American Institute of Mathematical Sciences

Advanced
May  2008, 7(3): 677-698. doi: 10.3934/cpaa.2008.7.677

Analysis of a biosensor model

Walter Allegretto 1, , Yanping Lin 2, and  Zhiyong Zhang 3,

1. 

Department of Mathematical Sciences, University of Alberta, Edmonton A B, Canada T6G 2G1
2. 

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
3. 

Department of Mathematics and Statistics, University of Alberta, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received  May 2007 Revised  September 2007 Published  February 2008

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: stability., finite volume method, Robin boundary condition, Biosensor model.
Mathematics Subject Classification: Primary: 35K55, 35D10; Secondary: 65M9.
Citation: Walter Allegretto, Yanping Lin, Zhiyong Zhang. Analysis of a biosensor model. Communications on Pure & Applied Analysis, 2008, 7 (3) : 677-698. doi: 10.3934/cpaa.2008.7.677
[1]

Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048
[2]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[3]

Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221
[4]

Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609
[5]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
[6]

Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic & Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865
[7]

Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129
[8]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393
[9]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003
[10]

Kei Fong Lam, Hao Wu. Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1847-1878. doi: 10.3934/dcds.2020096
[11]

Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913
[12]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195
[13]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020226
[14]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[15]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837
[16]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253
[17]

Nan Li, Song Wang, Shuhua Zhang. Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1349-1368. doi: 10.3934/jimo.2019006
[18]

Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006
[19]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153
[20]

Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences