Development in periodic series,method for resolving differential equations

TL;DR

This paper explores advanced series expansion methods, especially Fourier series, to solve a broad class of differential equations, including linear, nonlinear, and systems, by extending traditional techniques to non-orthogonal and non-sinusoidal bases.

Contribution

It introduces an extended methodology for solving diverse differential equations using Fourier series on non-orthogonal and non-sinusoidal bases, broadening applicability.

Findings

Applicable to all linear, homogeneous and non-homogeneous ODEs with constant coefficients

Extended to linear and nonlinear equations with variable coefficients

Includes solutions for systems of ODEs and integro-differential equations

Abstract

The development of functions of real variables in Taylor and Frobenius series (whole series which are formed in nonorthogonal, nonperiodic bases), in sinusoidal Fourier series (bases of orthogonal, periodic functions), in series of special functions (bases of orthogonal, nonperiodic functions), etc. is a commonly used method for solving a wide range of ordinary differential equations (ODEs) and partial differential equations (PDEs).In this article, based on an in-depth analysis of the properties of periodic sinusoidal Fourier series (SFS), we will be able to apply this procedure to a much broader category of ODEs (all linear, homogeneous and non-homogeneous equations with constant coefficients, a large category of linear and non-linear equations with variable coefficients, systems of ODEs, integro-differential equations, etc.). We will also extend this procedure and we use it to solve certain ODEs, on non-orthogonal periodic bases, represented by non sinusoidal periodic Fourier series (SFN).