(20 pts) The one dimensional wave equation is a second order PDE:\

(delu^(2))/(delt^(2))=a(del^(2)u)/(delx^(2)),

\ where

a>0

.\ (a)

(7pts)

Write the above equation as a system of first order PDEs in the form of\

(delU)/(delt)+A(delU)/(delx)=0

. Hint: introduce new variables

f=(delu)/(delt)

and

g=a(delu)/(delx)

.\ (b) (3 pts) What is the matrix

A

.\ (c) (5 pts) Find the eigenvalues of matrix

A

.\ (d) (5 pts) What is the type of the wave equation: hyperbolic, parabolic, or elliptic?

(20 pts) The one dimensional wave equation is a second order PDE: t2u2=ax22u, where a>0. (a) ( 7 pts) Write the above equation as a system of first order PDEs in the form of tU+AxU=0. Hint: introduce new variables f=tu and g=axu. (b) (3 pts) What is the matrix A. (c) (5 pts) Find the eigenvalues of matrix A. (d) (5 pts) What is the type of the wave equation: hyperbolic, parabolic, or elliptic