First Home work

1. Solve analytically the following Initial Value Problems:

   1. $$u_t(x,t) = u_{xx}(x,t), \ u(x,t=0)=\exp(-10(x-\frac\pi 2)^2), u_x(0,t)=u_x(\pi,t)=0.$$

   2. $$u_t(x,t) = u_{xx}(x,t), \ u(x,t=0)=\exp(-10(x-\frac\pi 2)^2), u_x(0,t)=u(\pi,t)=0.$$

   3. $$u_{tt}(x,t) = u_{xx}(x,t), \ u(x,t=0)=\exp(-10(x-\frac\pi 2)^2), u_t(x,t=0)=0, u_x(0,t)=u_x(\pi,t)=0.$$

   4. $$u_{tt}(x,t) = u_{xx}(x,t), \ u(x,t=0)=\exp(-10(x-\frac\pi 2)^2), u_t(x,t=0)=0, u_x(0,t)=u(\pi,t)=0.$$

2. Produce a movie of all four solutions as a function of time. This can be done, for example, in Mathematica by using the following template:

         Animate[Plot[Sin[x - t], {x, 0, 2 \pi}], {t, 0, 100}, 
         AnimationRate -> 0.4]
   

   Create one movie and upload it to LMS along with all your solutions.