Qualitative properties of partial differential equations and functional differential equations  Page description

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Details of project

  
Identifier
49819
Type K
Principal investigator Simon, László
Title in Hungarian Parciális differenciálegyenletek és funkcionál differenciálegyenletek kvalitatív vizsgálata
Title in English Qualitative properties of partial differential equations and functional differential equations
Panel Mathematics and Computing Science
Department or equivalent Applied Analysis and Computational Mathematics (Eötvös Loránd University)
Participants Besenyei, Ádám
Faragó, István
Garay, Barnabás
Simon, Péter
Starting date 2005-01-01
Closing date 2009-06-30
Funding (in million HUF) 9.750
FTE (full time equivalent) 4.76
state closed project





 

Final report

  
Results in Hungarian
1. A nemlineáris parabolikus parciális differenciálegyenletek témájában olyan egyenletek és rendszerek megoldásának létezését, egyértelműségét és kvalitatív tulajdonságait vizsgáltuk, amelyek az ismeretlen függvénytől "nem-lokálisan függő tagokat is tartalmaznak (pl. azok integrálját tartalmazzák). 2. A kutatási időszak első felében vizsgáltunk lángterjedést leíró reakció-diffúzió egyenleteket, melyek esetében főleg az utazó hullám megoldások létezését és stabilitását tanulmányoztuk. Az elliptikus egyenletekkel kapcsolatban a pozitív megoldások számát vizsgáltuk. Két nyitott problémát is sikerült megoldani a szinguláris egyenletekkel kapcsolatban. 3. Megvizsgáltuk a parabolikus típusú parciális differenciálegyenletek véges differenciás és lineáris véges elemes diszkretizációjának kvalitatív tulajdonságait.A Richardson extrapolácó segítségével új eredményeket értünk el az operátor szeletelés módszerében is, és elemeztük ezek alkalmazhatóságát modell jellegű, illetve valós légszennyezési feladatokon. 4. A numerikus dinamika területén elért eredmények a diszkretizációk kvalitatív-geometriai elméletét gazdagítják. Továbbá fontos eredményeket nyertünk a számítógéppel segített káosz bizonyításokban. Ennek során nem volt szükség a korábbi bizonyítások lényegi részét képező fokszám-, illetve indexelméleti megfontolásokra, csak a Brouwer-féle fixpont tételt alkalmaztuk.
Results in English
1. In the field of nonlinear parabolic partial differential equations we proved existence, uniqueness and qualitative properties of solutions to equations and systems, containing "non-local" terms (e.g. integrals of the unknown functions). 2. In the first part of the period of research we investigated reaction-diffusion equations describing flame propagation. We studied mainly the existence and stability of traveling wave solutions. Concerning elliptic equations we studied exact multiplicity of positive solutions. We could solve two open problems related to singular equations. 3. We have considered the qualitative properties of the discrete models for the linear parabolic equations, obtained by the finite difference and linear finite element discretizations. Using the Richardson extrapolation, we constructed several new operator splitting methods. We analyzed the applicability of these methods to model problems and to the real-life air pollution models. 4. In the field of numerical dynamics our results are connected with the qualitative-geometrical theory of discretizations. Further, we have given computer assisted proofs on chaotic behavior of certain systems. The proofs avoid of referring to any results of applied algebraic topology and rely only on the Brouwer fixed point theorem.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=49819
Decision
Yes





 

List of publications

  
I. Faragó, J. Geiser: Iterative operator-splitting methods for linear problems, Int. J. Comp. Sci. Eng., to appear, 2005
I. Dimov, I. Faragó, Á. Havasi, Z. Zlatev: Different splitting techniques with application to air pollution models, Int. J. Environmental Pollution, to appear, 2005
P.L. Simon, S. Kalliadasis, J.H. Merkin, S.K. Scott: The effect of a radial scavenger on the propagation of flames in an exothermic-endothermic system, J. Math. Chem. 38, 203-231., 2005
L. Simon, W. Jaeger: On a system of quasilinear parabolic functional differential equations, Acta Math. Hung., 112, 39-55, 2006
L. Simon: On quasilinear parabolic functional differential equations with discontinuous coefficients, Annales Univ. Sci. Budapest, 47, 211-229., 2005
J. Hernandez, J. Karátson, P.L. Simon: Multiplicity for semilinear elliptic equations involving singular nonlinearity, J. Nonlin. Anal., 65, 265-283, 2006
L. Simon: On some properties of nonlinear functional parabolic equations, Int. J. of Qualitative Theory of Diff. Equations and Appl. , to appear, 2008
I. Faragó, R. Horváth, S. Korotov: Discrete maximum principle for linear parabolic problems solved on hybrid meshes, Appl. Numer. Math. 43, 249-264, 2005
I. Faragó, Á Havasi, K. Georgiev (editors): Advances in Air Pollution Modelling for Environmental Security, NATO Science Series, 54, Springer, 406 p., 2005
I. Faragó, R. Horváth, W. Schilders: Investigation of numerical time integrations of the Maxwell equations using the staggered grid spatial discretization, Int. J. Num. Modelling, 18, 149-169, 2005
I. Faragó, J. Karátson: Gradient--finite element method for Saint-Venant model of elasto-plastic torsion in the hardening state, J. Comp. Meth. Sci. Eng., to appear, 2005
I. Faragó: Splitting methods for abstract Cauchy problems, Numerical Analysis and Its Application, Lect. Notes Comp. Sci. 3401, Springer, 35-45., 2005
I. Faragó: Lax equivalence theorem in Banach spaces with Parameter, Tübinger Berichte zur Funktionalanalysis, 14, 89-95., 2005
P.L. Simon: Classification of positive convex functions according to focal equivalence, IMA J. Appl. Math., 71, 519-533, 2006
L. Simon, W. Jaeger: On non-uniformly parabolic functional differential equations, Studia Sci. Math. Hung., 45, 285-300, 2008
I. Faragó, R. Horváth: A review of reliable numerical models for three-dimensional linear parabolic problems, Int. J. Numer. Meth. Engng., 70, 25-45, 2007
I. Faragó, Á. Havasi: Consistency analysis of operator splitting methods for Co-semigroups, Semigroup Forum, to appear, 2006
I. Faragó, Á. Havasi: Consistency analysis of operator splitting methods for Co-semigroups, Semigroup Forum, to appear, 2006
L. Simon: On a system with a singular parabolic equation, Folia FSN Universitatis Masarykianae Brunensis, Math., 16, 149-156, 2007
I. Faragó, R. Horváth: On the connections between the qualitative properties of the numerical solutions of linear parabolic problems, SIAM Sci. Comp. 28, 2313-2336, 2006
J. Englaender, P.L. Simon: Nonexistence of solutions to KKP-type equations of dimension greater than or equal to one, Electron. J. Diff. Eqns., 09, 1-6, 2006
P.L. Simon, A. Volford: Detailed study of limit cycles and global bifurcations in a circadian rhythm model, Int. J. Bif. Chaos 16, 349-367, 2006
P.L. Simon: Transverse instability of non-adiabatic flames, Lect. Ser. Computer and Computational Sc.., 1398-1401, 2006
Á. Besenyei: Existence of solutions of a nonlinear system modelling fluid flow in porous media, Electron. J. Diff. Eqsn., 153, 1-19, 2006
B. Bánhelyi, T. Csendes, B.M. Garay: A verified optimization technique to locate chaotic regions of a Henon system, J. Global Optimization 35, 145-160, 2006
B. Bánhelyi, T. Csendes, B.M. Garay: Optimization and the Miranda approach in detecting horseshoe-type chaos by computer, Int. J. of Bif. and Chaos, 17, 735-747, 2007
B.M. Garay, J. Vardai: Interpolation in dynamic equations on time scales, J. of Difference Equations and Appl., 13, 847-854, 2007
L. Simon: On nonlinear systems consisting of different types of differential equations, Periodica Math. Hung., 56, 143-156, 2008
L. Simon: Application of monotone type operators to parabolic and functional parabolic PDE's, Handbook of Differential Equations, Evolutionary Partial Differential Equations, Vol. 4, 267-321, Elsevier, 2008
L. Simon: On qualitative properties of a system containing singular partial functional equations, Electronic J. of Qualitative Theory of Diff. Equatsn., 20, 1-13, 2008
Á. Besenyei: Stabilization of solutions to a nonlinear system modelling fluid flow in porous media, Annales Univ. Sci. Budapest, 49, 115-136., 2006
Á. Besenyei: On a nonlinear system containing nonlocal terms related to a fluid flow model, Electronic J. of Qualitative Theory of Diff. Equatsn., 3, 1-13, 2008
B. Bánhelyi, T. Csendes, B.M. Garay: Rigorous lower bounds for the topological entropy via a verified optimization technique, Proceedings of the SCAN-2006 Conference, IEEE, pp. 10, 2007
I. Faragó, B. Gnandt, Á. Havasi: Additive and iterative splitting methods and their numerical investigation, Computers and Mathematics with Applications, 55, 2266-2279, 2008
P.L. Simon, J.H. Merkin, S.K. Scott: Bifurcations in non-adiabatic flame propagation models, Focus on Combustion Research, Nova Science Publishers, New York, 315-357, 2006
I.Z. Kiss, P.L. Simon, R.R. Kao: Epidemic outbreaks in preferentially mixed populations, Bull. Math. Biol., to appear, 2008
I. Faragó, P. Vabisevich, L. Vulkov (editors): Finite Difference Methods: Theory and Application, Rousse University Angel Kanchev, 352 p., 2007
I. Faragó, R. Horváth: Discrete Maximum principle and adequate discretizations of linear parabolic problems, SIAM Scientific Computing, 28, 2313-2336, 2006
I. Faragó: Application of the operator splitting method for real-life problems, Időjárás, Quart. J. HMS, 110, 379-395, 2006
I. Faragó: New operator splittings and their applications, T. Boyanov et al. eds., Numerical Methods and Application, Lect. Notes Comp. Sci. 4310, Springer, Nerlin, 443-450., 2007
Á. Elbert, B.M. Garay: Hungary, the extended first half of the 20th century, A Panorama of Hungarian Mathematics in the Twentieth Century, Springer, 245-294, 2005
B.M. Garay, L. Lóczi: Discretizing the fold bifurcation - a conjugacy result, Periodica Math. Hung., 56, 37-53, 2008
B.M. Garay, L. Lóczi: Discretizing the fold bifurcation - a conjugacy result, Periodica Math. Hung., 56, 37-53, 2008
B.M. Garay: A brief survay on the numerical dynamics for functional differential equations, Int. J. Bifurc. Chaos, 15, 729-742, 2005
B.M. Garay, B. Bánhelyi, T. Csendes, L. Hatvani: A computer-assisted proof for Sigma_3 chaos in the forced damped pendulum equation, SIAM J. on Appl. Dyn. Systems, 7, 843-867, 2008
B.M. Garay, B. Bánhelyi, T. Csendes, L. Hatvani: A computer-assisted proof for Sigma_3 chaos in the forced damped pendulum equation, SIAM J. on Appl. Dyn. Systems, 7, 843-867, 2008
B.M. Garay, P.L. Simon: Bifurcation analysis of an oscillatory CNN model with two cells, J. Appl. Math. Comp., 27, 199-210, 2008
B.M. Garay, R. Csikja, J. Tóth: Some chaotic properties of the beta-hysteresis transformation, Proc. of the NOLTA 2008 Conference, Budapest, 191-194, 2008
B.M. Garay, B. Bánhelyi, T. Csendes: A computer-assisted proof of chaotic behavior of the area-preserving Henon mapping, Proc. of the NOLTA 2008 Conference, Budapest, 596-599, 2008
B.M. Garay, G. Colombo, M. Feckan: Multivalued perturbations of a saddle dynamics, Diff. Eq. Dyn. Syst., to appear, 2008
I. Bartha, P.L. Simon, V. Müller: Has HIV evolved to induce immune pthogenesis?, Trends in Immunology, 29, 322-328, 2008
I. Bartha, P.L. Simon, V. Müller: Has HIV evolved to induce immune pthogenesis?, Trends in Immunology, 29, 322-328, 2008
T. Horváth, P.L. Simon: On the exact number of solutions of a singular boundary value problem, J. Diff. Int. Equ., to appear, 2008
L. Simon: On nonlinear functional parabolic equations, Proc. IMSE2008 Conference, to appear, 2009
Á. Besenyei: On nonlinear parabolic variational inequalities containing nonlocal terms, Acta Math. Hungar., 116, 145-162, 2007
I. Faragó, K. Georgiev, P.G. Thomsen, Z. Zlatev (editors9: Numerical Methods and Applications, Applied Math. Modelling, 32, N. 8, 2008
I. Faragó, Á. Havasi, Z. Zlatev (editors): Advanced Numerical Algorithms for Large-Scale Computations, An Int. J. Computers and Mathematics with Applications, 55, N. 10, 2008
I. Faragó, P. Simon, Z. Zlatev (editors): Large-Scale Scientific Computations, J. of Computational and Appl. Math., 226, N. 2, 2009
I. Faragó, J. Karátson: Gradient-finite element method for the Saint-Venant model of elasto-plastic torsion in the hardening state, Int. J. of Numerical Analysis and Modelling, 5, 206-222, 2008
I. Dimov, I. Faragó, Á. Havasi, Z. Zlatev: Different splitting techniques with application to air pollution models, Int. J. Environmental Pollution, 32, 174-199, 2008
P. Csomós, I. Faragó: Error analysis of the numerical solution of split differential equations, Mathematical and Computer Modelling, 48, 1090-1106, 2008
I. Faragó, Á. Havasi: Relationship between vanishing splitting errors and pairwise commutativity, Applied Math. Letters, 21, 10-14, 2008
I. Faragó: A modified iterated operator splittig method, Applied Math. Modelling, 32, 1542-1551., 2008
I. Faragó, P. Thomsen, Z. Zlatev: On the additive splitting procedures and their computer realization, Applied Math. Modelling, 32, 1552-1569., 2008
I. Faragó, R. Horváth: Qualitative properties of monotone linear operators, El. J. of Qualitative Theory of Diff. Equ., 8, 1-15, 2008
I. Faragó: Operator splitting procedures and their analysis, Alkalmazott Mat. Lapok, 25, 1-18, 2008
I. Faragó, G. Inzelt, M. Kornyik, Á. Kriston, T. Szabó: Stabilization of a numerical model through the boundary conditions for the real-time simulation of fuel cells, Innovations and Advanced Techniques in Systems, Computing Sciences and Software Engineering, Springer, 489-494, 2008
B.M. Garay, B. Bánhelyi, T. Csendes, Hatvani: Computer assisted proof of chaotic behavior of the forced dampted pendulum, Folia FSN Universitatis Brunensis 16, 9-20, 2006
B.M. Garay, B. Bánhelyi, T. Csendes: Rigirous lower bounds for the topological entropy via a verified optimization technique, Proc. Symp.Sci. Comp., Comp. Arithmetic and Validated Numerics, Duisburg, 2006, pp. 10, 2007
B.M. Garay, L. Hatvani, J. Kolumbán: The centennary of L. Fejér's Kolozsvár University habilitation lecture in stability theory (in Hungarian), Alk. Mat. Lapok 23, 163-189, 2006
B.M. Garay, Bánhelyi, Csendes: Rigorous lower bounds for the topological entropy via a verified optimization technique, Proc. Symp. Sci. Comp., Comp. Arihmetic and Validated Numerics, Duisburg, 2006, pp. 10, 2007
B.M. Garay, K.H. Lee: Attractorsand invariant manifolds under discretization with variable stepsize, Discrete Contin. Dyn. Syst. 13, 827-841, 2005
B.M. Garay, L.O. Chua: Isles of Eden and the ZUK Theorem in R^d, Internat. J. Bifurc. Chaos, 18, 2951-2963, 2008



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