Solution of parabolic equations by the least square method

In this paper, the parabolic approximation of wave equation will be solved by the method of least squares. At first, the radio wave propagation in homogeneous media will be considered. The electromagnetic field will be expanded by proper expansion functions, which satisfy the parabolic equation in homogeneous media. The expansion coefficients will be derived by the least square method for initial and boundary conditions. The least square functionals satisfy the initial and boundary conditions. Similar to the split step method, the field in the in-homogeneous media with known profile of refractive index can be obtained by proper phase shifting of the field in homogeneous media. This method is more reliable than the split step method and can be applied over rough boundary without any excess computations. In comparison with the finite difference method, the proposed method is very fast.