A numerical method and a code for seeking complex eigenvalues are given with the fundamental matrix method.The treatment of singularity,complex eigenvalue and numerical instability are discussed. The method is used to simulate the instability driven by a temperature gradient [η_i].Numerical results agree well with theoretical analyses.

It proposes three kinds of new difference schemes(exponential type,oscillatory type and exponential-oscillatory type) based on the coefficient structural characteristic and the behaviour of solution in the ODEs.The schemes can describe excellently dependent variables with steep gradient,shear layer and/orhighly oscillations.

The resonance interaction of plasma waves with the toroidal drifts of particles leads to the occurence of a class of high order resonance integrals, which are related closed with the plasma dispersion function. Ordinary integral forms of them in finite region have been obtained by suitable complex transformation, that is very useful for their numerical calculation in case of the imaginary part of the complex parameter being small.

Numerical method for solving the free boundary plasma equilibrium equation (a nonlinear partial differential equation of elliptic type) during the evolution of the external maintaining, field is presented. The direct method is employed to solve the corresponding set of difference equation and compared with the over relaxation method.It is shown that the direct method can give better computation precision and can save considerable computer time.