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b) Consider the wave equation azu azu at2 0 < x < 2, t>0, ar2 with...
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0)...
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0) = cos 3x, u(,0) = 21 b) Consider the heat equation диди 0<x<1, t > 0, at ax? with boundary conditions uz (0,t) = 0, uz(1,t) = 0, > 0, and initial conditions u(x,0) = { 0, 2.0, 0<r < 0.5, 0.5 <<1. Use the method of separation of variables to solve the equation.
au (x, at2 a (2,t), 0 < x < 57, to ac2 u(0,t) = 0, u(57,...
au (x, at2 a (2,t), 0 < x < 57, to ac2 u(0,t) = 0, u(57, t) = 0, t>0, u(3,0) = sin(4x), ut(x,0) 4 sin(5x), 0 < x < 57. u(x, t) =
Problem 1. Consider the wave equation ∂ 2 ∂t2 u = ∇2u with c 2 =...
Problem 1. Consider the wave equation ∂ 2 ∂t2 u = ∇2u with c 2 =
1 on a rectangle 0 < x < 2, 0 < y < 1 with u = 0 on the
boundary (fixed boundary condition). Find two independent
eigenfunctions um1,n1 (x, y, t) and um2,n2 (x, y, t) with either m1
6= m2 or n1 6= n2 (or both) which have the same eigenvalue
(frequency).
Problem 1. Consider the wave equation a2 u= at2 v4...
Solve the wave equation on the domain 0 < x < , t > 0 ?...
Solve the wave equation on the domain 0 < x < , t > 0 ? uxx Utt = with the boundary condition u (0, t) = 0 and the initial conditions u (x,0) = x2 u (x,0) = x
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.
Problem #9: Consider the below wave equation with the given conditions. clu olu 16 0<x< 5,...
Problem #9: Consider the below wave equation with the given conditions. clu olu 16 0<x< 5, t > 0, Ox2 u(0, 1) = u(5, 1) = 0, t > 0 700 u(x, 0) = 0, -0 = 7x(5– x) = Σ {1-(-1)"} sin(ntx/5), 0< x < 5. n=1 The solution to the above boundary-value problem is of the form U (x, t) = g(n, t) sin 97 * n=1 Find the function g(n, t). Problem #9: Enter your answer as...
3. Consider the damped wave equation with boundary conditions where 0 < β < 21tc/ L....
3. Consider the damped wave equation with boundary conditions where 0 < β < 21tc/ L. (i) Explain the physical meaning of the term-8ut. Why is β > 0? (ii) Explain the physical meaning of the boundary conditions. ii) Using separation of variables and superposition, solve the initial value problem (iv) What is the long-time behavior of the solution?
Consider the 1D wave equation Ye = a?yrz (1) with boundary conditions y(x 0,t) 0; y(x...
Consider the 1D wave equation Ye = a?yrz (1) with boundary conditions y(x 0,t) 0; y(x = L, t) = 0; y(r, t = 0) - f(x); y(r,t 0) 0. (2) Use separation of variables to convert the PDE into 2 ODEs. Clearly state the boundary conditions for the 2 ODEs
Consider the 1D wave equation Ye = a?yrz (1) with boundary conditions y(x 0,t) 0; y(x = L, t) = 0; y(r, t = 0) - f(x); y(r,t 0)...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx +...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
Find the solution to the heat equation on the infinite domain ∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1. in terms of the...
Find the solution to the heat equation on the infinite
domain
∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1.
in terms of the error function.
Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0) = cos 3x, u(,0) = 21 b) Consider the heat equation диди 0<x<1, t > 0, at ax? with boundary conditions uz (0,t) = 0, uz(1,t) = 0, > 0, and initial conditions u(x,0) = { 0, 2.0, 0<r < 0.5, 0.5 <<1. Use the method of separation of variables to solve the equation.
au (x, at2 a (2,t), 0 < x < 57, to ac2 u(0,t) = 0, u(57, t) = 0, t>0, u(3,0) = sin(4x), ut(x,0) 4 sin(5x), 0 < x < 57. u(x, t) =
Problem 1. Consider the wave equation ∂ 2 ∂t2 u = ∇2u with c 2 =
1 on a rectangle 0 < x < 2, 0 < y < 1 with u = 0 on the
boundary (fixed boundary condition). Find two independent
eigenfunctions um1,n1 (x, y, t) and um2,n2 (x, y, t) with either m1
6= m2 or n1 6= n2 (or both) which have the same eigenvalue
(frequency).
Problem 1. Consider the wave equation a2 u= at2 v4...
Solve the wave equation on the domain 0 < x < , t > 0 ? uxx Utt = with the boundary condition u (0, t) = 0 and the initial conditions u (x,0) = x2 u (x,0) = x
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.
Problem #9: Consider the below wave equation with the given conditions. clu olu 16 0<x< 5, t > 0, Ox2 u(0, 1) = u(5, 1) = 0, t > 0 700 u(x, 0) = 0, -0 = 7x(5– x) = Σ {1-(-1)"} sin(ntx/5), 0< x < 5. n=1 The solution to the above boundary-value problem is of the form U (x, t) = g(n, t) sin 97 * n=1 Find the function g(n, t). Problem #9: Enter your answer as...
3. Consider the damped wave equation with boundary conditions where 0 < β < 21tc/ L. (i) Explain the physical meaning of the term-8ut. Why is β > 0? (ii) Explain the physical meaning of the boundary conditions. ii) Using separation of variables and superposition, solve the initial value problem (iv) What is the long-time behavior of the solution?
Consider the 1D wave equation Ye = a?yrz (1) with boundary conditions y(x 0,t) 0; y(x = L, t) = 0; y(r, t = 0) - f(x); y(r,t 0) 0. (2) Use separation of variables to convert the PDE into 2 ODEs. Clearly state the boundary conditions for the 2 ODEs
Consider the 1D wave equation Ye = a?yrz (1) with boundary conditions y(x 0,t) 0; y(x = L, t) = 0; y(r, t = 0) - f(x); y(r,t 0)...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
Find the solution to the heat equation on the infinite
domain
∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1.
in terms of the error function.
Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
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