The *-product approach for linear ODEs: A numerical study of the scalar case

Abstract

Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be straightforwardly extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the method's efficacy and properties in the scalar case.

A new matrix equation expression for the solution of non‐autonomous linear systems of ODEs

“…where 4) hides an infinite series of nested integrals. However, as shown in [2], it is possible to approximate the ⋆-product by the usual matrix-matrix product in the scalar case. This approximation allows us to compute (4) more simply and cheaply.…”

“…Note that Θ(t − s) endows the condition t ≥ s in equation (2) and that U (t, s) is the bivariate function expressing the solutions of (1) for every initial time s ∈ I, with U (t, s) = 0 for t < s. From now on, we will denote with a tilde all the bivariate functions that are infinitely differentiable in both t and s over I, i.e., f ∈ C ∞ (I × I). Moreover, we define the following class of functions…”

“…The size of such systems is 2 k × 2 k for a sample with k spins and is usually sparse. In [2], we proposed a new method with spectral accuracy for solving scalar non-autonomous ordinary differential equations. In the present work, we extend this method to the case of systems of non-autonomous ODEs.…”

“…Thanks to this product, the solution of (1) can be expressed through the ⋆-product inverse and its formulation as a sequence of integrals and differential equations; see [4][5][6][7][8]. In [2], we illustrated that, by discretizing the ⋆-product with orthogonal functions, the solution of a scalar ODE is accessible by solving a linear system. In this work, we extend the results in [2], showing that, following the same principles, we can solve (1) through a linear system.…”

“…In [2], we illustrated that, by discretizing the ⋆-product with orthogonal functions, the solution of a scalar ODE is accessible by solving a linear system. In this work, we extend the results in [2], showing that, following the same principles, we can solve (1) through a linear system. Moreover, we show that the linear system solution can be expressed as the solution of a matrix equation with a rank one right-hand side.…”