Analytic solutions to nonlinear ODEs via spectral power series

2023/10/09 by Estelle Basor, Rebecca Morrison, Basor, Estelle +1 · 1 citation
Engineering · Physics and Astronomy · #Optical Network Technologies #Advanced Fiber Laser Technologies #Semiconductor Lasers and Optical Devices

paper · pdf · doi:10.48550/arxiv.2401.02401

Abstract

Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expensive to compute. In this paper, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a power series expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms.

Cited by

• Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations

Related

• On the Solution Calculation of Nonlinear Ordinary Differential Equations via Exact Quadratization
• A new matrix equation expression for the solution of non-autonomous linear systems of ODEs
• Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation
• Asymptotic Methods of ODEs: Exploring Singularities of the Second Kind
• A novel analytical operator method to solve linear ordinary differential equations with variable coefficients