Communications on Pure and Applied Analysis (CPAA)

Analysis of a biosensor model

Pages: 677 - 698, Volume 7, Issue 3, May 2008      doi:10.3934/cpaa.2008.7.677

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Walter Allegretto - Department of Mathematical Sciences, University of Alberta, Edmonton A B, Canada T6G 2G1, Canada (email)
Yanping Lin - Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1, Canada (email)
Zhiyong Zhang - Department of Mathematics and Statistics, University of Alberta, University of Alberta, Edmonton, Alberta T6G 2G1, Canada (email)

Abstract: In this paper we consider a biosensor model in $R^3$, consisting of a coupled parabolic differential equation with Robin boundary condition and an ordinary differential equation. Theoretical analysis is done to show the existence and uniqueness of a Holder continuous solution based on a maximum principle, weak solution arguments. The long-time convergence to a steady state is also discussed as well as the system situation. Next, a finite volume method is applied to the model to obtain an approximate solution. Drawing in part on the analytical results given earlier, we establish the existence, stability and error estimates for the approximate solution, and derive $L^2$ spatial norm convergence properties. Finally, some illustrative numerical simulation results are presented.

Keywords:  Biosensor model, Robin boundary condition, finite volume method, stability.
Mathematics Subject Classification:  Primary: 35K55, 35D10; Secondary: 65M99.

Received: May 2007;      Revised: September 2007;      Available Online: February 2008.