Question

The purpose of the next few modules is to develop a "spectral" method for solving the boundary value problem - T = f(x), 0 < x < 1 dr2 u(0)-0 a(1-0. , and other, similar, BVPs. As before, we represent this BVP as the operator equation Lu-f, where tp : Cる10,f] → C[0,f] is defined by

This is just a cutout from a set of lecture notes I've received. I understand that L_D is the second derivative operator, but I don't understand any of the other notation. Could you please tell me what L_D[0,l] -> C[0,l] means? Specifically I am not sure why the interval must be defined twice.

Also, I don't understand the notation C_D²[0,l] = { u E C²[0,l] | u(0) = u(l) = 0}

Answer 1

Hope this will work for you .

Here L_D denotes th operation, L_D = -T d^2/d x^2 which operates on 'u' which belongs to C^2 [0, l] and as given -T d^2 u/dx^2 = f (x), we know it (i.e., f (n)) belong to C [0, l] therefore L_D: C_2 [0, l] rightarrow C [0, l] here, u relementof C^2 [0, 1] means that 'u', is twice differentiable i.e. du/dx & d^2/dx^2 both exist and are continuous & f elementof C [0, l] means that 'f' is differentiable once, i.e., df/dx exist & continuous.