Abstract
We extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally, we show some numerical experiments confirming the theoretical results and a couple of applications in view of bifurcation analysis.
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Notes
Here, elements in \(\mathbb {R}^{d}\) are intended as row vectors.
In our case \(K^{*}(t,\sigma -t):=\omega ^{*}D_{2}K(\omega ^{*}(\sigma -t),v^{*}(\sigma ))\) for \(t\ge t_{0}\) and \(\sigma \in [t-r,t]\) with \(r:=\tau /\omega ^{*}\le 1\).
It is enough to consider the integrals at the right-hand side of the VIE and of its adjoint over the whole of \(\mathbb {R}\), by taking into account the definition of \(K_{0}^{*}\).
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Acknowledgements
The authors are members of INdAM Research group GNCS and of UMI Research group “Modellistica socio-epidemiologica”.
Funding
Open access funding provided by Università degli Studi di Udine within the CRUI-CARE Agreement. This work was partially supported by the Italian Ministry of University and Research (MUR) through the PRIN 2020 project (no. 2020JLWP23) “Integrated Mathematical Approaches to Socio-Epidemiological Dynamics” (CUP: E15F21005420006).
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Appendix
Appendix
This Appendix completes the proof of Propositon 4 by showing that \(\xi _1^*\) introduced in the proof cannot belong to R. To this aim, we observe that the monodromy operator in Propositon 4 can also be defined in \(L^1([-1,0],\mathbb {R}^d)\). The action of the operator remains the same, meaning that X—or better, the corresponding quotient space obtained by considering almost everywhere equality of functions—is invariant under such action. Thus, in the spectral decomposition, \(X=K\oplus R\) remains the same. Assume for a contradiction that \(\xi _1^*\in R\), i.e., that \(\mathfrak {M}_1^{*(0)}=\mathfrak {M}_1^*\in R\), and define \(Y:=L^{\infty }([0,1];\mathbb {R}^{d})\).Footnote 1 Consider the standard bilinear form \(\langle \cdot ,\cdot \rangle :Y\times X\rightarrow \mathbb {R}\) defined as
As a general fact, any left eigenvector of some operator A w.r.t. some eigenvalue \(\lambda \) is orthogonal—in the corresponding bilinear form—to any element in the range of the operator \(I-\lambda A\). This means that, by pairing \(\mathfrak {M}_1^*\) with any left eigenvector \(\psi \) of \(T^{*}(1,0)\) with respect to A1, we get
In order to obtain a contradiction with A2, we resort to adjoint theory for VIEs (see [34] as a general reference). Indeed, any RE of the form
with \(x_{t_{0}}=\alpha \) for some \(\alpha \in X:=L^{1}([-1,0];\mathbb {R}^{d})\)Footnote 2 and \(r\le 1\) can be written as the VIE
where
and
Existence and uniqueness [34, Chapter 9] allow to define the forward evolution family \(\{T(t,t_{0})\}_{t\ge t_{0}}\) on X through \(T(t,t_{0})\alpha =x_{t}\). From [34, Exercise 6, p.274], we have the adjoint VIEFootnote 3
with
for
and \(y^{s_{0}}=\psi \) for some \(\psi \in Y\), where we use the notation \(y^s(\eta ):=y(s+\eta )\) for \(\eta \in [0,r]\). Then, one defines the backward evolution family \(\{V(s,s_{0})\}_{s\le s_{0}}\) on Y through \(V(s,s_{0})\psi =y^{s}\).
From the theory of resolvents [34, Chapter 9, Section 3], we can express the solution of A3 and that of A4 respectively as
and
where \(R_{0}^{*}\) is the resolvent of A3.
Given \(t\in \mathbb {R}\), consider now the pairing \([\cdot ,\cdot ]_{t}:Y\times X\rightarrow \mathbb {R}\) defined as
Observe that such bilinear form is nondegenerate for all \(t\in \mathbb {R}\) whenever \(K^*\) (and thus \(K_0^*\)) is nontrivial. Indeed, assume by contradiction that there exists \(\psi \in Y\) such that \(\psi \) is nonzero, but \([\psi ,\cdot ]_t\) is constantly 0. By the nondegenerateness of the bilinear form \(\langle \cdot ,\cdot \rangle \), this means that the innermost integral is 0 for all \(\alpha \in X\) and almost all \(\eta \in [0,1]\). If \(\alpha :=x_t\), where x is the (unique modulo multiplication by constant) 1-periodic solution of the VIE, then such integral is equal to
Thus, \(x_{t+1}\) is almost everywhere equal to 0. Using periodicity, this means that x is almost everywhere 0, which is only possible if \(K^*\) is trivial, contradiction. Using similar arguments, one can prove that there is no nonzero \(\alpha \in X\) such that \([\cdot ,\alpha ]_t\) is constantly zero, after exchanging the order of integration in the definition of \([\cdot ,\cdot ]_t\).
We claim that the forward monodromy operator and the corresponding backward one are adjoint w.r.t. A7, i.e., that
Indeed, using A6, we have
for
and
As for A, we have
where the first equality comes from the definition of g, the second follows from the substitution \(\sigma \leftarrow t+\sigma \), the third is obtained by exchanging the order of integration between \(\eta \) and \(\sigma \), the fourth follows from the definition of \(\psi _{0}\), the fifth is obtained by exchanging the order of integration between \(\eta \) and \(\beta \), the sixth is due to the fact that \(K_{0}^{*}(t+\sigma ,t-1+\eta )\) vanishes for \(\eta <\sigma \), the seventh follows from the substitution \(\eta \leftarrow 1+\eta \), the eighth is obtained by just renaming the variables, the ninth is due to the fact that \(K_0^*(t+1+\beta ,t+\sigma )\) vanishes for \(\beta > \sigma \), the tenth is obtained by exchanging the order of integration between \(\beta \) and \(\sigma \), the eleventh follows from the substitution \(\sigma \leftarrow \sigma -t\), and the last comes from the definition of f. As for B, we have
where the first equality comes from the definition of g, the second is obtained by exchanging the order of integration between \(\theta \) and \(\sigma \), the third is due to the fact that \(R_{0}^{*}(\sigma ,t-1+\eta )\) vanishes if \(\sigma <t-1+\eta \) and \(K_0^*(t+\theta ,\sigma )\) vanishes if \(\sigma <t-1+\theta \), the sixth is obtained by changing the order of integration, the seventh follows from the 1-periodicity of \(R_0^*\), the eighth follows from the substitution \(\sigma \leftarrow \sigma -t\), the ninth is due to the fact that \(R_{0}^{*}(1+\beta ,\sigma )\) vanishes if \(\sigma >1+\beta \) and \(K_0^*(\sigma ,t+\theta )\) vanishes if \(\theta <\sigma -t-1\), the tenth follows from the substitutions \(\theta \leftarrow \theta -t\) and \(\beta \leftarrow \beta -t\), and the last comes from the definition of f. Eventually, using A5, we have
which proves A8.
Under the assumption that 1 is a simple eigenvalue, both the VIE and its adjoint have a unique 1-periodic solution modulo multiplication by some constant, say x and y respectively. Thus, the associated states \(y^t\) and \(x_t\) are respectively the left and right 1-eigenvectors of the operator \(T(t+1,t)\). Again, thanks to their uniqueness, we have \([y^t,x_t]_t\ne 0\) for all \(t\in \mathbb {R}\). Moreover, the continuity of the map \(t\mapsto [y^t,x_t]_t\) and the mean value theorem for definite integrals let us conclude that \(\int _0^1[y^t,x_t]_t\textrm{d}t\ne 0\). Finally, observe that
where \((v^*)'\) is indeed x. Thus
which contradicts A2 thanks to the fact that \(y^0\) is the left 1-eigenvector of \(T^*(1,0)\).
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Andò, A., Breda, D. Piecewise orthogonal collocation for computing periodic solutions of renewal equations. Adv Comput Math 49, 93 (2023). https://doi.org/10.1007/s10444-023-10094-4
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DOI: https://doi.org/10.1007/s10444-023-10094-4
Keywords
- Renewal equations
- Periodic solutions
- Boundary value problems
- Piecewise orthogonal collocation
- Finite element method
- Population dynamics