Is this the correct way to implement the following linear form in fenics

Question

Hello Everyone,

I have the following equation:

Is this the correct way of implementing it in fenics?

    ui = Expression('-40.*sin(pi*x[0])*sin(pi*x[1])*exp(-1000.*(pow(x[0]-0.5,2)+pow(x[1]-0.5,2)-c*t))'), c=c, t=0.0)
    dt = 0.005
    u0 = interpolate(ui, V)
    t = dt
    ui.t = t
    u1 = interpolate(ui, V)

    a =   1/dt**2*inner(u, v)*dx \
        + c**2*beta*inner(nabla_grad(v), nabla_grad(u))*dx

    L =   2.0/dt**2*inner(v,u1)*dx \
        - 1/dt**2*inner(v, u0)*dx \
        - (1.0 - 2.0*beta)*c**2*inner(nabla_grad(v), nabla_grad(u1))*dx \
        - beta*c**2*inner(nabla_grad(v), nabla_grad(u0))*dx

    S  = SystemAssembler(a, L, bc)        # Assemble linear system
    A  = PETScMatrix()
    b  = PETScVector()
    S.assemble(A)

    u = Function(V)        # The unknown u at the new time level

    solver = LUSolver()
    solver.set_operator(A)

    while t <= T:
      S.assemble(b)
      solver.solve(u.vector(), b)
      u0.assign(u1)
      u1.assign(u)
      file << (u1,t)
      t += dt