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(1 point) Consider the wave equation 1(1)utt = uzz for-oo < z < oo, t>0 with initial conditions ut (z,0-0 and u(z,0) = /(z), where (2) f(z) = 1 for 0 < z < 1, (3) f(z) =...
Find a formula for the solution of the initial value problem for for t>0, -oc < x < oo ut = uzz-u...
Find a formula for the solution of the initial value problem for for t>0, -oc < x < oo ut = uzz-u a(1:0) = g(z) -x < 1 < x where g is continuous and bounded.( Hint: use v(x, t) = et u(z. t).)
Find a formula for the solution of the initial value problem for for t>0, -oc
5] Consider the following initial value problem 9utt = uzz-9r sin(t), (x,0) u(x,0' -oo < x < oo, ...
5] Consider the following initial value problem 9utt = uzz-9r sin(t), (x,0) u(x,0' -oo < x < oo, t > 0, 0, otherwise 0, otherwise. Find the values of u(x,t) at the point x = 4, t = 3. Hint: Let u(x, t)- (x, t) + x sin(t). Write up the equation and the initial condi- tions satisfied by w. Find w(4,3) first
5] Consider the following initial value problem 9utt = uzz-9r sin(t), (x,0) u(x,0' -oo
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t)...
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t) u (x, y,0) uzz(z, y, t) + un, (x, y, t) = 0, 0 < x < 1, 0 < y < 1, 0, u(1,y,t)=0, u(z,0,t)=0, u(z, l,t) = 0 sin(r) sin (2my), ue (r, y,02 sin(2mx) sin(ry) t > 0, = =
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
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 solu...
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 nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0,...
(1 point) Solve the nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0, u(T,t) = 0 u(2,0) sin(52) u(x, t) = Steady State Solution lim oo u(a,t) =
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave speed? c (b) What is the domain of dependence for the point (r, t)...
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave speed? c (b) What is the domain of dependence for the point (r, t) (2,3)? (c) Using only characteristic lines and your knowledge of d'Alembert's solution, find the following values of u(r,t) c) u9,9)- (b) u2,5)
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck.
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...
5. (2+2+3 Points) Consider the linear wave equation for-oo <エ<oo,-oo < t <oo for - oo<<oo...
5. (2+2+3 Points) Consider the linear wave equation for-oo <エ<oo,-oo < t <oo for - oo<<oo for utt-c2uzz = f(x, t) tr(r,0)- (r) _oo < x < oo. a. Sketch the domain of dependence of the point (,t) (4, 1) in the (x, t)-plane in 1. case c = 2, Do the same in the case c b. State the general solution forrnula for this problem! in terms of the data f, φ, ψ. c. Now suppose ψ = 0...
Find a formula for the solution of the initial value problem for for t>0, -oc < x < oo ut = uzz-u a(1:0) = g(z) -x < 1 < x where g is continuous and bounded.( Hint: use v(x, t) = et u(z. t).)
Find a formula for the solution of the initial value problem for for t>0, -oc
5] Consider the following initial value problem 9utt = uzz-9r sin(t), (x,0) u(x,0' -oo < x < oo, t > 0, 0, otherwise 0, otherwise. Find the values of u(x,t) at the point x = 4, t = 3. Hint: Let u(x, t)- (x, t) + x sin(t). Write up the equation and the initial condi- tions satisfied by w. Find w(4,3) first
5] Consider the following initial value problem 9utt = uzz-9r sin(t), (x,0) u(x,0' -oo
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t) u (x, y,0) uzz(z, y, t) + un, (x, y, t) = 0, 0 < x < 1, 0 < y < 1, 0, u(1,y,t)=0, u(z,0,t)=0, u(z, l,t) = 0 sin(r) sin (2my), ue (r, y,02 sin(2mx) sin(ry) t > 0, = =
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
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 nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0, u(T,t) = 0 u(2,0) sin(52) u(x, t) = Steady State Solution lim oo u(a,t) =
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave speed? c (b) What is the domain of dependence for the point (r, t) (2,3)? (c) Using only characteristic lines and your knowledge of d'Alembert's solution, find the following values of u(r,t) c) u9,9)- (b) u2,5)
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck.
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...
5. (2+2+3 Points) Consider the linear wave equation for-oo <エ<oo,-oo < t <oo for - oo<<oo for utt-c2uzz = f(x, t) tr(r,0)- (r) _oo < x < oo. a. Sketch the domain of dependence of the point (,t) (4, 1) in the (x, t)-plane in 1. case c = 2, Do the same in the case c b. State the general solution forrnula for this problem! in terms of the data f, φ, ψ. c. Now suppose ψ = 0...
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