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418/518 Study Guide 418/518 Test #2 Study Guide Test #2 _________________________________________________________________________________ Problem #1: Use the method of separation of variables to solve the wave equation utt c2 uxx on the interval 0 x L subject to boundary conditions of the first or second kinds at x 0 and x L and given initial conditions u(x,0) f (x), ut (x,0) g(x) (0 x L) . Read Section 4.4 Review sections 2.3, 2.4.1, 2.4.2 and problem #3 of Test #1. Review problems 2.3.2 (a)-(f), and (2.3.3) (a)-(d) on page 39 of text. Understand and memorize the table on page 69. Review problems 2.4.1 and 2.4.2 on pages 69-70 of text. ___________________________________________________________________ Problem #2: Use the Method of Eigenfunction Expansion to solve the one dimensional heat equation ut kuxx q( x, t ) (0 x L, k constant, k 0) ut uxx q(x,t) (0 x L,t 0) subject to an initial condition u(x,0) f (x) (0 x L) , and homogeneous boundary conditions of the first or second kind. Read Method of Eigenfunction Expansion on pages 122-124. Do problems (3.4.8), (3.4.9), (3.4.10), (3.4.11), (3.4.12) on page 126. Read section 8.3 pp. 354-358, especially the Example on page 357. Do problem 8.3.1 (a) and (f). Also do problems (8.3.2) , (8.3.6) and (8.4.4) on page 364. Read supplement I. ________________________________________________________________________________ Problem #3: (a)-Find the equilibrium solution ue (x) of the following initial-boundary value problem: ut kuxx (0 x L,t 0, k 0) , u(x,0) f (x) (0 x L) , u(0,t) A, u(L,t) B or u(0,t) A, ux (L,t) B or ux (0,t) A, u(L,t) B where A and B are constants. (b) -Using the result of part (a) find the partial differential equation, the initial condition, and the boundary conditions satisfied by v(x,t) u(x,t) ue (x) where u(x,t) is the solution of the initial-boundary value problem stated in part (a). (c)- Find the ordinary differential equations, and initial conditons sautisfied by the functions bn (t) (n 1,2,3,....) where v(x,t) n b (t) (x) n n is the solution of the initial- n1 boundary value problem derived in part (b) found by applying the Method Of Eigenfunction Expansion. Read Method of Eigenfunction Expansion on pages 122-124. Read sections (8.1) and (8.2). Do problem (8.2.1) on page 352. ____________________________________________________________________ CONTINUED ON NEXT PAGE 418/518 Study Guide Test #2 418/518 Study Guide Test #2 Problem #4: Let u(x,t) solution of the following initial-boundary value problem: u(0,t) A(t), u(L,t) B(t) or u(0,t) A(t), ux ( L,t) B(t) or ux (0,t) A(t), u( L,t) B(t) (a)-Construct a function r(x,t) that satisfies the same boundary conditions as u(x,t) . Hint: Set r(x,t) (t) (t)x , and find (t) and (t) . (b) -Using the result of part (a) find the partial differential equation, the initial condition, and the boundary conditions satisfied by v(x,t) u(x,t) r(x,t) where u(x,t) is the solution of the initial-boundary value problem stated in part (a). (c)- Find the ordinary differential equations, and initial conditons satisfied by the functions bn (t) (n 1,2,3,....) where v(x,t) n b (t) (x) n n is the solution of the initial- n1 boundary value problem derived in part (b) found by using the Method of Eigenfunction Expansion. Read Method of Eigenfunction Expansion on pages 122-124. Read sections (8.1) and (8.2). Do problem (8.2.1) on page 352. ________________________________________________________________________ Problem #5: Given the graph of a piecewise smooth function f (x) defined on the interval 0 x L : (a)- Sketch the graph of f e (x) the even extension of f (x) on the interval L x L . (b)-Sketch the graph of f p (x) the periodic extension of f e (x) on the interval 3L x 3L . e n x be the Fourier cosine series of f (x) . To what value does this L n0 Fourier series converge at x 0, x L, x L, x 3L, x 3L, and at each point in the interval 0 x L where f (x) has a jump discontinuity? Given the graph of a piecewise smooth function f (x) defined on the interval 0 x L : (d)- Sketch the graph of f o (x) the odd extension of f (x) on the interval L x L . (c)-Let A cos n (e)-Sketch the graph of f p (x) the periodic extension of f o (x) on the interval 3L x 3L . o n x be the Fourier sine series of f (x) . To what value does this n L n0 Fourier series converge at x 0, x L, x L, x 3L, x 3L, and at each point in the interval 0 x L where f (x) has a jump discontinuity? Read section (3.3) pp. 96-109 and pp.109-111. Also read (3.3.5) pp. 111-113. Do problems (3.3.1)-(3.3.2), and (3.3.4). ______________________________________________________________________________ (f)-Let A sin CONTINUED ON NEXT PAGE 418/518 Study Guide Test #2 Problem #6: (a)-Compute the Fourier series of the function g(x) L x 0 f (x) h(x) 0 x L where g(x) ax b and h(x) cx d . (b)-Sketch the function to which the Fourier series of part (a) converges on the interval 3L x 3L . Be careful to indicate the values to which the series converges at x 3L, x 2L, x L, x 0, x L, x 2L, x 3L. Read section (3.2). Do problems (3.2.1), (3.2.2) on page 95. ___________________________________________________________________________