Homework Answers
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a)
14 points Consider the following equation : PDE: u+ 0 ,0
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
More details pls Consider the PDE 2u, - 3uy = 0 where u = u(x,y). It...
More details pls
Consider the PDE 2u, - 3uy = 0 where u = u(x,y). It can be shown (you don't have to do it) that product solutions to the PDE take the form u(x, y) = X(1)Y(y) = Cetrefty, (1) where C and can be any pair of constants. Here is what you need to work on: First, find infinitely many solutions to the PDE that look different from 1. Make sure to mention which method you used, if...
Assignment 0220 Marks) Solve the following IVBP: PDE : Uxx = (1/25) utt ICs: u (x,0)...
Assignment 0220 Marks) Solve the following IVBP: PDE : Uxx = (1/25) utt ICs: u (x,0) = x2 (nt - x), ut (x,0) = sin(x) BCs: u(0,t) = 0, u(nt,t) = 0 for 0<x<, t> 0. for 0<x<T. for t>0.
Solve the circularly symmetric vibrating membrane PDE given as u_tt = ∇^2*u BC : u(1, θ,...
Solve the circularly symmetric vibrating membrane PDE given
as
u_tt = ∇^2*u
BC : u(1, θ, 0) = 0, 0 < t < ∞
ICs :
u(r, θ, 0) = J_0*(2.4r) − 0.25*J_0*(14.93r), 0 ≤ r ≤ 1
u_t(r, θ, 0) = 0
Solve the circularly symmetric vibrating membrane PDE given as Utt = Dau BC : u(1,0,0) = 0, 0<t< oo ICs : u(r,0,0) = J.(2.4r) – 0.25J(14.93r), 0 <r <1 Ut(r,0,0) = 0
Using Laplace Equation PDE 40(a) Solve for u(x,y): x.x (b) From your solution, evaluate u(3, 1), let's say correct to two decimal places 40(a) Solve for u(x,y): x.x (b) From your solution, e...
Using Laplace Equation PDE
40(a) Solve for u(x,y): x.x (b) From your solution, evaluate u(3, 1), let's say correct to two decimal places
40(a) Solve for u(x,y): x.x (b) From your solution, evaluate u(3, 1), let's say correct to two decimal places
What type of PDE is this? Solve PDE using separation of variables (show all the work and logic) 05 x u(x,0) 4sin(37r),...
What type of PDE is this? Solve PDE using separation of variables (show all the work and logic) 05 x u(x,0) 4sin(37r), u,(x,0) 2sin(57) 0sx 1,t 2 0
Problem # 3 [20 Points] Solve PDE: ut = uxx - u, 0 < x < 1, 0 < t < ∞ BCs: u(0, t)=0 u(1, t)=0 0 < t < ∞ IC: u(x, 0) = sin(πx), 0 ≤ x ≤ 1 directly by separation of variabl...
Problem # 3 [20 Points] Solve PDE: ut = uxx - u, 0 < x < 1, 0 < t < ∞ BCs: u(0, t)=0 u(1, t)=0 0 < t < ∞ IC: u(x, 0) = sin(πx), 0 ≤ x ≤ 1 directly by separation of variables without making any preliminary trans- formation. Does your solution agree with the solution you would obtain if transformation u(x, t)= e(caret)(-t) w(x, t) were made in advance?
3. The Poisson equation is a PDE that occurs in many problems in science and engineering...
3. The Poisson equation is a PDE that occurs in many problems in science and engineering (such as compressible flow) and a simplified form of it is given by Uxx + uyy = u Solve this equation on the domain 0 < x < a and 0 Sy <b subject to the boundary conditions: u(0, y) = 0, u(a, y) = f(y), y(x,0) = 0, u(x,b) = g(x).
Show that the following PDE for u(x,y) is linear in u and homogeneous. ди ду ди...
Show that the following PDE for u(x,y) is linear in u and homogeneous. ди ду ди = 3- дх Ә2 и + sin(у) дх2
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a)
14 points Consider the following equation : PDE: u+ 0 ,0
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
More details pls
Consider the PDE 2u, - 3uy = 0 where u = u(x,y). It can be shown (you don't have to do it) that product solutions to the PDE take the form u(x, y) = X(1)Y(y) = Cetrefty, (1) where C and can be any pair of constants. Here is what you need to work on: First, find infinitely many solutions to the PDE that look different from 1. Make sure to mention which method you used, if...
Assignment 0220 Marks) Solve the following IVBP: PDE : Uxx = (1/25) utt ICs: u (x,0) = x2 (nt - x), ut (x,0) = sin(x) BCs: u(0,t) = 0, u(nt,t) = 0 for 0<x<, t> 0. for 0<x<T. for t>0.
Solve the circularly symmetric vibrating membrane PDE given
as
u_tt = ∇^2*u
BC : u(1, θ, 0) = 0, 0 < t < ∞
ICs :
u(r, θ, 0) = J_0*(2.4r) − 0.25*J_0*(14.93r), 0 ≤ r ≤ 1
u_t(r, θ, 0) = 0
Solve the circularly symmetric vibrating membrane PDE given as Utt = Dau BC : u(1,0,0) = 0, 0<t< oo ICs : u(r,0,0) = J.(2.4r) – 0.25J(14.93r), 0 <r <1 Ut(r,0,0) = 0
Using Laplace Equation PDE
40(a) Solve for u(x,y): x.x (b) From your solution, evaluate u(3, 1), let's say correct to two decimal places
40(a) Solve for u(x,y): x.x (b) From your solution, evaluate u(3, 1), let's say correct to two decimal places
What type of PDE is this? Solve PDE using separation of variables (show all the work and logic) 05 x u(x,0) 4sin(37r), u,(x,0) 2sin(57) 0sx 1,t 2 0
3. The Poisson equation is a PDE that occurs in many problems in science and engineering (such as compressible flow) and a simplified form of it is given by Uxx + uyy = u Solve this equation on the domain 0 < x < a and 0 Sy <b subject to the boundary conditions: u(0, y) = 0, u(a, y) = f(y), y(x,0) = 0, u(x,b) = g(x).
Show that the following PDE for u(x,y) is linear in u and homogeneous. ди ду ди = 3- дх Ә2 и + sin(у) дх2
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