = x+t, (t, x) ER [0, +co), u(0,x) = = cos(2), ut(0, 2) = e", u(t,0) = 1+t.
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
2. (P5, page 105, 10pts) Solve the following initial-boundary value problem: utt = 90 , tu(0,1) = u(2,1) = 0, u(3,0) = 0, ut(2,0) = 4(x) = e-, for 0<x<2,t> 0, for t>0, for 0<x<2, for 0 < x < 2.
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
2. Let u(z,t) be a differentiable function on R x [0, 0o). a) Show that the directional derivative of u at (x, t) = (zo, to) along v is Dvu(x, t) = ▽u(ro, to) , v b) Solve the following homogeneous linear transport equation ul + uz = 0, u(x,0) =-2 cosx c) Solve the following non-homogeneous equation ut-2uz--2 cos (x-t), u(x, 0) = sin x d) Solve the following second-order homogeneous linear euqation u(z,0) = sin x, ut (z,...
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