I presume that P_0 and H_0 are known functions and that you are solving for Q_{E10} as a function of phi and lambda? Looks like a Reynolds equation. I can't say that the following will definitely work as I haven't tried it out, but  you could split  Q_{E10} into its real and and imaginary parts and solve the resulting coupled Poisson-like equations using your favourite method, such as SOR or multigrid. Or, given that it is a linear equation, merely work with complex variables as it stands. The latter is the quicker route; give it a go and see what happens!

If your independent variables \phi and \lambda are real, then your problem can be simplified by separating real and imaginary parts of the of the differential equations and solve them separately by real means. For the dependent variables, let  H_0 = H_0re + i H_0im and like wise for P_0 and Q_E10. and perform complex operations and separate real and imaginary components. If any of your dependent variables are real, use it to simplify your results.  Hope this helps.

Thanks D. Andrew S. Rees , you are true it is a Reynolds equation (first order for small perturbation) and H_0 & P_O are the static solution of Reynolds and the are real and calculated using FDM; but (Q_E10) is complex. I'll try your suggestion and let you know about the results.

Thanks Nasir Haniffa, yes the independent variables are real and the dependent also are real except (Q_E10). so I'll separating the real and imaginary parts and solve them. I'll let you know about the results.