Moving boundary problems arise in many important applications to biology and chemistry. Comparing to the fixed boundary problem, moving boundary problem is more reasonable. To the best of our knowledge, there’s few results on the moving boundary for nonlinear first-order hyperbolic initial-boundary value problems. In the present paper, we mainly clarify the problem and show the existence and uniqueness of the solution for such kind of problems. We take a classical transform to straighten the moving boundary and develop a monotone approximation, based on upper and lower solutions technique, for solving a class of first-order hyperbolic initial-boundary value problems of moving boundary. Such an approximation results in the existence and uniqueness of the solution for the problem. The idea behind such a method is to replace the actual solution in all the nonlinear and nonlocal terms with some previous guess for the solution, then solve the resulting linear model to obtain a new guess for the solution. Iteration of such a procedure yields the solution of the original problem upon passage to the limit. A novelty of such a technique is that an explicit solution representation for each of these iterates is obtained, and hence an efficient numerical scheme can be developed. The key step is a comparison principle between consecutive guesses.
| Published in | American Journal of Applied Mathematics (Volume 7, Issue 3) |
| DOI | 10.11648/j.ajam.20190703.12 |
| Page(s) | 80-89 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2019. Published by Science Publishing Group |
Hyperbolic IBVP, Moving Boundary, Upper-lower Solutions, Monotone Approximation
| [1] | J. Crank. Free and Moving Boundary Problems. Mathematics of Computation, 46 (174): 429-500, 1986. |
| [2] | Purlis Emmanuel and V. O. Salvadori. Bread baking as a moving boundary problem. Part 1: Mathematical modelling. Journal of Food Engineering, 91 (3): 428-433, 2009. |
| [3] | H. Chen, W. Lv and S. Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic and Related Models, 8 (4): 667-684, 2018. |
| [4] | Pavel Drbek and J. Milota. Methods of Nonlinear Analysis. Automatic Control IEEE Transactions on, 17 (6): 848-849, 1973. |
| [5] | G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala. Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985. |
| [6] | C. V. Pao. Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
| [7] | A. S. Ackleh and K. Deng. Monotone method for nonlinear nonlocal hyperbolic problems. Electronic Journal of Differential Equations, Conference (10): 11-22, 2003. |
| [8] | A. S. Ackleh. Modeling biological process in aggregation of phytoplankton, PH. D. Thesis, University of Tennessee, Knoxville, TN, 1993. |
| [9] | D. P. Aikman. Modeling of growth and competition in plant monocultures. In Individual-Based Models and Approaches in Ecology, Edited by D. L. Deangelis and L. J. Gross, Chapman and Hall, London, pages 472-491, 1992. |
| [10] | E. D. Ford and K. A. Sorrensen. Theory and models of inter-plant competetion as a spatial process. In IndividualBased Models and Approaches in Ecology, Edited by D. L. Deangelis and L. J. Gross, Chapman and Hall, London, pages 363-407, 1992. |
| [11] | A. S. Ackleh and K. Deng. Monotone scheme for nonlinear first-order hyperbolic initial-boundary value problems. Applied Mathematics Letters, 13 (5): 111-119, 2000. |
| [12] | H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer Science and Business Media, 2010. |
| [13] | W. Lv and S. Wu. A free boundary problem for a class of nonlinear nonautonomous size-structured population model. Applicable Analysis, 97 (16): 2852-2864, 2018. |
| [14] | A. S. Ackleh and K. Deng. Existence-uniqueness of solutions for a nonlinear nonautonomous size-structured population model: an upper-lower solution approach. Canadian Appled Mathematics Quarterly, 8: 1–15, 2000. |
| [15] | H. T. Banks and K. Kunisch. Estimation techniques for distributed parameter systems. Estimation Techniques for Distributed Parameter Systems. Birkh¨auser Boston, 1989. |
| [16] | A. Calsina and J. Saldana. A model of physiologically structured population dynamics with a nonlinear individual growth rate. Journal of Mathematical Biology, 33 (4): 335-364, 1995. |
APA Style
Shaohua Wu, Di Chi. (2019). Monotone Method for Nonlinear First-order Hyperbolic Initial-boundary Value Problems of Moving Boundary. American Journal of Applied Mathematics, 7(3), 80-89. https://doi.org/10.11648/j.ajam.20190703.12
ACS Style
Shaohua Wu; Di Chi. Monotone Method for Nonlinear First-order Hyperbolic Initial-boundary Value Problems of Moving Boundary. Am. J. Appl. Math. 2019, 7(3), 80-89. doi: 10.11648/j.ajam.20190703.12
AMA Style
Shaohua Wu, Di Chi. Monotone Method for Nonlinear First-order Hyperbolic Initial-boundary Value Problems of Moving Boundary. Am J Appl Math. 2019;7(3):80-89. doi: 10.11648/j.ajam.20190703.12
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Download@article{10.11648/j.ajam.20190703.12,
author = {Shaohua Wu and Di Chi},
title = {Monotone Method for Nonlinear First-order Hyperbolic Initial-boundary Value Problems of Moving Boundary},
journal = {American Journal of Applied Mathematics},
volume = {7},
number = {3},
pages = {80-89},
doi = {10.11648/j.ajam.20190703.12},
url = {https://doi.org/10.11648/j.ajam.20190703.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190703.12},
abstract = {Moving boundary problems arise in many important applications to biology and chemistry. Comparing to the fixed boundary problem, moving boundary problem is more reasonable. To the best of our knowledge, there’s few results on the moving boundary for nonlinear first-order hyperbolic initial-boundary value problems. In the present paper, we mainly clarify the problem and show the existence and uniqueness of the solution for such kind of problems. We take a classical transform to straighten the moving boundary and develop a monotone approximation, based on upper and lower solutions technique, for solving a class of first-order hyperbolic initial-boundary value problems of moving boundary. Such an approximation results in the existence and uniqueness of the solution for the problem. The idea behind such a method is to replace the actual solution in all the nonlinear and nonlocal terms with some previous guess for the solution, then solve the resulting linear model to obtain a new guess for the solution. Iteration of such a procedure yields the solution of the original problem upon passage to the limit. A novelty of such a technique is that an explicit solution representation for each of these iterates is obtained, and hence an efficient numerical scheme can be developed. The key step is a comparison principle between consecutive guesses.},
year = {2019}
}
TY - JOUR T1 - Monotone Method for Nonlinear First-order Hyperbolic Initial-boundary Value Problems of Moving Boundary AU - Shaohua Wu AU - Di Chi Y1 - 2019/08/30 PY - 2019 N1 - https://doi.org/10.11648/j.ajam.20190703.12 DO - 10.11648/j.ajam.20190703.12 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 80 EP - 89 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20190703.12 AB - Moving boundary problems arise in many important applications to biology and chemistry. Comparing to the fixed boundary problem, moving boundary problem is more reasonable. To the best of our knowledge, there’s few results on the moving boundary for nonlinear first-order hyperbolic initial-boundary value problems. In the present paper, we mainly clarify the problem and show the existence and uniqueness of the solution for such kind of problems. We take a classical transform to straighten the moving boundary and develop a monotone approximation, based on upper and lower solutions technique, for solving a class of first-order hyperbolic initial-boundary value problems of moving boundary. Such an approximation results in the existence and uniqueness of the solution for the problem. The idea behind such a method is to replace the actual solution in all the nonlinear and nonlocal terms with some previous guess for the solution, then solve the resulting linear model to obtain a new guess for the solution. Iteration of such a procedure yields the solution of the original problem upon passage to the limit. A novelty of such a technique is that an explicit solution representation for each of these iterates is obtained, and hence an efficient numerical scheme can be developed. The key step is a comparison principle between consecutive guesses. VL - 7 IS - 3 ER -
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