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Relevant bibliographies by topics / Singularly perturbed problem

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Academic literature on the topic 'Singularly perturbed problem'

Author: Grafiati

Published: 4 June 2021

Last updated: 19 February 2022

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Journal articles on the topic "Singularly perturbed problem"

1

Vrábeľ, Róbert. "Quasilinear and quadratic singularly perturbed Neumann's problem." Mathematica Bohemica 123, no. 4 (1998): 405–10. http://dx.doi.org/10.21136/mb.1998.125970.

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2

Yang, Lufeng. "The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data." Mathematical Problems in Engineering 2021 (February 26, 2021): 1–11. http://dx.doi.org/10.1155/2021/8844163.

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A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. A novel rational spectral collocation combined with a singularity-separated method for this problem is presented. The solution is expressed as u = w + v , where w is the solution of corresponding auxiliary boundary value problem and v is a singular correction with direct expressions. The rational spectral collocation method combined with a sinh transformation is applied to solve the weakened singularly boundary value problem. According to the asymptotic analysis, the sinh transformation parameters can be determined by the width and position of the boundary layers. The parameters in the singular correction can be determined by the boundary conditions of the original problem. Numerical experiment supports theoretical results and shows that compared with previous research results, the novel method has the advantages of a high computational accuracy in singularly perturbed reaction-diffusion problems with nonsmooth data.

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Vrábeľ, Róbert. "Upper and lower solutions for singularly perturbed semilinear Neumann's problem." Mathematica Bohemica 122, no. 2 (1997): 175–80. http://dx.doi.org/10.21136/mb.1997.125912.

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Zholtikov, Vitaliy P., and Vladislav V. Efendiev. "Singularly Perturbed Control with Delay Problem." Journal of Automation and Information Sciences 29, no. 1 (1997): 40–43. http://dx.doi.org/10.1615/jautomatinfscien.v29.i1.50.

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5

Gol’dberg, V. N. "Stability of a singularly perturbed problem." Differential Equations 48, no. 4 (2012): 524–37. http://dx.doi.org/10.1134/s0012266112040076.

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Li, Gongbao, Peng Luo, Shuangjie Peng, Chunhua Wang, and Chang-Lin Xiang. "A singularly perturbed Kirchhoff problem revisited." Journal of Differential Equations 268, no. 2 (2020): 541–89. http://dx.doi.org/10.1016/j.jde.2019.08.016.

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Zhumanazarova, Assiya, and Young Im Cho. "Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem." Mathematics 8, no. 2 (2020): 213. http://dx.doi.org/10.3390/math8020213.

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In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.

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Grossmann, Christian, Lars Ludwig, and Hans-Görg Roos. "Layer-adapted methods for a singularly perturbed singular problem." Computational Methods in Applied Mathematics 11, no. 2 (2011): 192–205. http://dx.doi.org/10.2478/cmam-2011-0010.

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Abstract In the present paper we analyze linear finite elements on a layer adapted mesh for a boundary value problem characterized by the overlapping of a boundary layer with a singularity. Moreover, we compare this approach numerically with the use of adapted basis functions, in our case modified Bessel functions. It turns out that as well adapted meshes as adapted basis functions are suitable where for our one-dimensional problem adapted bases work slightly better.

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9

Shi, Y. M. "A Singular Singularly Perturbed Nonlinear Vector Boundary Value Problem." Journal of Mathematical Analysis and Applications 187, no. 3 (1994): 919–42. http://dx.doi.org/10.1006/jmaa.1994.1398.

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Wang, Ai-feng, and Ming-kang Ni. "Contrast structure for singular singularly perturbed boundary value problem." Applied Mathematics and Mechanics 35, no. 5 (2014): 655–66. http://dx.doi.org/10.1007/s10483-014-1819-7.

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Dissertations / Theses on the topic "Singularly perturbed problem"

1

Kunert, Gerd. "A note on the energy norm for a singularly perturbed model problem." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100062.

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A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.

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2

Adkins, Jacob. "A Robust Numerical Method for a Singularly Perturbed Nonlinear Initial Value Problem." Kent State University Honors College / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=ksuhonors1513331499579714.

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3

Kunert, Gerd. "Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100011.

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Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

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Grosman, Serguei. "Robust local problem error estimation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600475.

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Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reaction-diffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.

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Kunert, Gerd. "A posteriori H^1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100730.

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The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

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Kunert, Gerd. "Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes." Universitätsbibliothek Chemnitz, 2000. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200000867.

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We consider a singularly perturbed reaction-diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.

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Meng, Jianzhong. "Some singular singularly perturbed problems." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp04/mq24686.pdf.

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Kunert, Gerd. "A posteriori error estimation for convection dominated problems on anisotropic meshes." Universitätsbibliothek Chemnitz, 2002. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200200255.

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A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.

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Jung, Chang-Yeol. "Numerical approximation of two dimensional singularly perturbed problems." [Bloomington, Ind.] : Indiana University, 2006. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3215193.

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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2006.<br>Source: Dissertation Abstracts International, Volume: 67-04, Section: B, page: 2029. Adviser: Roger Temam. "Title from dissertation home page (viewed June 20, 2007)."

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Franz, Sebastian. "Singularly perturbed problems with characteristic layers : Supercloseness and postprocessing." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2008. http://nbn-resolving.de/urn:nbn:de:bsz:14-ds-1218629566251-73654.

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In this thesis singularly perturbed convection-diffusion equations in the unit square are considered. Due to the presence of a small perturbation parameter the solutions of those problems exhibit an exponential layer near the outflow boundary and two parabolic layers near the characteristic boundaries. Discretisation of such problems on standard meshes and with standard methods leads to numerical solutions with unphysical oscillations, unless the mesh size is of order of the perturbation parameter which is impracticable. Instead we aim at uniformly convergent methods using layer-adapted meshes combined with standard methods. The meshes considered here are S-type meshes--generalisations of the standard Shishkin mesh. The domain is dissected in a non-layer part and layer parts. Inside the layer parts, the mesh might be anisotropic and non-uniform, depending on a mesh-generating function. We show, that the unstabilised Galerkin finite element method with bilinear elements on an S-type mesh is uniformly convergent in the energy norm of order (almost) one. Moreover, the numerical solution shows a supercloseness property, i.e. the numerical solution is closer to the nodal bilinear interpolant than to the exact solution in the given norm. Unfortunately, the Galerkin method lacks stability resulting in linear systems that are hard to solve. To overcome this drawback, stabilisation methods are used. We analyse different stabilisation techniques with respect to the supercloseness property. For the residual-based methods Streamline Diffusion FEM and Galerkin Least Squares FEM, the choice of parameters is addressed additionally. The modern stabilisation technique Continuous Interior Penalty FEM--penalisation of jumps of derivatives--is considered too. All those methods are proved to possess convergence and supercloseness properties similar to the standard Galerkin FEM. With a suitable postprocessing operator, the supercloseness property can be used to enhance the accuracy of the numerical solution and superconvergence of order (almost) two can be proved. We compare different postprocessing methods and prove superconvergence of above numerical methods on S-type meshes. To recover the exact solution, we apply continuous biquadratic interpolation on a macro mesh, a discontinuous biquadratic projection on a macro mesh and two methods to recover the gradient of the exact solution. Special attentions is payed to the effects of non-uniformity due to the S-type meshes. Numerical simulations illustrate the theoretical results.

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Books on the topic "Singularly perturbed problem"

1

Boglaev, Igor. Domain decomposition in boundary layers for a singularly perturbed parabolic problem. Faculty of Information and Mathematical Sciences, Massey University, 1997.

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2

Gheorghe, Moroșanu, ed. Singularly perturbed boundary-value problems. Birkhäuser Verlag, 2007.

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3

Barbu, Luminiţa, and Gheorghe Moroşanu. Singularly Perturbed Boundary-Value Problems. Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8331-2.

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4

Mazia, V. G. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Springer Basel, 2000.

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5

Weak convergence methods and singularly perturbed stochastic control and filtering problems. Birkhäuser, 1990.

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Maz’ya, Vladimir, Serguei Nazarov, and Boris A. Plamenevskij. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhäuser Basel, 2000. http://dx.doi.org/10.1007/978-3-0348-8432-7.

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Maz’ya, Vladimir, Serguei Nazarov, and Boris A. Plamenevskij. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhäuser Basel, 2000. http://dx.doi.org/10.1007/978-3-0348-8434-1.

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8

Kushner, Harold J. Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4612-4482-0.

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9

Mazʹi︠a︡, V. G. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Birkhäuser Verlag, 2000.

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10

Roos, Hans-Görg. Numerical methods for singularly perturbed differential equations: Convection-diffusion and flow problems. Springer-Verlag, 1996.

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Book chapters on the topic "Singularly perturbed problem"

1

Wasow, Wolfgang. "A Singularly Perturbed Turning Point Problem." In Linear Turning Point Theory. Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4612-1090-0_11.

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2

Cai, Chenxiao, Zidong Wang, Jing Xu, and Yun Zou. "The Sensitivity-Shaping Problem for Singularly Perturbed Systems." In Finite Frequency Analysis and Synthesis for Singularly Perturbed Systems. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-45405-4_6.

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Filar, Jerzy A., and Ke Liu. "Hamiltonian cycle problem and singularly perturbed Markov decision process." In Institute of Mathematical Statistics Lecture Notes - Monograph Series. Institute of Mathematical Statistics, 1996. http://dx.doi.org/10.1214/lnms/1215453564.

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Rai, Pratima, and Kapil K. Sharma. "Singularly Perturbed Convection-Diffusion Turning Point Problem with Shifts." In Mathematical Analysis and its Applications. Springer India, 2015. http://dx.doi.org/10.1007/978-81-322-2485-3_31.

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Banasiak, Jacek, and Mirosław Lachowicz. "Asymptotic Expansion Method in a Singularly Perturbed McKendrick Problem." In Methods of Small Parameter in Mathematical Biology. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05140-6_5.

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Kushner, Harold J. "The Nonlinear Filtering Problem." In Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4612-4482-0_6.

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O’Riordan, E., and J. Quinn. "Numerical Method for a Nonlinear Singularly Perturbed Interior Layer Problem." In Lecture Notes in Computational Science and Engineering. Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-19665-2_20.

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Haber, S., and N. Levinson. "A Boundary Value Problem for A Singularly Perturbed Differential Equation." In Selected Papers of Norman Levinson. Birkhäuser Boston, 1998. http://dx.doi.org/10.1007/978-1-4612-5332-7_35.

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Levinson, N. "A Boundary Value Problem for A Singularly Perturbed Differential Equation." In Selected Papers of Norman Levinson. Birkhäuser Boston, 1998. http://dx.doi.org/10.1007/978-1-4612-5332-7_36.

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Haber, S., and N. Levinson. "A Boundary Value Problem for a Singularly Perturbed Differential Equation." In Selected Papers of Norman Levinson Volume 1. Birkhäuser Boston, 1998. http://dx.doi.org/10.1007/978-1-4612-5341-9_35.

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Conference papers on the topic "Singularly perturbed problem"

1

Dalla Riva, M., and M. Lanza de Cristoforis. "Singularly perturbed loads for a nonlinear traction boundary value problem on a singularly perturbed domain." In Proceedings of the 7th International ISAAC Congress. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814313179_0004.

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2

Sobolev, Vladimir. "Decomposition of Traveling Wave Existence Problem for Singularly Perturbed Equations." In 2020 International Conference on Information Technology and Nanotechnology (ITNT). IEEE, 2020. http://dx.doi.org/10.1109/itnt49337.2020.9253204.

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3

Myshkov, Stanislav. "On the minimax approach in a singularly perturbed control problem." In 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA). IEEE, 2017. http://dx.doi.org/10.1109/cnsa.2017.7973993.

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Myshkov, Stanislav K., and Vladimir V. Karelin. "Minimax control in the singularly perturbed linear-quadratic stabilization problem." In 2015 International Conference "Stability and Control Processes" in Memory of V.I. Zubov (SCP). IEEE, 2015. http://dx.doi.org/10.1109/scp.2015.7342130.

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BOGLAEV, IGOR. "FINITE DIFFERENCE DOMAIN DECOMPOSITION FOR A SINGULARLY PERTURBED PARABOLIC PROBLEM." In Proceedings of the Fourth International Conference. WORLD SCIENTIFIC, 1999. http://dx.doi.org/10.1142/9789814291071_0050.

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WARD, MICHAEL J. "SPIKES FOR SINGULARLY PERTURBED REACTION-DIFFUSION SYSTEMS AND CARRIER’S PROBLEM." In Differential Equations & Asymptotic Theory in Mathematical Physics. WORLD SCIENTIFIC, 2004. http://dx.doi.org/10.1142/9789812702395_0003.

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Kuznetsov, Evgenii, Sergey Leonov, and Katherine Tsapko. "On the exact solution of a singularly perturbed aerodynamic problem." In COMPUTATIONAL MECHANICS AND MODERN APPLIED SOFTWARE SYSTEMS (CMMASS’2019). AIP Publishing, 2019. http://dx.doi.org/10.1063/1.5135674.

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Tulegenova, Bibigul, Zeynep Kambarova, and Aygerim Shaldanbaeva. "Solution of a singularly perturbed Cauchy problem by the similitude method." In INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2014). AIP Publishing LLC, 2014. http://dx.doi.org/10.1063/1.4893867.

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Bisht, Shilpi, and Riyaz A. Khan. "Parametric Quintic Spline Solution for Second Order Singularly Perturbed Boundary Value Problem." In 2011 International Conference on Communication Systems and Network Technologies (CSNT). IEEE, 2011. http://dx.doi.org/10.1109/csnt.2011.130.

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Sagara, Muneomi, Hiroaki Mukaidani, and Toru Yamamoto. "Numerical computation of linear quadratic control problem for singularly perturbed stochastic systems." In 2009 International Conference on Networking, Sensing and Control (ICNSC). IEEE, 2009. http://dx.doi.org/10.1109/icnsc.2009.4919367.

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Reports on the topic "Singularly perturbed problem"

1

Ferguson, Warren E., and Jr. Analysis of a Singularly-Perturbed Linear Two-Point Boundary-Value Problem. Defense Technical Information Center, 1986. http://dx.doi.org/10.21236/ada172582.

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2

Garbey, M., and H. G. Kaper. Heterogeneous domain decomposition for singularly perturbed elliptic boundary value problems. Office of Scientific and Technical Information (OSTI), 1995. http://dx.doi.org/10.2172/510563.

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3

Adjerid, Slimane, Mohammed Aiffa, and Joseph E. Flaherty. High-Order Finite Element Methods for Singularly-Perturbed Elliptic and Parabolic Problems. Defense Technical Information Center, 1993. http://dx.doi.org/10.21236/ada290410.

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4

Kushner, Harold J. Functional Occupation Measures and Ergodic Cost Problems for Singularly Perturbed Stochastic Systems. Defense Technical Information Center, 1989. http://dx.doi.org/10.21236/ada208578.

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Flaherty, Joseph E., and Robert E. O'Malley. Asymptotic and Numerical Methods for Singularly Perturbed Differential Equations with Applications to Impact Problems. Defense Technical Information Center, 1986. http://dx.doi.org/10.21236/ada169251.

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