# American Institute of Mathematical Sciences

May  2012, 17(3): 735-757. doi: 10.3934/dcdsb.2012.17.735

## Two theorems on singularly perturbed semigroups with applications to models of applied mathematics

 1 Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland 2 European Actuarial Services, Ernst & Young Business Advisory Sp. z o.o. i Wspólnicy sp.k., Rondo ONZ 1, 00-124, Warsaw, Poland

Received  May 2011 Revised  October 2011 Published  January 2012

We present two theorems on convergence of semigroups related to singularly perturbed abstract Cauchy problems, and apply them to some recent models of applied mathematics. The semigroups considered are related to piecewise deterministic Markov processes jumping between several copies of a rectangle in $\mathbb{R}^M$ and moving along deterministic integral curves of some ODEs between jumps. Our theorems describe limit behavior of the processes in the cases of frequent jumps and of fast motions in the direction of chosen variables. These results are motivated by Kepler--Elston's model of gene regulation and Lipniacki's model of gene expression. Application to other models, including those of mathematical economics, are also discussed.
Citation: Adam Bobrowski, Radosław Bogucki. Two theorems on singularly perturbed semigroups with applications to models of applied mathematics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 735-757. doi: 10.3934/dcdsb.2012.17.735
##### References:
 [1] J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis: A study of a singularly perturbed abstract telegraph system, J. Evol. Equ., 9 (2009), 293-314. doi: 10.1007/s00028-009-0009-7. [2] A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127. [3] A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583. [4] A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in, 77 (2008), 520-521. [5] A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336. doi: 10.1007/s00233-006-0676-4. [6] A. Bobrowski and R. Bogucki, Semigroups generated by convex combinations of several Feller generators in models of mathematical biology, Studia Math., 189 (2008), 287-300. doi: 10.4064/sm189-3-6. [7] A. Bobrowski, T. Lipniacki, K. Pichór and R. Rudnicki, Asymptotic behaviour of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769. doi: 10.1016/j.jmaa.2006.11.043. [8] B. M. Brown, D. Elliot and D. F. Paget, Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory, 49 (1987), 196-199. doi: 10.1016/0021-9045(87)90087-6. [9] M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. With discussion, J. R. Stat. Soc. Ser. B, 46 (1984), 353-388. [10] M. H. A. Davis, "Lectures on Stochastic Control and Nonlinear Filtering," Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 75, Published for the Tata Institute of Fundamental Research, Bombay, by Springer-Verlag, Berlin, 1984. [11] M. H. A. Davis, "Markov Processes and Optimization," Chapman and Hall, 1993. [12] R. M. Dudley, "Real Analysis and Probability," Revised reprind of the 1989 original, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2002. [13] S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. [14] W. H. Fleming and Q. Zhang, Risk-sensitive production planning of a stochastic manufacturing system, SIAM J. Control Optim., 36 (1998), 1147-1170. doi: 10.1137/S036301299631034X. [15] B. Hat, P. Paszek, M. Kimmel, K. Piechór and T. Lipniacki, How the number of alleles influences gene expression, J. Statist. Phys., 128 (2007), 511-533. doi: 10.1007/s10955-006-9218-4. [16] A. Haurie and F. Moresino, Singularly perturbed piecewise deterministic games, SIAM J. Control Optim., 47 (2008), 73-91. doi: 10.1137/050627599. [17] A. Haurie, A two-timescale stochastic game framework for climate change policy assessment, in "Dynamic Games: Theory and Applications" (eds. A. Haurie and G. Zaccour), GERAD 25th Anniv. Ser., 10, Springer, New York, (2005), 193-211. [18] E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups," rev. ed., American Mathematical Society Colloquium Publications, vol. 31, Amer. Math. Soc., Providence, R.I., 1957. [19] F. Hoppensteadt, Stability in systems with parameter, J. Math. Anal. Appl., 18 (1967), 129-134. doi: 10.1016/0022-247X(67)90187-4. [20] T. B. Kepler and T. C. Elston, Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations, Biophys. J., 81 (2001), 3116-3136. doi: 10.1016/S0006-3495(01)75949-8. [21] V. S. Korolyuk and A. F. Turbin, "Mathematical Foundations of the State Lumping of Large Systems," Translated from the 1978 Russian original by V. V. Zayats and Y. A. Atanov and revised by the authors, Mathematics and its Applications, 264, Kluwer Academic Publishers Group, Dordrecht, 1993. [22] T. G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Funct. Anal., 3 (1969), 354-375. doi: 10.1016/0022-1236(69)90031-7. [23] T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), 55-67. doi: 10.1016/0022-1236(73)90089-X. [24] T. G. Kurtz, Applications of an abstract perturbation theorem to ordinary differential equations, Houston J. Math., 3 (1977), 67-82. [25] P. D. Lax, "Linear Algebra and Its Applications," 2 edition, Pure and Applied Mathematics (Hoboken), Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2007. [26] T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A. R. Brasier and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol., 238 (2006), 348-367. doi: 10.1016/j.jtbi.2005.05.032. [27] T. Lipniacki, P. Paszek, A. R. Brasier, B. Luxon and M. Kimmel, Stochastic regulation in early immune response, Biophys. J., 90 (2006), 725-742. doi: 10.1529/biophysj.104.056754. [28] G. G. Lorentz, "Bernstein Polynomials," 2nd edition, Chelsea Publishing Co., New York, 1986. [29] J. L. Massera, On Liapounoff's conditions of stability, Ann. Math. (2), 50 (1949), 705-721. doi: 10.2307/1969558. [30] R. E. Megginson, "An Introduction to Banach Space Theory," Graduate Texts in Mathematics, 183, Springer-Verlag, New York, 1998. [31] C. Meyer, "Matrix Analysis and Applied Linear Algebra," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512. [32] J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory," Series on Advances in Mathematics for Applied Sciences, 34, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. [33] M. Rausand and A. Høyland, "System Reliability Theory: Models, Statistical Methods, and Applications," Second edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2004. [34] L. Saloff-Coste, Lectures on Finite Markov Chains, in "Lectures on Probability Theory and Statistics" (Saint-Flour, 1996), Lecture Notes in Mathematics, 1665, Springer, Berlin, 1997. [35] V. V. Sarafyan and A. V. Skorokhod, On fast switching dynamical systems, Theory Probab. Appl., 32 (1987), 595-607. doi: 10.1137/1132092. [36] V. V. Sarafyan, Weak convergence of measures and asymptotic behavior of solutions of the Cauchy problem for systems of differential equations with a small parameter, Theory Probab. Appl., 34 (1989), 351-357. doi: 10.1137/1134035. [37] S. Sethi, H. Zhang and Q. Zhang, "Average-Cost Control of Stochastic Manufacturing Systems," Stochastic Modelling and Applied Probability, 54, Springer-Verlag, New York, 2005. [38] M. Sova, Convergence d'opérations linéaires non bornées, (French) Rev. Roumaine Math. Pures Appl., 12 (1967), 373-389. [39] A. N. Tikhonov, Systems of differential equations containing small parameters in the derivatives, (Russian) Mat. Sb. N. S., 31(73) (1952), 575-586. [40] M. E. Taylor, "Partial Differential Equations. Basic Theory," Texts in Applied Mathematics, 23, Springer-Verlag, New York, 1996. [41] A. B. Vasil'eva and B. F. Butuzov, "Asymptotic Expansions of the Solutions of Singularly Perturbed Equations," (Russian) Izdat. "Nauka," Moscow, 1973. [42] W. Walter, "Differential and Integral Inequalities," Translated from German by Lisa Rosenblatt and Lawrence Shampine, Ergebnisse d. Mathematik u. ihrer Granzgebiete, Vol. 55, Springer-Verlag, New York-Berlin, 1970. [43] W. Walter, Differential inequalities and maximum principles: Theory, new methods and applications, in "Preceedings of the Second World Congress of Nonlinear Analysts, Part 8" (Athens, 1996), Nonlinear Anal., 30 (1997), 4695-4711. doi: 10.1016/S0362-546X(96)00259-3. [44] W. Walter, "Ordinary Differential Equations," Translated from the sixth German (1996) edition by Russell Thompson, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998. [45] G. G. Yin and Q. Zhang, "Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach," Applications of Mathematics (New York), 37, Springer-Verlag, New York, 1998.

show all references

##### References:
 [1] J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis: A study of a singularly perturbed abstract telegraph system, J. Evol. Equ., 9 (2009), 293-314. doi: 10.1007/s00028-009-0009-7. [2] A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127. [3] A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583. [4] A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in, 77 (2008), 520-521. [5] A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336. doi: 10.1007/s00233-006-0676-4. [6] A. Bobrowski and R. Bogucki, Semigroups generated by convex combinations of several Feller generators in models of mathematical biology, Studia Math., 189 (2008), 287-300. doi: 10.4064/sm189-3-6. [7] A. Bobrowski, T. Lipniacki, K. Pichór and R. Rudnicki, Asymptotic behaviour of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769. doi: 10.1016/j.jmaa.2006.11.043. [8] B. M. Brown, D. Elliot and D. F. Paget, Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory, 49 (1987), 196-199. doi: 10.1016/0021-9045(87)90087-6. [9] M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. With discussion, J. R. Stat. Soc. Ser. B, 46 (1984), 353-388. [10] M. H. A. Davis, "Lectures on Stochastic Control and Nonlinear Filtering," Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 75, Published for the Tata Institute of Fundamental Research, Bombay, by Springer-Verlag, Berlin, 1984. [11] M. H. A. Davis, "Markov Processes and Optimization," Chapman and Hall, 1993. [12] R. M. Dudley, "Real Analysis and Probability," Revised reprind of the 1989 original, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2002. [13] S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. [14] W. H. Fleming and Q. Zhang, Risk-sensitive production planning of a stochastic manufacturing system, SIAM J. Control Optim., 36 (1998), 1147-1170. doi: 10.1137/S036301299631034X. [15] B. Hat, P. Paszek, M. Kimmel, K. Piechór and T. Lipniacki, How the number of alleles influences gene expression, J. Statist. Phys., 128 (2007), 511-533. doi: 10.1007/s10955-006-9218-4. [16] A. Haurie and F. Moresino, Singularly perturbed piecewise deterministic games, SIAM J. Control Optim., 47 (2008), 73-91. doi: 10.1137/050627599. [17] A. Haurie, A two-timescale stochastic game framework for climate change policy assessment, in "Dynamic Games: Theory and Applications" (eds. A. Haurie and G. Zaccour), GERAD 25th Anniv. Ser., 10, Springer, New York, (2005), 193-211. [18] E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups," rev. ed., American Mathematical Society Colloquium Publications, vol. 31, Amer. Math. Soc., Providence, R.I., 1957. [19] F. Hoppensteadt, Stability in systems with parameter, J. Math. Anal. Appl., 18 (1967), 129-134. doi: 10.1016/0022-247X(67)90187-4. [20] T. B. Kepler and T. C. Elston, Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations, Biophys. J., 81 (2001), 3116-3136. doi: 10.1016/S0006-3495(01)75949-8. [21] V. S. Korolyuk and A. F. Turbin, "Mathematical Foundations of the State Lumping of Large Systems," Translated from the 1978 Russian original by V. V. Zayats and Y. A. Atanov and revised by the authors, Mathematics and its Applications, 264, Kluwer Academic Publishers Group, Dordrecht, 1993. [22] T. G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Funct. Anal., 3 (1969), 354-375. doi: 10.1016/0022-1236(69)90031-7. [23] T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), 55-67. doi: 10.1016/0022-1236(73)90089-X. [24] T. G. Kurtz, Applications of an abstract perturbation theorem to ordinary differential equations, Houston J. Math., 3 (1977), 67-82. [25] P. D. Lax, "Linear Algebra and Its Applications," 2 edition, Pure and Applied Mathematics (Hoboken), Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2007. [26] T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A. R. Brasier and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol., 238 (2006), 348-367. doi: 10.1016/j.jtbi.2005.05.032. [27] T. Lipniacki, P. Paszek, A. R. Brasier, B. Luxon and M. Kimmel, Stochastic regulation in early immune response, Biophys. J., 90 (2006), 725-742. doi: 10.1529/biophysj.104.056754. [28] G. G. Lorentz, "Bernstein Polynomials," 2nd edition, Chelsea Publishing Co., New York, 1986. [29] J. L. Massera, On Liapounoff's conditions of stability, Ann. Math. (2), 50 (1949), 705-721. doi: 10.2307/1969558. [30] R. E. Megginson, "An Introduction to Banach Space Theory," Graduate Texts in Mathematics, 183, Springer-Verlag, New York, 1998. [31] C. Meyer, "Matrix Analysis and Applied Linear Algebra," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512. [32] J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory," Series on Advances in Mathematics for Applied Sciences, 34, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. [33] M. Rausand and A. Høyland, "System Reliability Theory: Models, Statistical Methods, and Applications," Second edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2004. [34] L. Saloff-Coste, Lectures on Finite Markov Chains, in "Lectures on Probability Theory and Statistics" (Saint-Flour, 1996), Lecture Notes in Mathematics, 1665, Springer, Berlin, 1997. [35] V. V. Sarafyan and A. V. Skorokhod, On fast switching dynamical systems, Theory Probab. Appl., 32 (1987), 595-607. doi: 10.1137/1132092. [36] V. V. Sarafyan, Weak convergence of measures and asymptotic behavior of solutions of the Cauchy problem for systems of differential equations with a small parameter, Theory Probab. Appl., 34 (1989), 351-357. doi: 10.1137/1134035. [37] S. Sethi, H. Zhang and Q. Zhang, "Average-Cost Control of Stochastic Manufacturing Systems," Stochastic Modelling and Applied Probability, 54, Springer-Verlag, New York, 2005. [38] M. Sova, Convergence d'opérations linéaires non bornées, (French) Rev. Roumaine Math. Pures Appl., 12 (1967), 373-389. [39] A. N. Tikhonov, Systems of differential equations containing small parameters in the derivatives, (Russian) Mat. Sb. N. S., 31(73) (1952), 575-586. [40] M. E. Taylor, "Partial Differential Equations. Basic Theory," Texts in Applied Mathematics, 23, Springer-Verlag, New York, 1996. [41] A. B. Vasil'eva and B. F. Butuzov, "Asymptotic Expansions of the Solutions of Singularly Perturbed Equations," (Russian) Izdat. "Nauka," Moscow, 1973. [42] W. Walter, "Differential and Integral Inequalities," Translated from German by Lisa Rosenblatt and Lawrence Shampine, Ergebnisse d. Mathematik u. ihrer Granzgebiete, Vol. 55, Springer-Verlag, New York-Berlin, 1970. [43] W. Walter, Differential inequalities and maximum principles: Theory, new methods and applications, in "Preceedings of the Second World Congress of Nonlinear Analysts, Part 8" (Athens, 1996), Nonlinear Anal., 30 (1997), 4695-4711. doi: 10.1016/S0362-546X(96)00259-3. [44] W. Walter, "Ordinary Differential Equations," Translated from the sixth German (1996) edition by Russell Thompson, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998. [45] G. G. Yin and Q. Zhang, "Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach," Applications of Mathematics (New York), 37, Springer-Verlag, New York, 1998.
 [1] Marek Bodnar. Distributed delays in Hes1 gene expression model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2125-2147. doi: 10.3934/dcdsb.2019087 [2] Somkid Intep, Desmond J. Higham. Zero, one and two-switch models of gene regulation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 495-513. doi: 10.3934/dcdsb.2010.14.495 [3] William Chad Young, Adrian E. Raftery, Ka Yee Yeung. A posterior probability approach for gene regulatory network inference in genetic perturbation data. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1241-1251. doi: 10.3934/mbe.2016041 [4] Pavol Bokes. Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5539-5552. doi: 10.3934/dcdsb.2019070 [5] Yun Li, Fuke Wu, George Yin. Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4417-4443. doi: 10.3934/dcdsb.2019125 [6] Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934/mbe.2017065 [7] Pavol Bokes. Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2129-2145. doi: 10.3934/dcdsb.2021126 [8] Jian Ren, Feng Jiao, Qiwen Sun, Moxun Tang, Jianshe Yu. The dynamics of gene transcription in random environments. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3167-3194. doi: 10.3934/dcdsb.2018224 [9] Vincent Renault, Michèle Thieullen, Emmanuel Trélat. Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks and Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019 [10] Ö. Uğur, G. W. Weber. Optimization and dynamics of gene-environment networks with intervals. Journal of Industrial and Management Optimization, 2007, 3 (2) : 357-379. doi: 10.3934/jimo.2007.3.357 [11] Erik Kropat, Gerhard-Wilhelm Weber, Erfan Babaee Tirkolaee. Foundations of semialgebraic gene-environment networks. Journal of Dynamics and Games, 2020, 7 (4) : 253-268. doi: 10.3934/jdg.2020018 [12] Ying Hao, Fanwen Meng. A new method on gene selection for tissue classification. Journal of Industrial and Management Optimization, 2007, 3 (4) : 739-748. doi: 10.3934/jimo.2007.3.739 [13] Qi Wang, Lifang Huang, Kunwen Wen, Jianshe Yu. The mean and noise of stochastic gene transcription with cell division. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1255-1270. doi: 10.3934/mbe.2018058 [14] Kai Wang, Hongyong Zhao, Hao Wang. Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022122 [15] Baltazar D. Aguda, Ricardo C.H. del Rosario, Michael W.Y. Chan. Oncogene-tumor suppressor gene feedback interactions and their control. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1277-1288. doi: 10.3934/mbe.2015.12.1277 [16] Zizi Wang, Zhiming Guo, Huaqin Peng. Dynamical behavior of a new oncolytic virotherapy model based on gene variation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1079-1093. doi: 10.3934/dcdss.2017058 [17] Feng Jiao, Qiwen Sun, Genghong Lin, Jianshe Yu. Distribution profiles in gene transcription activated by the cross-talking pathway. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2799-2810. doi: 10.3934/dcdsb.2018275 [18] Kunwen Wen, Lifang Huang, Qiuying Li, Qi Wang, Jianshe Yu. The mean and noise of FPT modulated by promoter architecture in gene networks. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2177-2194. doi: 10.3934/dcdss.2019140 [19] Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313 [20] Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049

2021 Impact Factor: 1.497