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3. This question is about non-homogeneous boundary conditions (a) Consider Laplace's equation on a rectangle, with...
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...
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x...
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
Thank you. 5. Find the solution u(x, y) of Laplace's equation in the rectangle 0<<a, 0<y<b...
Thank you.
5. Find the solution u(x, y) of Laplace's equation in the rectangle 0<<a, 0<y<b that satisfies the boundary conditions u(0, y = 0, u(a, y) = 0,uy 3,0) = 0, 2,b) = g(1), where J2 0<x<a/2 g(x) = - 0<a/2 <<a
3. Consider the Laplace's equation on a rectangular domain subject to the following boundary conditions that...
3. Consider the Laplace's equation on a rectangular domain subject to the following boundary conditions that represents the steady-state heating of a plate. A temperature probe shows that (1/2, 1/4) = 0. Solve this problem using the method of separation of variables. (7) byllyy = 0 0 <I<41 and O y <21 U-(0,y)=0, 1-(41, y) = cos(2), 4(1,0) = cos(2), 4(1,2)=0. (total 25 marks
Problem 2 (Chapter 7; 60 points) Solve the following Laplace's equation in a parallelepiped, care...
Problem 2 (Chapter 7; 60 points) Solve the following Laplace's equation in a parallelepiped, carefully explaining all steps. Make sure to check for zero eigenvalues. Note that the single non-homogeneous boundary condition is on the y-plane (y D). The boundary conditions are simple enough so that all Fourier coefficient integrals are easily calculated. You don't have to give details in the solution of the two LHBVPs, but make sure to check all boundary conditions after you find the solution: ou...
Problem 2. (15 points) Solve the following Laplace's equation in a cube as outlined below. au...
Problem 2. (15 points) Solve the following Laplace's equation in a cube as outlined below. au au au 2,2 + a2 + a2 = 0, on 0<x<1, 0<y<1, 0<?<1, (0, y, z) = (1, y, z) = 0, (x, 0, 2) = u(x, 1, ) = 0, (x, y,0) = 0, u(x,y, 1) = x. (a) Seek a solution of the form u(x, y, z) = F(x) G(v) H(-). Show that with the appropriate choice of separation constants, you can...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a given periodic function with satisfying f (0) = f (2π) and Moreover, u(r,0 is bounded for r s a Which of the following is the (general) solution Select one: A. where for B. where )cos(n)de and for C. where and 2m for n- 1,2,3, D. where Co E R f(0) cos(n0)de and for
Let a >0 Solve the following Laplace's equation...
(1 point) Solve the heat problem with non-homogeneous boundary conditions ди (x, t) at = a2u...
(1 point) Solve the heat problem with non-homogeneous boundary conditions ди (x, t) at = a2u (2,t), 0 < x < 5, t> 0 ar2 u(0,t) = 0, u5,t) = 3, t>0, u(x,0) = **, 0<x< 5. Recall that we find h(x), set v(x, t) = u(x, t) – h(x), solve a heat problem for v(x, t) and write u(x, t) = v(x, t) +h(x). Find h(c) h(x) = The solution u(x, t) can be written as u(x,t) =h(x) +...
(1 point) Solve the heat problem with non-homogeneous boundary conditions du (x, 1) = ot (x,1),...
(1 point) Solve the heat problem with non-homogeneous boundary conditions du (x, 1) = ot (x,1), 0<x<2, t> 0 dx (0,t) = 0, (2, 1) = 2, t> 0, u(x,0) = 0<x<2. Recall that we find h(x), set u(x, t) = u(x, t)-h(x), solve a heat problem for u(x, t) and write u(x, t) = u(x, t) + h(x). Find h(x) h(x) = The solution u(x, t) can be written as u(x, t) = h(x) + u(x, t), where u(x,...
1. Consider the following inhomogeneous wave equation on (0,7) : utt - 4uxx = (1 -...
1. Consider the following inhomogeneous wave equation on (0,7) : utt - 4uxx = (1 - x) cost Uz(0,t) = cost-1, uz(7,t) = cost u(3,0) = 2(7,0) = cos 3x (a) Convert the PDE to an equation with homogeneous boundary conditions by making an appropriate substitution u(x, t) = u(x, t) - p(x, t), implying u(,t) = v(x, t) + p(2,t) for an appropriate function p(x, t). (b) Finish solving the PDE using the Method of Eigenfunction expansion.
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...
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
Thank you.
5. Find the solution u(x, y) of Laplace's equation in the rectangle 0<<a, 0<y<b that satisfies the boundary conditions u(0, y = 0, u(a, y) = 0,uy 3,0) = 0, 2,b) = g(1), where J2 0<x<a/2 g(x) = - 0<a/2 <<a
3. Consider the Laplace's equation on a rectangular domain subject to the following boundary conditions that represents the steady-state heating of a plate. A temperature probe shows that (1/2, 1/4) = 0. Solve this problem using the method of separation of variables. (7) byllyy = 0 0 <I<41 and O y <21 U-(0,y)=0, 1-(41, y) = cos(2), 4(1,0) = cos(2), 4(1,2)=0. (total 25 marks
Problem 2 (Chapter 7; 60 points) Solve the following Laplace's equation in a parallelepiped, carefully explaining all steps. Make sure to check for zero eigenvalues. Note that the single non-homogeneous boundary condition is on the y-plane (y D). The boundary conditions are simple enough so that all Fourier coefficient integrals are easily calculated. You don't have to give details in the solution of the two LHBVPs, but make sure to check all boundary conditions after you find the solution: ou...
Problem 2. (15 points) Solve the following Laplace's equation in a cube as outlined below. au au au 2,2 + a2 + a2 = 0, on 0<x<1, 0<y<1, 0<?<1, (0, y, z) = (1, y, z) = 0, (x, 0, 2) = u(x, 1, ) = 0, (x, y,0) = 0, u(x,y, 1) = x. (a) Seek a solution of the form u(x, y, z) = F(x) G(v) H(-). Show that with the appropriate choice of separation constants, you can...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a given periodic function with satisfying f (0) = f (2π) and Moreover, u(r,0 is bounded for r s a Which of the following is the (general) solution Select one: A. where for B. where )cos(n)de and for C. where and 2m for n- 1,2,3, D. where Co E R f(0) cos(n0)de and for
Let a >0 Solve the following Laplace's equation...
(1 point) Solve the heat problem with non-homogeneous boundary conditions ди (x, t) at = a2u (2,t), 0 < x < 5, t> 0 ar2 u(0,t) = 0, u5,t) = 3, t>0, u(x,0) = **, 0<x< 5. Recall that we find h(x), set v(x, t) = u(x, t) – h(x), solve a heat problem for v(x, t) and write u(x, t) = v(x, t) +h(x). Find h(c) h(x) = The solution u(x, t) can be written as u(x,t) =h(x) +...
(1 point) Solve the heat problem with non-homogeneous boundary conditions du (x, 1) = ot (x,1), 0<x<2, t> 0 dx (0,t) = 0, (2, 1) = 2, t> 0, u(x,0) = 0<x<2. Recall that we find h(x), set u(x, t) = u(x, t)-h(x), solve a heat problem for u(x, t) and write u(x, t) = u(x, t) + h(x). Find h(x) h(x) = The solution u(x, t) can be written as u(x, t) = h(x) + u(x, t), where u(x,...
1. Consider the following inhomogeneous wave equation on (0,7) : utt - 4uxx = (1 - x) cost Uz(0,t) = cost-1, uz(7,t) = cost u(3,0) = 2(7,0) = cos 3x (a) Convert the PDE to an equation with homogeneous boundary conditions by making an appropriate substitution u(x, t) = u(x, t) - p(x, t), implying u(,t) = v(x, t) + p(2,t) for an appropriate function p(x, t). (b) Finish solving the PDE using the Method of Eigenfunction expansion.
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