A Closed Form Solution to One Dimensional Robin Boundary Problems

Chang-li YANG, Ai-lin ZHU

Acta Mathematicae Applicatae Sinica(English Series) ›› 2012, Vol. 28 ›› Issue (3) : 549-556.

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Acta Mathematicae Applicatae Sinica(English Series) ›› 2012, Vol. 28 ›› Issue (3) : 549-556. DOI: 10.1007/s10255-012-0156-4
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A Closed Form Solution to One Dimensional Robin Boundary Problems

  • Chang-li YANG, Ai-lin ZHU
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Abstract

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). In this paper, we will use reflected and absorbed Brownian motion and stochastic differential equations to construct a closed form solution to one dimensional Robin boundary problems. Meanwhile, we will give a reasonable explanation to the closed form solution from a stochastic point of view. Finally, we will extend the problem to Robin boundary problem with two boundary conditions and give a specific solution by resorting to a stopping time.

Key words

partial differential equations / Robin boundary problem / diffusion processes / Girsanov’s formula

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Chang-li YANG, Ai-lin ZHU. A Closed Form Solution to One Dimensional Robin Boundary Problems. Acta Mathematicae Applicatae Sinica(English Series), 2012, 28(3): 549-556 https://doi.org/10.1007/s10255-012-0156-4

References

[1] Andrews, S.S., Bray, D. Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Physical Biology, 1: 137-151 (2004)
[2] Erban, R., Chapman, S.J. Reactive boundary conditions for stochastic simulations of reaction diffusion processes. Physical Biology, 4: 16-28 (2007)
[3] Hattne, J., Fagne, D., Elf, J. Stochastic reaction-diffusion simulation with MesoRD. Bioinformatics, 21(12): 2923-2924 (2005)
[4] Isaacson, S.A., Peskin, C.S. Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations. SIAM Journal on Scientific Computing, 28(1): 47-74 (2006)
[5] Karatzas, I., Shreve, S.E. Brownian Motion and Stochastic Calculus. Springer-Verlag, 1988
[6] Lyons, T.J., Zheng, W.A. On conditional diffusion processes. In: Proceedings of the Royal Society of Edinburgh, Section A. Mathematics 115: 243-255 (1990)
[7] Qian, Z.M., Zheng, W.A. Sharp bounds for transition probability densities of a class of diffusions. Comptes Rendus Mathematique, 335: 953-957 (2002)
[8] Revuz, D., Yor, M. Continuous Martingales and Brownian Motion. Springer-Verlag, 1991
[9] Stundzia, A., Lumsden, C. Stochastic simulation of coupled reaction-diffusion processes. Journal of Computational Physics, 127: 196-207 (1996)
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