(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii. Find the corresponding eigenvalue relation and eigenfunctions. Note that you do not have to normalise the eigenfunctions. (b) Solve the heat equation (2) 0<х<1 t>0 Ut u(0, t) u(1,t) u(x, 0) sin(2тx) + 1 (a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii....
Q2-2 y (0) = 0, y' (t) = 0 6) u" - x?u'+ dy=0 | u (0) = u (1) = 0 Q2: Find eigenvalues and eigenfunctions of the Sturm-Liouville system 1) "") + y = 0, y'(0) = 0 , y' (1) = 0 dx ² 2) u" + au = 0 , u(a) = u (b ) = 0 3) " +(-4 + 1 ) = 0 , u'(0) = 0 , u(1) = 0 4) y" +...
1. Use a Fourier series to solve the IVP IMG 9174.JPG 3,024x 4,032 IC u(x, 0) = |x-1/2], 0 < x < 1 BC 14(0,1) = 0 = ur(1,1), t E R+ If we consider the resulting Fourier series as a function on the whole x- axis what kind of function is it? IMG 9179 IPG 1. Use a Fourier series to solve the IVP IMG 9174.JPG 3,024x 4,032 IC u(x, 0) = |x-1/2], 0
1. Solve fully the heat equation problem: Ut = 5ucx u(0,t) = u(1, t) = 0 u(x,0) = x – 23 (Provide all the details of separation of variables as well as the needed Fourier expansions.)
1 & 5 Solve the following heat equations using Fourier series ux Ut, 0 <x <1,t>0, u (0,t) = 0 = u(1,t), u(x,0) = x/2 1/ 2/ Ux=Ut, 0<x< m ,t>0 ,u(0,t) = 0 = u( 1, t), u(x, O) = sinx- sin3x 3/ usxut, O <x < 1 ,t>0, u(0,t) = 0 = u,(1, t), u(x,0) = 1 -x2 Ux=Ut,O<x <m ,t>0, u(0, t) = 0 = u,( rt , t) , u(x, 0) = (sinxcosx)2 4/ 5/Solve the...
(4 points) Use the Fourier integral transformations to solve the heat equation д2u du 0 < x u(x, 0) = 0, 100, a(0,t) (Please use "alpha" for the variable α.) n(x, t) = Jo (4 points) Use the Fourier integral transformations to solve the heat equation д2u du 0
Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12. Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12.
1 point) Solve the nonhomogeneous heat problem ut=uxx+4sin(2x), 0<x<π,ut=uxx+4sin(2x), 0<x<π, u(0,t)=0, u(π,t)=0u(0,t)=0, u(π,t)=0 u(x,0)=5sin(5x)u(x,0)=5sin(5x) u(x,t)=u(x,t)= Steady State Solution limt→∞u(x,t)=limt→∞u(x,t)= Please show all work. (1 point) Solve the nonhomogeneous heat problem Ut = Uxx + 4 sin(2x), 0< x < , u(0,1) = 0, tu(T, t) = 0 u(x,0) = 5 sin(52) u(a,t) Steady State Solution limt u(x, t) = Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts...
1. Solve fully the heat equation problem: ut = 5u: u(0,t) = u(1,t) = 0 (3,0) = 2 - 3 (Provide all the details of separation of variables as well as the needed Fourier expansions.)
FInd u(x,t) and lim u(x,t) Solve the heat problem Ut = Uzx + 5 sin(4x) - sin(2x), 0 < x <7, u(0,1) = 0, u(,t) = 0 u(x,0) = 0