QUESTION 5 (5.1) Compute the Fourer cosine series for the function (5) 0 те (-т, т/2) f(+) 3D {1 z€ |-п/2, т/2] 0 те (т...
subject is differential equations please hurry up (5) (12 poins)(a) Compute the sine series for the function f such that f(z) r(4-z) on the interval l0,4 (b) Compute the solution u(z,t) for the partial differential equation with z in the interval (0,4) and t > 0: 3tit tigr with u(0,t) = u(4,t)-0 for t > 0 u(z, 0)(4-a) for 0 <4 (boundary conditions) (initial conditions) (5) (12 poins)(a) Compute the sine series for the function f such that f(z) r(4-z)...
(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...
(4) (a) Compute the Fourier series for the function f(s)-- interval [-T, on the (b) Compute the solution u(t,a) for the partial differential equation on the interval [o, ) luWith u(t, 0) u(t,1)-0 for t>0 (boundary conditions) u(o,z)-3 sin(2x)-5 sin(5z) + sin(6z), for O < < 1 (initial conditions) (20 points) (4) (a) Compute the Fourier series for the function f(s)-- interval [-T, on the (b) Compute the solution u(t,a) for the partial differential equation on the interval [o, )...
4. (a) Expand the given function in an appropriate cosine or sine series. (x) , , -1<x<0 05x< (6 marks) (b) Find the product solutions for the given partial differential equation by using separation of variables. U, +3u, = 0 (6 marks)
subject is differential equations please hurry up (5) (12 poins)(a) Compute the sine series for the function f such that f(z) r(4-z) on the interval l0,4 (b) Compute the solution u(z,t) for the partial differential equation with z in the interval (0,4) and t > 0: 3tit tigr with u(0,t) = u(4,t)-0 for t > 0 u(z, 0)(4-a) for 0 <4 (boundary conditions) (initial conditions)
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...
Question 4. Calculate the Fourier sine series and the Fourier cosine series of the function f(x) = sin(x) on the interval [0, 1]. Hint: For the cosine series, it is easiest to use the complex exponential version of Fourier series. Question 5. Solve the following boundary value problem: Ut – 3Uzx = 0, u(0,t) = u(2,t) = 0, u(x,0) = –2? + 22 Question 6. Solve the following boundary value problem: Ut – Uxx = 0, Uz(-7,t) = uz (77,t)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
which part b uses the answer from part a. 4. (35 pts) Let f(x) = x(1-x) for 0 < x < 1. (a) (15 pts) Compute the Fourier cosine series FCS f(x). (b) (5 pta) Find the formal solution of the problem BC u,(O, t)-u(1,t)-0, (c) (5 pts) Show that there can be no solution of problem (A) which is Ca for 0 S S 1 and (d) (10 pts) Show that there is a Co solution of the DE...
where M=7 322-M2 4) Find the inverse - transform of F(z) = (2-1)(2-2M)' (15 marks) 0 t<-M/2 M <t< - 5) Show that the Fourier transform of function f(t) sin 7 s (10 marks) au 6) Show that u = ln(x2 + xy + y2) satisfies the partial differential equation x x ди +y 2. (7 marks) au 7) Solve the partial differential equation = e-cos(x) where at du x = 0, at =tet ax at and t = 0,...