The one-dimensional heat equation is solved on a grid consisting of \(N\) spatial grid points and \(M\) time layers. The boundary conditions are of Dirichlet type. Calculate, in terms of \(N\) and \(M\), the total number of the arithmetic operations required to complete the solution. Do this for:

a) Simple explicit scheme (7.28).

b) Simple implicit scheme (7.32). Here, assume that the system of linear algebraic equations is solved at each time step using the Thomas algorithm.

c) Runge-Kutta method of the fourth order (7.52). Here, assume that the spatial derivative is discretized by the central difference of the second order.