Keywords: cyclic splitting scheme, operator theory, spectral theory, iterative splitting scheme, nonlinear differential equations, Schrödinger equation.

This paper employs a novel idea to approximate the solutions of arbitrary linear and time dependent operator equations with the help of iterative schemes. The idea is based on an infinite operator series (the cyclic operator decomposition) and when we consider partial differential equations, we apply an integral formulation instead of a differential formulation, without neglecting the problem of reducing the sparsity of the derived linearized matrix system. The efficiency of such a method is based on the idea of deriving a Neumann series instead of the inverse operator. From a theoretical point of view, this method allows selectively choosing the cyclic operators and the corresponding generating function in such a way that the splitting scheme is an interesting way to derive analytical solutions. Nonlinear operators are linearized using iterative splitting methods which are based on alternating Picard fixed point iteration schemes, so that the cyclic splitting scheme can be extended to nonlinear applications. Numerical simulations are done to validate the test problems and the method is applied to the Schrödinger equation.

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