Stieltjes Bochner spaces and applications to the study of parabolic equations

Abstract

This work is devoted to the mathematical analysis of Stieltjes Bochner spaces and their applications to the resolution of a parabolic equation with Stieltjes time derivative. This novel formulation allows us to study parabolic equations that present impulses at certain times or lapses where the system does not evolve at all and presents an elliptic behavior. We prove several theoretical results related to existence of solution, and propose a full algorithm for its computation, illustrated with some realistic numerical examples related to population dynamics.

Numerical Solution of Stieltjes Differential Equations

“…The ideas we developed in this work can be used to design an algorithm that improves the numerical approximation of the linear Stieltjes parabolic problems given in [13]. In this work, the authors obtained a numerical approximation of (130) by truncating the Fourier series of the solution and using the exact solution (120) to obtain each of the coefficients of the aforementioned series.…”

“…) (0D model) can also be appreciated in the 2D model based on a parabolic partial differential equation of the Stieltjes type (cf [13]. (Section 5)): t) − ∇ • (η∇u) = f (t, u(t), u), in [(0, T) \ C g ] × Ω, η∇u • n = 0, on (0, T) × ∂Ω, u(0, x) = u 0 (x), in Ω,…”

“…where [13] for a description of the Stieltjes-Bochner space L 2 g ([0, T); L 2 (Ω))). We can apply Method (132) to the integral expression form of the solution of (133), and we obtain the following numerical method: given…”

“…[14] (Chapter 7) for the mathematical analysis of the previous approximation in the case g(t) = t). An analogous discretization can be applied to System (130), and the resulting scheme can be compared with the scheme given in [13]. In future works, we intend to mathematically study the approximation (134) even in the case of considering a finite-dimensional subspace V h ⊂ H 1 (Ω).…”

Existence and Uniqueness of Solution for Stieltjes Differential Equations with Several Derivators

“…The usual setting for Stieltjes differential equations in the literature involves a single derivator either in its theoretical -see for example [4,6,9,11,15,16,22]-or numerical studies [4,5]. This is also the case for other differential problems involving Stieltjes derivatives such as in [13,17,21], or even the corresponding integral counterparts.…”