## Abstract

A mathematical model in the form of two coupled diffusion equations is provided

for a competitive chemical reaction between an antigen and a labelled antigen for antibody sites on a cell wall; boundary conditions are such that the problem is both nonlinear and nonlocal. This is then re-characterized first as a pair of coupled singular integro-differential equations and then as a system of four Volterra integral equations. The latter permits a proof of existence and uniqueness of the solution of the original problem. Small and large time asymptotic solutions are derived and, from the first characterization, a regular perturbation solution is obtained. Numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin.

for a competitive chemical reaction between an antigen and a labelled antigen for antibody sites on a cell wall; boundary conditions are such that the problem is both nonlinear and nonlocal. This is then re-characterized first as a pair of coupled singular integro-differential equations and then as a system of four Volterra integral equations. The latter permits a proof of existence and uniqueness of the solution of the original problem. Small and large time asymptotic solutions are derived and, from the first characterization, a regular perturbation solution is obtained. Numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin.

Original language | English |
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Pages (from-to) | 1081-1112 |

Number of pages | 32 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 72 |

Issue number | 4 |

Early online date | 15 Aug 2012 |

DOIs | |

Publication status | Published - 2012 |

## Keywords

- mathematical model
- capillary-fill device
- antibody-antigen
- Volterra equations
- existence and uniqueness
- asymptotic results
- regular perturbation
- numerical approximation