Dirac’s formalism combined with complex Fourier operational matrices to solve initial and boundary value problems

• Methods based on series approximations are useful for differential equations.
• Projections in a Hilbert space basis, combined with optimization algorithms, provide precise solutions.
• Dirac’s formalism allows extensions for complex functions.
• Fourier decomposition provides a simple and precise algorithm to solve differential equations.
• Precision is considerably improved by using this kind of method.

Approximations of functions in terms of orthogonal polynomials have been used to develop and implement numerical approaches to solve spectrally initial and boundary value problems. The main idea behind these approaches is to express differential and integral operators by using matrices, and this, in turn, makes the numerical implementation easier to be expressed in computational algebraic languages. In this paper, the application of the methodology is enlarged by using Dirac’s formalism, combined with complex Fourier series.