A modified spectral method for solving operator equations

In this paper we introduce a modified spectral method for solving the linear operator equation Lu=f,L:D(L)⊆H1→H2, where H1H1 and H2H2 are normed vector spaces with norms ‖.‖1‖.‖1 and ‖.‖‖.‖, respectively and D(L)D(L) is the domain of LL. Also for each h∈H2h∈H2, ‖h‖2=(h,h)‖h‖2=(h,h) where (.,.)(.,.) is an inner product on H2H2. In this method we make a new set {ψn}n=0∞ for H1H1 using LL and two sets in H1H1 and H2H2. Then using the new set {ψn}n=0∞ we solve this linear operator equation. We show that this method does not have some shortcomings of spectral method, also we prove the stability and convergence of the new method. After introducing the method we give some conditions that under them the nonlinear operator equation Lu+Nu=fLu+Nu=f can be solved. Some examples are considered to show the efficiency of method.