3150 Review Problems for Final Exam. (1) Find the Fourier series of the 2π-periodic function whose values are given on [0, 2π) by cos(x) 0 x π f(x) =

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1 350 Review Problems for Final Eam () Find the Fourier series of the 2π-periodic function whose values are given on [0, 2π) by cos() 0 π f() = 0 π < < 2π (2) Let F and G be arbitrary differentiable functions of one variable. Then u(, t) = F ( + ct) + G( ct) is a solution to the wave equation 2 u t = 2 u 2 c2, provided that F and G are sufficiently smooth. Use this result to solve the wave equation 2 with initial data as follows. (a) u(, 0) = e 2, u(, 0) = 0, < <. t (b) u(, 0) = 0, u(, 0) =, < <. t ( 2 + ) 4 (c) u(, 0) = e 2, u(, 0) =, < <. t ( 2 + ) 4 (Use the superposition principle along with the results of the previous two parts.) (3) Find the sine series epansion of f() = on [0, 2π). (4) Find the requested Fourier series of the given function: (a) The Fourier series of the 6π-periodic function f() = 2 + cos(3) + sin(9) (b) The Fourier sine series of the 3-periodic function f() = 4 sin( 2π ) + 7 sin(8π) 3 (c) The Fourier cosine series of the 2π-periodic function f() = 5 7 cos() + 3 cos(3) 7 cos(4) (d) The comple Fourier series of the 2π-periodic function f() = 5 7e 3i + 9e 5i (e) The Fourier double sine series of the (2, 5)-periodic function f(, y) = 8 sin() sin( 4π y) + 9 sin(2) sin( 6π y) 5 5 (5) Find a period of the given function. (a) cos(6) (b) sin(5π) (c) sin( 4 ) 5 (d) cos( 5π ) p (6) Let f be the 5-periodic function defined on [0, 5) by 2 0 < 2 f() = 2 2 < < 5 To what does the Fourier series of f converge when = 2? (7) (a) Solve the equation 7u + u y = 0 by an appropriate change of variables. (b) Find the solution u that is equal to /(5 + 2 ) along the -ais. (8) Consider the equation 6u t + 0u = 0 for u = u(, t). (a) Verify that u(, t) = f(3 5t) is a solution, where f is any differentiable function of a single variable. (b) Find a solution of the equation which equals sin() on the -ais. (9) Find the Fourier series of the 2π-periodic function f() = sin( ) cos( 3 ). Hint: use trig identities. 2 2 (0) Use the method of characteristic curves to find solutions u = u(, y) to the equation 3y 2 u 2u y = 0. Check your answer by plugging back into the equation. () (a) Solve the equation 3u + 5u t = 9 by an appropriate change of variables. (b) Find the solution that equals e t2 along the t-ais (2) Let g() = 3 on the interval 0 2π. (a) Sketch the graph of the odd 4π-periodic etension of g. (b) Find the Fourier sine series of g(). (c) By evaluating at = π, deduce a series identity. (3) Let g() = 3 on the interval 0 2π. (a) Sketch the graph of the even 4π-periodic etension of g. (b) Find the Fourier cosine series of g(). (4) Verify that u = e 2t cos(3) is a solution of the heat equation for a suitable value of c. (5) (a) Find the Fourier series of the 2-periodic function whose values on [, ] are given by f() = 2. (b) Derive the value of series

2 2 (6) Determine if the given function is piecewise continuous, piecewise smooth, or neither. Here, all functions are defined on [, ] and have f(0) = 0. (a) f() = cos(/ 2 ) (b) f() = cos(/) (c) f() = sin()/ (d) f() = 2 cos(/) (7) Solve the equation 5u t + 4u = 3u by an appropriate change of variables. (8) Show that if f() has period T, then f(3) has period T/3. (9) Let u and u 2 be two solutions of 3u + 2 u y = 0. Show that u = au + bu 2 is also a solution for any constants a and b. (20) Solve the boundary value wave problem where L = π, f() = sin() cos(), g() = 0, c = 5. (2) Find and plot the nodal lines for the function u(, y, t) = sin( 4π ) sin( π y) sin( 3t), 0 < < 7, 0 < y < (22) (a) Solve the boundary value heat problem u(, 0) = f(), 0 < < L where L = π, c =, f() = 6 sin() 4 sin(4). (b) Solve the same problem, but with non-homogeneous boundary conditions and temperature distribution as follows: u(0, t) = 0, u(π, t) = 00, t > 0 f() = 20 sin() + 30 sin(4)) + 00 π (c) Solve the same problem, but with insulated endpoints and temperature distribution as follows: u (0, t) = 0, u (π, t) = 0, t > 0 f() = 20 cos() + 2 cos(4) (23) Verify that the function u(, y, t) = sin(2) sin(3y)e 65t satisfies the two dimensional heat equation u t = 5 2 u. (24) Use d Alembert s solution of the vibrating string problem to solve where L = 2π, f() = 0, g() = sin() 9 sin(3), c =. Completely describe f and G (an antiderivative of g ). Also, determine u(π/4, π/8). (25) Find the comple Fourier series of the 2π-periodic function given by f() = e 3 on the interval π < π. (26) Solve the boundary value problem u tt = c 2 (u + u yy) u(, 0, t) = 0, u(, b, t) = 0, t > 0 u(0, y, t) = 0, u(a, y, t) = 0, t > 0 u(, y, 0) = f(, y), u t(, y, 0) = g(, y), (, y) R if a =, b = 4, c = 2, f(, y) = ( 2 ) sin( 5π y), and g(, y) = 5 sin(3π) sin(πy). 4 2 (27) Solve the boundary value heat problem u(, 0) = f(), 0 < < L where L = π, c =, f() = 20 sin() + 30 sin(4).

3 350 Review Problems for Final Eam 3 (28) Use Fourier series methods to solve the boundary value problem where L = 5, f() = 0, g() = 3 sin 2π 2 sin 6π, c = 2. Also, determine the deflection at the mid point of the string 5 5 at time t = 7.37 seconds. (29) Determine whether the given partial differential equation and boundary or initial conditions are linear or nonlinear, and, if linear, whether they are homogeneous or nonhomogeneous. (a) u + y 2 u y = 2, u(0, y) = 0, u (0, y) = 0. (b) u + u 2 t = 5, u(0, t) = u(, t). (c) e u + e y u y = e y u y, u(, 0) =. + 2 (d) u 5u tt = 4u cos(t), u(, 0) = u(, 2). (e) 2y(u + 3u y) = 4u t, u(, y, 0) = + y 2. (f) (u + 3u y)u t = 0, u (0, y, t) = 0, u (, y, t) = 0. (g) u ttt = 6(u + u yyy), u(0, y, t) = 0, u(2, y, t) = y, u(, 0, t) = 0, u(, 5, t) = + 2. (30) Use d Alembert s solution of the vibrating string problem to solve where L =, f() = sin(2π), g() = 0, c = 2. Completely describe f and G (an antiderivative of g ). Also, determine u(/4, 5). (3) (a) Solve the boundary value heat problem u(, 0) = f(), 0 < < L where L =, c =, f() = 0. π (b) Solve the same problem, but with non-homogeneous boundary conditions and temperature distribution as follows: u(0, t) = 0, u(π, t) = 5, t > 0 f() = 0 (c) Solve the same problem, but with insulated endpoints and temperature distribution as follows: u (0, t) = 0, u (π, t) = 0, t > 0 f() = 0 (32) Solve the boundary value problem u tt = u + u u(0, t) = 0, u(π, t) = 0, t > 0 u(, 0) = sin(3), u t(, 0) = 0, 0 < < π (Use the method of separation of variables.) (33) Verify that the given function satisfies the two dimensional Laplace equation. (a) u = 3 3y 2 (b) u = e 3y sin(3) (34) Find and plot the nodal lines for the function u(, y, t) = sin(3π) sin(5πy) cos( 7π t), 0 < <, 0 < y <. 2 (35) Find the comple Fourier series of the 2π-periodic function given by f() = cos(5) on the interval π < π. (36) Solve the boundary value wave problem u tt = c 2 (u + u yy) u(, 0, t) = 0, u(, b, t) = 0, t > 0 u(0, y, t) = 0, u(a, y, t) = 0, t > 0 u(, y, 0) = f(, y), u t(, y, 0) = g(, y), (, y) R if a =, b =, c = 2, f(, y) = 8 sin(π) sin(2πy), and g(, y) = 0 sin(3π) sin(πy).

4 4 (37) Solve the boundary value heat problem u(, 0, t) = 0, u(, b, t) = 0, t > 0 u(0, y, t) = 0, u(a, y, t) = 0, t > 0 u(, y, 0) = f(, y), (, y) R if a =, b =, c = 2, f(, y) = 8 sin(π) sin(2πy). (38) Let b() be a function with graph y (a) Describe in qualitative terms what would happen if an elastic string of length with c = had initial displacement f() = b() and initial velocity g() = 0. (b) For what initial displacement and velocity would the solution model a single blip propagating back and forth across the string. (c) If two blips on an elastic string moving in opposite directions collided, what would happen? < 2 (39) Let h() = 0 otherwise. (a) Find the Fourier integral representation of h. (b) Epress h() as a comple Fourier integral. (c) Suppose an infinite wire with thermal diffusivity c 2 = has initial temperature h(). Obtain an epression for the temperature u(, t) in integral form. (d) Find the solution of the wave equation on an infinite string, < <, t > 0 u(, 0) = f() u t(, 0) = g() where c = 5, f() = 0, g() = h(). (40) Solve the vibrating membrane problem u tt = c 2 ( u rr + r ur ) where a = 3, c =, f(r) =.5, and g(r) = 0. (4) Find solutions u(, y) to 2yu u y = 0 by separating variables. 2 < < 2 (42) Let f() = 0 otherwise (a) Sketch the graph of f. (b) Find the fourier integral representation of f. ( 2 sin(2w) (c) Deduce the identity π = 4 cos(2w) ) sin(2w) dw. 0 w 2 w ( 2 sin(2w) (d) What is the value of 4 cos(2w) ) sin(37w) dw? 0 w 2 w 2e 2 0 (43) Let f() = 0 < 0. (a) Find the Fourier transform of f(). (b) Epress f() as a (comple) Fourier integral. (c) By evaluating the Fourier integral of f() at = 0 and at = 37, derive two integral identities.

5 350 Review Problems for Final Eam 5 e 2 0 (d) Let g() = and h() = 2π 2 e 2 0. Show that f g() = h(). 0 < 0 0 < 0 (e) Use the above result to find the Fourier transform of h(), given that F (g()) (ω) = 2π (2 + iω). (44) Consider a vibrating string with length L = 6, ends fied, and c = 7, corresponding to the wave equation u tt = 49u. (a) Find u(, t) if the initial velocity is zero and the initial displacement is f() = 3 3 = 2 π 2 [sin( π6 ) 9 sin( 3π6 ) + 25 sin( 5π6 ) ]. (b) Find u(, t) if the initial displacement is zero and the initial velocity is g() = 3 3 = 2 π 2 [sin( π6 ) 9 sin( 3π6 ) + 25 sin( 5π6 ) ]. (45) Using Fourier methods, find the solution of the wave equation on an infinite string, < <, t > 0 u(, 0) = f() u t(, 0) = g() where c = 2, f() = e 2, g() = 0. (46) Find the temperature u(, t) in a wire (length 8 cm, thermal diffusivity c 2 = 4 cm2 /sec) whose ends are perfectly insulated, assuming that u(, 0) = f() = 5 cos( 2π 3π ) 7 cos( 8 8 ). (47) Solve the vibrating membrane problem u tt = c 2 ( u rr + r ur ) where a = 2, c = 0, f(r) = 4 r 2, and g(r) = 0. (48) (a) Solve the equation 5u + 4u y = 0 by an appropriate change of variables. (b) Find the solution u that is equal to e along the -ais. (49) Using Fourier methods, find the solution of the heat equation on an infinite rod where c = 5, f() = 2 +. (50) Solve the vibrating membrane problem, < <, t > 0 u(, 0) = f() u tt = c 2 ( u rr + r ur ) where a =, c =, f(r) =, and g(r) = 5. (5) Two circular drumheads have the same tension and density, but the second has a radius one third as large as the first. How much higher is the frequency of the fundamental mode of the second drumhead? Eplain carefully. (52) Let g() = on the interval 0 2π. (a) Find the Fourier sine series of g(). (b) Find the double Fourier sine series of f(, y) = 9y on the rectangle 0 2π, 0 y 2π. (c) Find the displacement u(, y, t) of a square vibrating membrane of sides a = b = 2π and c = 2 if the initial displacement is 9y and the initial velocity is 0. (53) Compute the Laplacian in an appropriate coordinate system and decide if the function is harmonic, i.e. satisfies Laplace s equation 2 u = 0: (54) (a) Let f() = e 2 0 u(, y) = 2 + y 2.. Find the Fourier transform of f. 0 < 0

6 6 (b) Show that the convolution h() = f f() is h() = 2π e < 0. (c) Find the Fourier transform of h in three was: a) directly, b) using the behavior of Fourier transform under convolution, and c) using the rule for F (f()). (55) Find the deflection u(r, t) of the circular membrane of radius a = 2 if c =, the initial velocity is zero, and the initial deflection is a function f(r) with Fourier-Bessel series f(r) = J 0( α 2 r) + α2 J0( 4 2 r) + α3 J0( 9 2 r) +. (56) Compute the Laplacian in an appropriate coordinate system and decide if the function satisfies Laplace s equation 2 u = 0: u(, y, z) = y + z 2 + y. 2 (57) Verify that u = e 2t cos(3) is a solution of the heat equation for a suitable value of c. (58) Solve the vibrating membrane problem ( u tt = c 2 u rr + ) r ur where a =, c = 2, f(r) = ( r 2 ) 2, and g(r) = 0. (59) Use the method of separation of variables to find solutions u = u(, t) to the equation 3t 2 u 4u t = 0 on 0 2π satisfying u(0, t) = u(π, t) = 0 and u(, 0) = 3 sin(2) + sin(5). (60) Find the Fourier integral representation of the function e > 2 f() = 0 otherwise