Non-classical diffusion equations related to birth-death processes with two boundaries
Authors:
Z. G. Mansourati and L. L. Campbell
Journal:
Quart. Appl. Math. 54 (1996), 423-443
MSC:
Primary 60J80; Secondary 35B20, 35K55, 60J70
DOI:
https://doi.org/10.1090/qam/1402403
MathSciNet review:
MR1402403
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Abstract: A pair of forward and backward diffusion equations is considered. In the forward equation, boundary values appear in the differential equation, and in the backward equation, boundary values are related to average values of the solution in the interior of the domain. The forward equation can be regarded as a diffusion approximation to a type of birth-death process with returns to the interior, or as a heat equation in one dimension where heat flowing out from the boundaries is returned to the interior. Existence and uniqueness theorems are proved, and some properties of the associated eigenvalues and eigenfunctions are deduced. An expression for the steady-state solution is obtained. Some information on the goodness of the diffusion approximation is also obtained.
F. Baccelli and G. Fayolle, Analysis of models reducible to a class of diffusion processes in the positive quarter plane, SIAM J. Appl. Math. 47, 1367–1385 (1987)
P. G. Buckholtz, L. L. Campbell, R. D. Milbourne, and M. T. Wasan, Analysis of transient behaviour of certain processes with return to a central state, J. Appl. Probab. 20, 61–70 (1983)
L. L. Campbell, Transient analysis of birth-death processes with two boundaries, Canad. J. Stat. 13, 123–130 (1985)
J. R. Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Appl., vol. 23, Addison-Wesley, Reading, MA, 1984
S. Chaimsiri and M. S. Leonard, A diffusion approximation for bulk queues, Management Sci. 27, 1188–1199 (1981)
R. H. Cole, General boundary conditions for an ordinary linear differential system, Trans. Amer. Math. Soc. 111, 521–550 (1964)
W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 41, 468–475 (1983)
W. Feller, The parabolic differential equations and the associated semi-group of transformations, Ann. Math. 55, 468–519 (1952)
W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77, 1–31 (1954)
A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44, 401–407 (1986)
P. W. Glynn, Diffusion approximations, in Stochastic Models, Chapter 4, edited by D. P. Heyman and M. J. Sobel, North-Holland, New York, 1990
Z. G. Mansourati, Non-classical diffusion equations related to a class of birth-death processes with two boundaries, Ph.D. Thesis, Queen’s University at Kingston, January, 1990
G. Pujolle, Réseaux de files d’attente: Méthode des diffusions, Editions Hommes et Techniques, Paris, 1980, pp. 41–72
C. E. Wilder, Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points, Trans. Amer. Math. Soc. 18, 415–442 (1917)
F. Baccelli and G. Fayolle, Analysis of models reducible to a class of diffusion processes in the positive quarter plane, SIAM J. Appl. Math. 47, 1367–1385 (1987)
P. G. Buckholtz, L. L. Campbell, R. D. Milbourne, and M. T. Wasan, Analysis of transient behaviour of certain processes with return to a central state, J. Appl. Probab. 20, 61–70 (1983)
L. L. Campbell, Transient analysis of birth-death processes with two boundaries, Canad. J. Stat. 13, 123–130 (1985)
J. R. Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Appl., vol. 23, Addison-Wesley, Reading, MA, 1984
S. Chaimsiri and M. S. Leonard, A diffusion approximation for bulk queues, Management Sci. 27, 1188–1199 (1981)
R. H. Cole, General boundary conditions for an ordinary linear differential system, Trans. Amer. Math. Soc. 111, 521–550 (1964)
W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 41, 468–475 (1983)
W. Feller, The parabolic differential equations and the associated semi-group of transformations, Ann. Math. 55, 468–519 (1952)
W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77, 1–31 (1954)
A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44, 401–407 (1986)
P. W. Glynn, Diffusion approximations, in Stochastic Models, Chapter 4, edited by D. P. Heyman and M. J. Sobel, North-Holland, New York, 1990
Z. G. Mansourati, Non-classical diffusion equations related to a class of birth-death processes with two boundaries, Ph.D. Thesis, Queen’s University at Kingston, January, 1990
G. Pujolle, Réseaux de files d’attente: Méthode des diffusions, Editions Hommes et Techniques, Paris, 1980, pp. 41–72
C. E. Wilder, Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points, Trans. Amer. Math. Soc. 18, 415–442 (1917)
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Article copyright:
© Copyright 1996
American Mathematical Society
