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Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) ...
Can anyone help with this question please? Given a domain Ω c R2 and a smooth function f,uo : Ω-+ R consider the proble...
Can anyone help with this question please?
Given a domain Ω c R2 and a smooth function f,uo : Ω-+ R consider the problem Uz (x, t)-Au (x, t) + u(x, t) u(x, t = f(x) Y(x, t) E Ω × (0, oo), V(x, t) E 2 x (0, 00), Assume that u(z, t) is a smooth solution and that v(x) is a smooth stationary (i.e., time-independent) solution. Derive a PDE problem for the difference w(x, t)u(x, t)(x) By multiplying...
Problem 1 Consider the heat PDE in d 3: Write down the unique solution to (1) using a 3-dimension...
Problem 1 Consider the heat PDE in d 3: Write down the unique solution to (1) using a 3-dimensional Brownian motion B. Write down your answer as expectation, as an integral over Ω, and as an integral over Rd.
Problem 1 Consider the heat PDE in d 3: Write down the unique solution to (1) using a 3-dimensional Brownian motion B. Write down your answer as expectation, as an integral over Ω, and as an integral over Rd.
2. Consider the initial-boundary value problem 0 0, t >0 Uz uェェ, Uz(0, t) _ u(0, t) = 0. a(z, 0...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
This is PDE problem. Please show all steps in detail with neat handwriting. Problem . Consider...
This is PDE problem. Please show all steps in detail with neat
handwriting.
Problem . Consider the function a) Find the full Fourier Series of F(x) a(0, y, t) = u(a, y, t) 0 u(z, 0, t ) = u(z, b, l) = 0 u(z,y,0) = f(z,y), u(x, y,0)-g(x,y), 0<y< b,t0 a) b) Solve the initial-boundary value problem for 2D wave equation. What is the physical interpretation of these boundary conditions
PDE question Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt...
PDE
question
Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
2. Solve the one-dimensional heat equation problem for a unit length bar with insulated ends with...
2. Solve the one-dimensional heat equation problem for a unit length bar with insulated ends with a prescribed initial linear temperature distribution: c2uxx = 111 , l4 (0,t)-14 (l,t)-0, 0 < x < 1 20-x) , Last Name A-M 3x, Last Name N -Z u(x,0) = The general solution to this problem is given in Example 4, page 563 in the text in terms of a Fourier Cosine Series. Write out the solution steps and evaluate the Fourier coefficients by...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
(2) [15pts] Write the solution of the PDE for 0
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(2) [15pts] Write the solution of the PDE for 0 <<1 and t>0 u(0,t) 0 u(1,t) = 0 u(x,0)-0 Make sure you simplify as much as possible by using the fact that the source term f depends only on z and not on t. What is the solution at long time lim,-+oo u(x, t) if f()T) [Hint: You can obtain this by solving a single (trivial) ODE]
(2) [15pts] Write the solution...
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 sol...
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...
Can anyone help with this question please?
Given a domain Ω c R2 and a smooth function f,uo : Ω-+ R consider the problem Uz (x, t)-Au (x, t) + u(x, t) u(x, t = f(x) Y(x, t) E Ω × (0, oo), V(x, t) E 2 x (0, 00), Assume that u(z, t) is a smooth solution and that v(x) is a smooth stationary (i.e., time-independent) solution. Derive a PDE problem for the difference w(x, t)u(x, t)(x) By multiplying...
Problem 1 Consider the heat PDE in d 3: Write down the unique solution to (1) using a 3-dimensional Brownian motion B. Write down your answer as expectation, as an integral over Ω, and as an integral over Rd.
Problem 1 Consider the heat PDE in d 3: Write down the unique solution to (1) using a 3-dimensional Brownian motion B. Write down your answer as expectation, as an integral over Ω, and as an integral over Rd.
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
This is PDE problem. Please show all steps in detail with neat
handwriting.
Problem . Consider the function a) Find the full Fourier Series of F(x) a(0, y, t) = u(a, y, t) 0 u(z, 0, t ) = u(z, b, l) = 0 u(z,y,0) = f(z,y), u(x, y,0)-g(x,y), 0<y< b,t0 a) b) Solve the initial-boundary value problem for 2D wave equation. What is the physical interpretation of these boundary conditions
PDE
question
Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
2. Solve the one-dimensional heat equation problem for a unit length bar with insulated ends with a prescribed initial linear temperature distribution: c2uxx = 111 , l4 (0,t)-14 (l,t)-0, 0 < x < 1 20-x) , Last Name A-M 3x, Last Name N -Z u(x,0) = The general solution to this problem is given in Example 4, page 563 in the text in terms of a Fourier Cosine Series. Write out the solution steps and evaluate the Fourier coefficients by...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
Can you help me with this question please?
(2) [15pts] Write the solution of the PDE for 0 <<1 and t>0 u(0,t) 0 u(1,t) = 0 u(x,0)-0 Make sure you simplify as much as possible by using the fact that the source term f depends only on z and not on t. What is the solution at long time lim,-+oo u(x, t) if f()T) [Hint: You can obtain this by solving a single (trivial) ODE]
(2) [15pts] Write the solution...
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...
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