An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations

Maset, Stefano

doi:10.1007/s00211-015-0754-1

Cited by 11 publications

(106 citation statements)

References 50 publications

(98 reference statements)

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“…The convergence analysis of the numerical method described in Section 2 follows the abstract approach, 10 intended for neutral functional differential equations. Note that REs can be treated as equations of this kind by interpreting the relevant solutions as derivatives of other functions.…”

Section: Convergence Analysismentioning

confidence: 99%

“…The general fixed‐point problem described in Reference 10 consists in finding

(v^{*}, β^{*}) \in 𝕍 \times 𝔹

with

v^{*} : = 𝒢 (u^{*}, α^{*})

and

(u^{*}, α^{*}, β^{*}) \in 𝕌 \times 𝔸 \times 𝔹

such that

(u^{*}, α^{*}, β^{*}) = Φ (u^{*}, α^{*}, β^{*})

for

Φ : 𝕌 \times 𝔸 \times 𝔹 \to 𝕌 \times 𝔸 \times 𝔹

given by

Φ (u, α, β) : = (\begin{matrix} ℱ false(𝒢 false(u, α false), u, β false) \\ false(α, β false) prefix- ℬ false(𝒢 false(u, α false), u, β false) \end{matrix}),

where

ℱ : 𝕍 \times 𝕌 \times 𝔹 \to 𝕌

in the first line defines the right‐hand side of the functional equation of the relevant BVP and

ℬ : 𝕍 \times 𝕌 \times 𝔹 \to 𝔸 \times 𝔹

in the second one represents the boundary conditions. The solution

v = 𝒢

…”

Section: Convergence Analysismentioning

confidence: 99%

“…In Reference 7 an extension of the Floquet theory to REs was proposed, allowing the study of local asymptotic stability of periodic solutions, while 8 provides a theoretical analysis of convergence of the piecewise collocation method described in Reference 9 to actually compute the periodic solutions. Such analysis is based on the abstract approach in Reference 10 for general boundary value problems (BVPs).…”

Section: Introductionmentioning

confidence: 99%

“…Section 2 will describe both the BVP formulation and the relevant numerical method. Section 3 will go through the steps of the convergence analysis, by stating the propositions that ensure the validity of both the theoretical assumptions (Subsection 3.1) and the numerical ones (Subsection 3.2) required in Reference 10. As it will be shown, some parts of the proofs of the latter can be notably simplified by resorting to the BVP with finite‐dimensional boundary condition.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite‐dimensional boundary conditions

Andò

2021

Comp and Math Methods

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The problem of computing periodic solutions can be expressed as a boundary value problem and solved numerically via piecewise collocation. Here, we extend to renewal equations the corresponding method for retarded functional differential equations in (K. Engelborghs et al., SIAM J Sci Comput., 22 (2001), pp. 1593–1609). The theoretical proof of the convergence of the method has been recently provided in (A. Andò and D. Breda, SIAM J Numer Anal., 58 (2020), pp. 3010–3039) for retarded functional differential equations and in (A. Andò and D. Breda, submitted in 2021) for renewal equations and consists in both cases in applying the abstract framework in (S. Maset, Numer Math., 133 (2016), pp. 525–555) to a reformulation of the boundary value problem featuring an infinite‐dimensional boundary condition. We show that, in the renewal case, the proof can also be carried out and even simplified when considering the standard formulation, defined by boundary conditions of finite dimension.

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Section: Convergence Analysismentioning

confidence: 99%

“…The general fixed‐point problem described in Reference 10 consists in finding

(v^{*}, β^{*}) \in 𝕍 \times 𝔹

with

v^{*} : = 𝒢 (u^{*}, α^{*})

and

(u^{*}, α^{*}, β^{*}) \in 𝕌 \times 𝔸 \times 𝔹

such that

(u^{*}, α^{*}, β^{*}) = Φ (u^{*}, α^{*}, β^{*})

for

Φ : 𝕌 \times 𝔸 \times 𝔹 \to 𝕌 \times 𝔸 \times 𝔹

given by

Φ (u, α, β) : = (\begin{matrix} ℱ false(𝒢 false(u, α false), u, β false) \\ false(α, β false) prefix- ℬ false(𝒢 false(u, α false), u, β false) \end{matrix}),

where

ℱ : 𝕍 \times 𝕌 \times 𝔹 \to 𝕌

in the first line defines the right‐hand side of the functional equation of the relevant BVP and

ℬ : 𝕍 \times 𝕌 \times 𝔹 \to 𝔸 \times 𝔹

in the second one represents the boundary conditions. The solution

v = 𝒢

…”

Section: Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite‐dimensional boundary conditions

Andò

2021

Comp and Math Methods

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“…For this reason, in the last years the interest in the study of the dynamics of delay models has been increasing and important challenges, in particular numerical, have been identified. Indeed, delay equations describe infinite-dimensional dynamical systems, and theoretical results should be complemented with efficient numerical methods to approximate solutions of initial value problems [3,4,5,7,17,15,16,19,44,43], boundary value problems [48,49,50], and to investigate the stability of equilibria and periodic solutions [10,11,12,13,14,47,52,53,66]. In applications the attention is focused not only on the approximation of the dynamical properties for some given parameter values, but also on how such properties change when varying some parameters.…”

mentioning

confidence: 99%

Pseudospectral discretization of delay differential equations in <i>sun-star</i> formulation: Results and conjectures

Diekmann

Scarabel

Vermiglio

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the *-space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the *-shift operator plays a fundamental role. By applying the pseudospectral discretization technique we derive a system of ordinary differential equations, whose dynamics can be efficiently analyzed by existing bifurcation tools. To better understand to what extent the resulting finite-dimensional system "mimics" the dynamics of the original infinite-dimensional one, we study the pseudospectral approximations of the *-shift operator and of the *-generator in the supremum norm, which is the natural choice for delay differential equations, when the discretization parameter increases. In this context there are still open questions. We collect the most relevant results from the literature and we present some conjectures, supported by various numerical experiments, to illustrate the behavior w.r.t. the discretization parameter and to indicate the direction of ongoing and future research.

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Collocation Techniques for Structured Populations Modeled by Delay Equations

Andò

Breda

2020

SEMA SIMAI Springer Series

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As of 2013, the SIMAI Springer Series opens to SEMA in order to publish a joint series aiming to publish advanced textbooks, research-level monographs and collected works that focus on applications of mathematics to social and industrial problems, including biology, medicine, engineering, environment and finance. Mathematical and numerical modeling is playing a crucial role in the solution of the complex and interrelated problems faced nowadays not only by researchers operating in the field of basic sciences, but also in more directly applied and industrial sectors. This series is meant to host selected contributions focusing on the relevance of mathematics in real life applications and to provide useful reference material to students, academic and industrial researchers at an international level. Interdisciplinary contributions, showing a fruitful collaboration of mathematicians with researchers of other fields to address complex applications, are welcomed in this series.

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An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations

Cited by 11 publications

References 50 publications

Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite‐dimensional boundary conditions

Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite‐dimensional boundary conditions

Pseudospectral discretization of delay differential equations in <i>sun-star</i> formulation: Results and conjectures

Collocation Techniques for Structured Populations Modeled by Delay Equations

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