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5. Solve the initial boundary problem Uz(x,t) = urr(x, t), 0 < x < 2, 0 < t < oo t4(0,t) = ur(2, t) = 0, 0...
3. In class we discussed the heat conduction problem with the boundary conditions a(0, t) 0, t4(1,t)-0, t > 0 and the initial condition u(r,0) f(a) We found the solution to be of the form where (2n-1)n 1,2,3,. TL 20 Now consider the heat conduction problem with the boundary conditions u(0, t) 1,u(T, t)0, t>0 and the initial condition ur,0) 0. Find u(r,t). Hint: First you must find the steady state. 3. In class we discussed the heat conduction problem...
Problem #2115 Points) Solve the following initial-boundary-value prob- lem: u(0,t) = 1 uz(1,1) + ßu(1,1) = 1 BCs: 0 < t < oo 0 IC: u(z,0)= sin(nx)+x, 1 x by transforming it into homogeneous BCs and then solving the transformed problem Problem #2115 Points) Solve the following initial-boundary-value prob- lem: u(0,t) = 1 uz(1,1) + ßu(1,1) = 1 BCs: 0
PDE questions. Please show all steps in detail. 2. Consider the initial-boundary value problem 0
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr) Let u be the solution to the initial boundary value problem for the...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1. Let...
(1 point) Solve the heat problem U4 = Uxx, 0 < x < 1, uz (0,t) = 0, uz(t,t) = 0 u(x,0) = cos? (x) (THINK) u(x, t) =
1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20 exp(-2), 0<x<!5 (a) Solve using separation of variables. You may leave the eigenfunction expansion coef (b) Plot the solution at t-1,3,5 and 30, along with the initial condition and steady state ficients in inner product form. solution, using 15 terms in your truncated expansion. You may use mupad to evaluate the eigenfunction expansion coefficients from part (a) which you left in inner product form...
1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20 exp(-2), 0<x<!5 (a) Solve using separation of variables. You may leave the eigenfunction expansion coef (b) Plot the solution at t-1,3,5 and 30, along with the initial condition and steady state ficients in inner product form. solution, using 15 terms in your truncated expansion. You may use mupad to evaluate the eigenfunction expansion coefficients from part (a) which you left in inner product form...
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.
Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0 Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0