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A Fixed Point Approach to Simulation of Functional Differential Equations with a Delayed Argument

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Abstract

A computational method is developed for a family of functional differential equations in one independent variable with a single deviating argument, assumed to be a delay. These equations are mapped into a finite system of ODEs, a subsystem of which involves only the function controlling the delay. Key features include efficiency during method of steps, freedom from Jacobians and root finding techniques, and computing a continuous approximation. The nonlinear differential equations may be retarded, neutral or advanced. The method is established for state dependent delays, stiff equations, discontinuous initial history, a specific loss of monotonicity in the delay and extended naturally to distributed delays. When restricted to ODEs, the method extends naturally to PDEs; the hope is the delayed version here can eventually be extended to delayed PDEs. Conditions for convergence of the approximation are established, and results of numerical experiments are reported to indicate robustness of the implementation.

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Acknowledgements

The author would like to thank Dr. James Sochacki and Dr. Anthony Tongen for their contributions which enabled this manuscript to be produced. The author would also like to thank Dr. Lisette de Pillis, who made the author aware of the blood, spleen and tumor problem. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The author would also like to thank the (hard working) referees for their helpful suggestions, which ultimately made this a better paper.

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  1. Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN, 47809, USA

    Vincenzo M. Isaia

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Correspondence to Vincenzo M. Isaia.

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Appendix

Appendix

For a full discussion of the BST model, see [19] and references therein. This is the list of 9 components from the full system of DEs in the BST problem, see page 35, which are all functions of t

$$\begin{aligned} T, D_B, D_S, D_T, E^a_S, E^m_S, E^a_B, E^m_B, E^a_T \end{aligned}$$

This is the list of parameters (all constant) from the same system:

$$\begin{aligned}&\mu _B, \mu _{TB}, \mu _{BB}, \max D, \mu _{BS}, a_D, b_{DE}, \mu _{BSE}, b_a, a_{E_aS}, E_{naive}, r_{am}, b_p, \theta _D, a_{E_m}, \\&\quad a_{E_aT}, c, r, k, q, m, d, s, l, \alpha , \mu _{SB}^{normal}, \mu ^*_{SB} \end{aligned}$$

The functions \(v_{blood}(t)\) and \(v_{tumor}(t)\) are nonhomogeneous (forcing) terms. This is a list of known functions in the same system:

$$\begin{aligned}&\mu _{SB}(D_S) \equiv \mu ^*_{SB} + \frac{\mu _{SB}^{normal} - \mu ^*_{SB}}{1 + \frac{D_S}{\theta _{shut}}} \qquad DC_{on} \equiv \left\{ \begin{array}{cl} 0 &{}\hbox { if }D_S(t) = 0 \\ 1 &{}\hbox { if }D_S(t) > 0 \\ \end{array} \right. \\&\mu _{BTE} \equiv \mu _{BB}\left( \frac{T}{\alpha + T}\right) \qquad {\mathscr {D}} \equiv d\frac{\left( \frac{E^a_T}{T}\right) ^l}{s+\left( \frac{E^a_T}{T}\right) ^l} \end{aligned}$$

The full system of equations, using the above and prime for derivative, is

$$\begin{aligned}&\left\{ \begin{array}{rcl} D'_B &{} = &{} -\mu _B D_B + \mu _{TB} D_T + v_{blood}(t) \\ (E^a_B)' &{} = &{} \mu _{SB}(D_S) E^a_S - \mu _{BB} E^a_B \\ (E^m_B)' &{} = &{} \mu _{SB}(D_S) E^m_S - \mu _{BB} E^m_B \\ \end{array} \right. \\&\left\{ \begin{array}{rcl} D'_S &{} = &{} \max D \left( 1 - e^{\frac{-\mu _{BS} D_B}{\max D}} \right) - a_D D_S - b_{DE}E^a_S D_S \\ (E^a_S)' &{} = &{} \mu _{BSE}E^a_B - \mu _{SB}(D_S) E^a_S +b_a D_S E^m_S + a_{E_aS}(DC_{on} E_{naive} - E^a_S) \\ &{} &{} - r_{am}E^a_S + b_p \frac{D_S(t - \tau _D) E^a_S(t-\tau _D)}{\theta _D + D_S(t-\tau _D)} \\ (E^m_S)' &{} = &{} r_{am} E^a_S - (a_{E_m} + b_a D_S +\mu _{SB}(D_S)) E^m_S + \mu _{BSE} E^m_B \\ \end{array} \right. \\&\left\{ \begin{array}{rcl} (E^a_T)' &{} = &{} \mu _{BTE}(T) E^a_B - a_{E_aT} E^a_T - cE^a_T T \\ T' &{} = &{} rT(1-\frac{T}{k}) - {\mathscr {D}}T \\ D'_T &{} = &{} \frac{mT}{q+T} - (\mu _{TB}+a_D)D_T + v_{tumor}(t) \\ \end{array} \right. \end{aligned}$$

Auxiliary variables were used for \(\mu _{SB}\), \(\mu _{BTE}\), \(E^a_T T^{-1}\), \({\mathscr {D}}\), \(1 - e^{\frac{-\mu _{BS} D_B}{\max D}}\), \((\theta _D + D_S(t-\tau _D))^{-1}\), \((q + T)^{-1}\), and \((\cdot )^{l}\) since \(l = 2/3\), to polynomial ones.

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Isaia, V.M. A Fixed Point Approach to Simulation of Functional Differential Equations with a Delayed Argument. J Dyn Diff Equat 35, 1047��1082 (2023). https://doi.org/10.1007/s10884-021-10081-7

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