Mathematical Model of a Chemical Reaction Within a Small Cell, withApplications in Biosciences

This work is concerned with modelling the evolution of competitive chemical reactions within a small cell with a labelled and unlabelled antigen reacting with a specific antibody on the side wall. A model consisting of coupled heat conduction equations with nonlinear and nonlocal boundary conditions is considered and shown to be equivalent to a system of Volterra integral equations with weakly singular kernel. This work generalizes some previous work done on the case of the single heat equation ([1], [2]). We prove the existence and uniqueness of the nonlinear system on $[0, 1)$. The asymptotic behavior of the solution as t tends to $0$ and $t$ tends to infinity is obtained and other properties of the solution, e.g., monotonicity, are investigated. In order to obtain a numerical solution of the system of VIES we use the technique of subtracting out singularities to derive explicit and implicit Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence. Numerical results are presented. This is a joint work with T. Diogo and S. McKee.

References

1. S. Jones, B. Jumarhon, S. McKee, J. A. Scott. A mathematical model of a biosensor, Journal of Engineering Mathematics 30, Netherlands, (1996) 321-337.
2. B. Jumarhon, S. McKee. On the heat equation with nonlinear an nonlocal boundary conditions, Journal of Mathematical Analysis and Applications 190, (1995) 806-820.