Consider the 1D wave equation d^2u/dt^2 = c^2( d^2u/dx^2) with the following boundary conditions: u(0, t)...

Consider the 1D wave equation d^2u/dt^2 = c^2( d^2u/dx^2) with the following boundary conditions: u(0, t) = ux (L, t) = 0 . (a) Use separation of variables technique to calculate the eigenvalues, eigenfunctions and general solution. (b) Now, assume L = π and c = 1. With initial conditions u(x, 0) = 0 and ut(x, 0) = 1, calculate the solution for u(x, t). (c) With initial conditions u(x, 0) = sin(x/2) and ut(x, 0) = 2 sin(x/2) − 3 sin(5x/2) calculate the solution for u(x, t).

Expert Solution

genius_generous answered 1 year ago

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0...

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0. ∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1) ∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2) u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2. (a) Compute the coefficients a0, a1, a2, ....

Consider the following linear operator: L u = d^2u/dx^2+ 2 du/dx + u. Consider the eigenvalue...

Consider the following linear operator: L u = d^2u/dx^2+ 2 du/dx + u. Consider the eigenvalue problem L u + λu = 0, x ∈ (0, π), u(0) = 0, u(π) = 0. (a) Determine the possible eigenvalues and eigenfunctions. (b) Use an integrating factor to put the eigenvalue problem in Sturm-Liouville form. (c) What is the appropriate inner product for this system?

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0)...

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0) = 0 ut(x, 0) = 0.1x(π − x) u(0,t) = u(π,t) = 0 (a) Solve the problem using the separation of variables. (b) Solve the problem using D’Alembert’s solution. Hint: I would suggest doing an odd expansion of ut(x,0) first; the final solution should be exactly like the one in (a).

Solve the wave equation ∂2u/∂t2 = 4 ∂2u/∂x2 , 0 < x < 2, t >...

Solve the wave equation ∂2u/∂t2 = 4 ∂2u/∂x2 , 0 < x < 2, t > 0 subject to the following boundary and initial conditions. u(0, t) = 0, u(2, t) = 0, u(x, 0) = { x, 0 < x ≤ 1 2 − x, 1 < x < 2 , ut(x, 0) = 0

Solve the wave equation, a2 ∂2u ∂x2 = ∂2u ∂t2 0 < x < L, t...

Solve the wave equation, a2 ∂2u ∂x2 = ∂2u ∂t2 0 < x < L, t > 0 (see (1) in Section 12.4) subject to the given conditions. u(0, t) = 0, u(π, t) = 0, t > 0 u(x, 0) = 0.01 sin(9πx), ∂u ∂t t = 0 = 0, 0 < x < π u(x, t) = + ∞ n = 1

1. Consider the following second-order differential equation. d^2x/dt^2 + 3 dx/dt + 2x − x^2 =...

1. Consider the following second-order differential equation. d^2x/dt^2 + 3 dx/dt + 2x − x^2 = 0 (a) Convert the equation into a first-order system in terms of x and v, where v = dx/dt. (b) Find all of the equilibrium points of the first-order system. (c) Make an accurate sketch of the direction field of the first-order system. (d) Make an accurate sketch of the phase portrait of the first-order system. (e) Briefly describe the behavior of the first-order...

Consider the ODE u" + lambda u=0 with the boundary conditions u'(0)=u'(M)=0, where M is a...

Consider the ODE u" + lambda u=0 with the boundary conditions u'(0)=u'(M)=0, where M is a fixed positive constant. So u=0 is a solution for every lambda, Determine the eigen values of the differential operators: that is a: find all lambda such that the above ODE with boundary conditions has non trivial sol. b. And, what are the non trivial eigenvalues you obtain for each eigenvalue

Use a LaPlace transform to solve d^2x/dt^2+dx/dt+dy/dt=0 d^2y/dt^2+dy/dt-4dy/dt=0 x(0)=1,x'(0)=0 y(0)=-1,y'(0)=5

6. Consider the one dimensional wave equation with boundary conditions and initial conditions: PDE : utt...

6. Consider the one dimensional wave equation with boundary conditions and initial conditions: PDE : utt = c 2 uxx, BC : u(0, t) = u(L, t) = 0, IC : u(x, 0) = f(x), ut(x, 0) = g(x) a) Suppose c = 1, L = 1, f(x) = 180x 2 (1 − x), and g(x) = 0. Using the first 10 terms in the series, plot the solution surface and enough time snapshots to display the dynamics of the...

Consider the equation: ?̇ +2? = ?(?) with initial condition x(0) = 2 (a) If u(t)...

Consider the equation: ?̇ +2? = ?(?) with initial condition x(0) = 2 (a) If u(t) = 0, find the solution ?(?). What is ?(?) as t -> ∞? (b) If u(t) = 4+t, find the solution ?(?). What is ?(?) as t -> ∞? (c) If u(t) = ?3?, find the solution ?(?). What is ?(?) as t -> ∞? (d) If u(t) = δ(t), find the solution ?(?). What is ?(?) as t -> ∞?

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