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3. Consider the damped wave equation with boundary conditions where 0 < β < 21tc/ L....
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)...
b) Consider the wave equation azu azu at2 0 < x < 2, t>0, ar2 with...
b) Consider the wave equation azu azu at2 0 < x < 2, t>0, ar2 with boundary conditions u(0,t) = 0, u(2, t) = 0, t> 0, and initial conditions u(x,0) = x(2 – x), ut(x,0) = = 0, 0 < x < 2. Use the method of separation of variables to determine the general solution of this equation. (15 marks)
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann b...
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
=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...
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...
2. 24 pts. Boundary conditions for EM wave reflection and transmission. The boundary conditions i...
2. 24 pts. Boundary conditions for EM wave reflection and transmission. The boundary conditions incident on a boundary can be written as: conditions on electromagnetic waves (ii) E- E0 (ii) B-B: =0 (a) 10 pts. Corresponding Maxwell Equations. For each equation (i, ii, iii, iv) above write the corresponding Maxwell equation in BOTH operator form AND in integral form from which the boundary condition can be derived. What terms in Maxwell's equations are assumed to be zero in the above?...
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0...
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0 < x < 4, t > 0, u(4,t) = 0, 1 > 0 op u(x,0) = 0, ди at = 6x(4- x) = 384 ${1 - (-1)"} sin(npox/4), 0< x < 4. n=1 The solution to the above boundary-value problem is of the form u(x,t) = 8(n, t) sin "* n=1 Find the function g(n,1).
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
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0)...
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0) = sin(x), 0<x<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
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)...
b) Consider the wave equation azu azu at2 0 < x < 2, t>0, ar2 with boundary conditions u(0,t) = 0, u(2, t) = 0, t> 0, and initial conditions u(x,0) = x(2 – x), ut(x,0) = = 0, 0 < x < 2. Use the method of separation of variables to determine the general solution of this equation. (15 marks)
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
=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...
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...
2. 24 pts. Boundary conditions for EM wave reflection and transmission. The boundary conditions incident on a boundary can be written as: conditions on electromagnetic waves (ii) E- E0 (ii) B-B: =0 (a) 10 pts. Corresponding Maxwell Equations. For each equation (i, ii, iii, iv) above write the corresponding Maxwell equation in BOTH operator form AND in integral form from which the boundary condition can be derived. What terms in Maxwell's equations are assumed to be zero in the above?...
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0 < x < 4, t > 0, u(4,t) = 0, 1 > 0 op u(x,0) = 0, ди at = 6x(4- x) = 384 ${1 - (-1)"} sin(npox/4), 0< x < 4. n=1 The solution to the above boundary-value problem is of the form u(x,t) = 8(n, t) sin "* n=1 Find the function g(n,1).
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
9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0) = sin(x), 0<x<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
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