(Solved): The forced wave equation is, using subscripts to denote differentiation with respect to the subscr ...

The forced wave equation is, using subscripts to denote differentiation with respect to the subscript variable, \[ u_{t t}=c^{2} u_{x x}+F(t, x), \] where \( F(t, x) \) is the forcing term, \( u(t, x) \) is displacement at position \( x \) and time \( t \) and \( c \) is the wave speed. You are to develop a Matlab code to solve this equation numerically, using the method of lines, for various wave speeds \( c \), on \( 0 \leq x \leq 1 \), with boundary conditions \( u(t, 0)=u(t, 1)=0 \), initial conditions \( u(0, x)=u_{t}(0, x)=0 \), and \( F(x, t)=3 \operatorname{sign}\left(x-\frac{1}{2}\right) \sin (\pi t) \). (a) Using a centred space discretisation, give the system of ODEs and associated initial conditions you will solve along with any algebraic equations. (b) Explain/show the matrix system that will be solved.