Exam 10: Partial Differential Equations and Fourier Series

The ends of a rod 40 cm in length are connected to reservoirs that maintain the temperature at 12°C at x = 0 and 14°C at x = 40. The initial boundary value problem governing how heat conducts through the rod is as follows: What is the solution u(x, t) of the initial boundary value problem?

Consider the function f (x) = 5 + 3 . Which of the following is the odd periodic extension of f (x)?

Consider the following periodic function with period 4: Which of these is the Fourier representation for f (t)?

Consider the following periodic function with period : F (t) = F = f (t) The Fourier representation has the form Which of these are the coefficients ?

Suppose the following initial boundary value problem governs the shape of a string 140 cm long, where t is measured in minutes: What is the initial displacement of the string at x = 110?

The ends of a rod 45 cm in length are connected to reservoirs that maintain the temperature at 13°C at x = 0 and 18°C at x = 45. The initial boundary value problem governing how heat conducts through the rod is as follows: What is the transient temperature w(x, t) portion of the solution of the initial value boundary value problem?

Assume that $\lambda$ > 0. Consider the boundary value problem + ?y = 0, y (0) = 0, (6) = 0 Which of the following are eigenvectors for this boundary value problem? Select all that apply.

Suppose f (x) is defined by f (x) = 4 on the interval [0, 9]. Consider the function Compute F .

Consider the function f (x) = 7 + 3 . Which of the following is the even periodic extension of f (x)?

Consider the following periodic function with period 2?: F (t) = F (t + 2 $\pi$ ) = f (t) Which of these is the Fourier representation for f (t)?

Find the Fourier series for f (x) = 4, 0 < x <

Suppose the following initial boundary value problem governs the shape of a string 60 cm long, where t is measured in minutes: What is the speed of wave propagation along the string?

Determine the function u(x, y) satisfying Laplace's equation + = 0 in the rectangle , and satisfying the boundary conditionsu

Which of the following statements are true? Select all that apply.

Consider the following periodic function with period 6: f (t) = f (t + 6) = f (t) To what value does the Fourier series for f (t) converge for t = 2?

Consider the following function: F (x) = What is the Fourier cosine series for f (x)?

Consider the following periodic function with period 6: H(t) = H(t + 12) = H(t) The Fourier series representation for H(t) has the form ‪ where 12 is the period of H(t). What is the coefficient ? Express your answer as a simplified fraction.

What is the steady state solution for the heat conduction equation Equipped with the following boundary conditions: U(0, t) - (0, t) = 5 U(40, t) + (40, t) = 4

Consider the following periodic function with period : f (t) = f = f (t) The Fourier representation has the form Compute f , where m is an odd integer.

Consider the following periodic function with period 8: Which of these is the Fourier representation for f (x)?