Solve the non-homogeneous wave equation utt = uxx u(0, t) =...

Solve the non-homogeneous wave equation $u_{tt} = u_{xx}$ $u(0, t) = te^{-t}$ $u(x, 0) = 0$ $u(10, t) = 0$ $u_t(x, 0) = 0$ (a) Separate the problem into boundary and interior parts. (b) Find all the eigenfunctions, and transform the interior PDE. (c) Solve the resulting ODEs and invert to write the solution as a series. (d) Plot a 5-term series solution for times $t = \{1, 2, 3, 4, 5\}$.

Transcript

00:01 So, here we have to solve the given initial value problem.

00:05 So, that we have been given del square u by del of t square plus 2 times of del u by del t plus u is equals to the del square u by del x square where x is varying from minus infinity to infinity while t greater than to 0.

00:34 And initial values have been given u of x comma 0 is equals to 1 over 1 plus x square and u of t x comma 0 is equals to 1.

00:46 So, let us start with this.

00:47 So, basically this we have given the term.

00:50 So, in a part we have to find it out it for the v of x comma t which is given equals to e to the power t u of x comma t.

01:02 So, we have to find the ivp satisfying the v of x comma t.

01:07 So, here let us suppose u of x comma t that will be equals to the v of x comma t and this e to the power will go to that side become e to the power minus t.

01:21 So, we can in simple manner u of x is equals to v of x e to the power minus t can be written and hence u of x x will be equals to v of x e to the power minus of t, but basically here we have to add the one more x.

01:39 Now, in terms of t u of t would be equals to v of t e to the power minus of t minus of e to the power minus of t v then u t t would be equals to e to the power minus t v of t t minus of v t plus v of t minus of v into the minus of e to the power minus t.

02:02 We are just differentiating this functions and getting these values.

02:07 So, u of t t would be equals to u of minus t v of t t minus of 2 v of t plus v.

02:14 So, put these values in differential equation...

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