  We study a new method - called Schrodingerisation introduced in [Jin, Liu,
Yu, arXiv: 2212.13969] - for solving general linear partial differential
equations with quantum simulation. This method converts linear partial
differential equations into a `Schrodingerised' or Hamiltonian system, using a
new and simple transformation called the warped phase transformation. Here we
provide more in-depth technical discussions and expand on this approach in a
more detailed and pedagogical way. We apply this to more examples of partial
differential equations, including heat, convection, Fokker-Planck, linear
Boltzmann and Black-Scholes equations. This approach can also be extended to
Schrodingerise general linear partial differential equations, including the
Vlasov-Fokker-Planck equation and the Liouville representation equation for
nonlinear ordinary differential equations.
