(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(...
Prove that the following two-point boundary-value problem has a UNIQUE solution. Thank you Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00<s< 00. Assume that on this domain THEOREM4 11. Prove that the following two-point boundary-value problem has a unique solution: "(t3 5)x +sin t Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
Let u be the solution to the initial boundary value problem for the Heat Equation, 0,uột, 2) = 40ều(t, z), t + (0, 0, z + (0,5); with initial condition u(0, x) = f(x), where f(0) = 0 and f'(5) = 0, and with boundary conditions u(t,0) = 0, 0,ult, 5) = 0. Using separation of variables, the solution of this problem is u(t, 2) = Čem () w.(2), n= 1 with the normalization conditions 0,() = 1, W. (2–...
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R. Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R.
Let u be the solution to the initial boundary value problem for the Heat Equation, tE (0, o0), т€ (0, 3)%; дди(t, г) — 4 0?и(t, a), with initial condition E0, , u(0, x) f(x) 3 and with boundary conditions д,u(t, 3) — 0. и(t,0) — 0, Find the solution u using the expansion и(t, 2) 3D У с, чп (t) w,(m), n-1 with the normalization conditions Vn (0) 1, 1. Wn _ (2n 1) a. (3/10) Find the functions...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
7. Consider these two boundary-value problems: r"-f (t, x, x') 1. Show that if x is a solution of boundary-value problem ii, then the function yt) x((t -a)/h) solves boundary-value problem i, where h b-a 7. Consider these two boundary-value problems: r"-f (t, x, x') 1. Show that if x is a solution of boundary-value problem ii, then the function yt) x((t -a)/h) solves boundary-value problem i, where h b-a
Problem 4 (4 points each). Let S = R {0}. (a) Let f: S R be f(x) = cos(1/x). Show that lim-0 f(x) does not exist. (b) For any fixed a > 0, let f: S+R be f(x) = rºcos(1/x). Show that lim -- f(x) = 0. (c) Find a value be R for which the function f: R+R given by f(x) = { 2" cos(1/x) if r +0, if x = 0, is continuous at 0. Is this b...
Let u be the solution to the initial boundary value problem for the Heat Equation дли(t, 2) — 4 әғи(t, 2), te (0, o0), те (0,1); with initial condition , u(0, a)f() and with boundary conditions 0. u(t, 0)0 u(t, 1) Find the solution u using the expansion и(t, г) "(2)"т (?)"а " n 1 with the normalization conditions 1 Vn (0) 1, wn 2n a. (3/10) Find the functions wn, with index n> 1. Wn b. (3/10) Find the...
Let u be the solution to the initial boundary value problem for the Heat Equation 202u(t, ) te (0, o0) (0,3); дли(t, 2) хе _ with boundary conditions ut, 0) 0 u(t, 3) 0 and with initial condition 3 9 u(0, ar) f(x){ 5, | 4' 4 0, Те The solution u of the problem above, with the conventions given in class, has the form ()n "(2)"п (г)"а "," n-1 with the normalization conditions 3 Wn 2n vn (0) 1,...