Fractional differential equations and inverse problems.

Vista sencilla de ítem
dc.contributor.advisorMejía-Salazar, Carlos Enrique
dc.contributor.authorEcheverry Franco, Manuel Danilo
dc.contributor.researchgroupComputación Científicaspa
dc.date.accessioned2021-08-30T14:45:39Z
dc.date.available2021-08-30T14:45:39Z
dc.date.issued2020
dc.descriptionDiagramasspa
dc.description.abstractOur goal is the study of identification problems in the framework of transport equations with fractional derivatives. We consider time fractional diffusion equations and space fractional advection dispersion equations. The majority of inverse problems are ill-posed and require regularization. In this thesis we implement one and two dimensional discrete mollification as regularization procedures. The main original results are located in chapters 4 and 5 but chapter 2 and the appendices contain other material studied for the thesis, including several original proofs. The selected software tool is MATLAB and all the routines for numerical examples are original. Thus, the routines are part of the original results of the thesis. Chapters 1, 2 and 3 are introductions to the thesis, inverse problems and fractional derivatives respectively. They are survey chapters written specifically for this thesis.eng
dc.description.abstractNuestro objetivo es el estudio de problemas de identificación en el marco de ecuaciones de transporte con derivadas fraccionarias. Consideramos ecuaciones difusivas con derivada temporal fraccionaria y ecuaciones de advección dispersión con derivada espacial fraccionaria. La mayoría de los problemas inversos son mal condicionados y requieren regularización. En esta tesis implementamos procedimientos de regularización basados en molificación discreta en una y dos dimensiones. Los principales resultados originales se encuentran en los capítulos 4 y 5 pero el capítulo 2 y los apéndices contienen material adicional estudiado para la tesis incluídas varias demostra- ciones originales. La herramienta de software escogida es MATLAB y todas las rutinas para los ejemplos numéricos son originales, de manera que las rutinas son parte de los resultados originales de la tesis. Los capítulos 1, 2 y 3 son introductorios a la tesis, a los problemas inversos y a las derivadas fraccionarias respectivamente. Se trata de capítulos monográficos escritos especialmente para esta tesis. (Texto tomado de la fuente)spa
dc.description.degreelevelDoctoradospa
dc.description.degreenameDoctor en Ciencias - Matemáticasspa
dc.description.researchareaAnálisis Numéricospa
dc.description.sponsorshipConvocatoria 647 de Colcienciasspa
dc.format.extentix, 80 páginasspa
dc.format.mimetypeapplication/pdfspa
dc.identifier.instnameUniversidad Nacional de Colombiaspa
dc.identifier.reponameRepositorio Institucional Universidad Nacional de Colombiaspa
dc.identifier.repourlhttps://repositorio.unal.edu.co/spa
dc.identifier.urihttps://repositorio.unal.edu.co/handle/unal/80051
dc.language.isoengspa
dc.publisherUniversidad Nacional de Colombiaspa
dc.publisher.branchUniversidad Nacional de Colombia - Sede Medellínspa
dc.publisher.departmentEscuela de matemáticasspa
dc.publisher.facultyFacultad de Cienciasspa
dc.publisher.placeMedellínspa
dc.publisher.programMedellín - Ciencias - Doctorado en Ciencias - Matemáticasspa
dc.relation.referencesC. D. Acosta, R. Bürger, and C. E. Mejía. Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations. Numerical Methods for Partial Differential Equations, 28(1):38–62, 2012.spa
dc.relation.referencesC. D. Acosta, R. Bürger, and C. E. Mejía. A stability and sensitivity analysis of parametric functions in a sedimentation model. DYNA, 81(183):22–30, 2014.spa
dc.relation.referencesC. D. Acosta, R. Bürger, and C. E. Mejía. Efficient parameter estimation in a mocro- scopic traffic flow model by discrete mollification. Transportmetrica A: Transport Sci- ence, 11(8):702–715, 2015.spa
dc.relation.referencesC. D. Acosta and C. E. Mejía. Stabilization of explicit methods for convection diffusion equations by discrete mollification. Comput. Math. Appl., 55:368–380, 2008.spa
dc.relation.referencesC.D. Acosta and R. Burger. Difference schemes stabilized by discrete mollification for degenerate parabolic equations in two space dimensions. IMA J. Numer. Anal., 32:1509–1540, 2012.spa
dc.relation.referencesC.D. Acosta and C.E. Mejía. Stable computations by discrete mollification. Universidad Nacional de Colombia, 2014.spa
dc.relation.referencesA. Aldoghaither and T. Laleg-Kirati. Parameter and differentiation order estimation for a two dimensional fractional partial differential equation. Journal of Computational and Applied Mathematics, 369:112570, 2020.spa
dc.relation.referencesE.J. Anderson and M.S. Phanikumar. Surface storage dynamics in large rivers: Com- paring three-dimensional particle transport, one-dimensional fractional derivative and multirate transient storage models. Water Resources Research, 47:W09511, 2011.spa
dc.relation.referencesIsaac Asimov. Asimov’s Biographical Encyclopedia Of Science And Technology. Dou- bleday, 2nd edition, 1982.spa
dc.relation.referencesD. Benson, M. Meerschaert, and J. Revielle. Fractional calculus in hydrologic modeling: A numerical perspective. Advances in Water Resources, 51:479–497, 2013.spa
dc.relation.referencesD.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert. Application of a fractional advection-dispersion equation. Water Resources Research, 36(6):1403–1412, 2000.spa
dc.relation.referencesH. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer New York, 2010.spa
dc.relation.referencesR. Brociek, A. Chmielowska, and D. Slota. Comparison of the probabilistic ant colony optimization algorithm and some iteration method in application for solving the inverse problem on model with the caputo type fractional derivative. Entropy, 22:555, 2020.spa
dc.relation.referencesT. Chan, G. Golub, and P. Mulet. A nonlinear primal-dual method for total variation- based image restoration. SIAM Journal on Scientific Computing, 20(6):1964–1977, 1999.spa
dc.relation.referencesK. Diethelm. The Analysis of Fractional Differential Equations: An Application- Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2010.spa
dc.relation.referencesM.D. Echeverry and Mejía C.E. A two dimensional discrete mollification oper- ator and the numerical solution of an inverse source problem. AXIOMS, xx,5; doi:10.3390/axiomsxx010005:11, 2018.spa
dc.relation.referencesS. Fomin and T. Chugunov, V. ahd Hashida. Application of fractional differential equations for modeling the anomalous diffusion of contaminant from fracture into porous rock matrix with bordering alteration zone. Transport in Porous Media, 81(2):187–205, Jan 2010.spa
dc.relation.referencesD. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer Berlin Heidelberg, 2001.spa
dc.relation.referencesR. Gorenflo and F. Mainardi. Fractional calculus: integral and differential equations of fractional order. arXiv:0885.3823v1, 2008.spa
dc.relation.referencesC. Groetsch. Inverse Problems in the Mathematical Sciences. Vieweg, 1993.spa
dc.relation.referencesP. Hansen. Rank-Deficient and Discrete Ill-Posed Problems. Society for Industrial and Applied Mathematics, 1998.spa
dc.relation.referencesP. Hansen. Discrete inverse problems, insight and algorithms. SIAM, 2010.spa
dc.relation.referencesG. Huang, Q. Huang, and H. Zhan et al. Modeling contaminant transport in homo- geneous porous media with fractional advection-dispersion equation. Science in China Ser. D Earth Sciences, 48(Supp. II):295–302, 2005.spa
dc.relation.referencesV. Isakov. Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences. Springer International Publishing, 2017.spa
dc.relation.referencesJ.C. Lagarias, J.A. Reeds, M.H. Wright, and P.E. Wright. Convergence properties of the nelder-mead simplex method in low dimensions. SIAM Journal of Optimization, 9:112–147, 1998.spa
dc.relation.referencesP. Linz. Analytical and Numerical Methods for Volterra Equations. Society for Indus- trial and Applied Mathematics, 1985.spa
dc.relation.referencesF. Liu, Anh V.V., I. Turner, and P. Zhuang. Time fractional advection-dispersion equation. J. Appl. Math. and Computing, 13:233–245, 2003.spa
dc.relation.referencesY. Ma, P. Prakash, and A. Deiveegan. Generalized tikhonov methods for an inverse source problem of the time-fractional diffusion equation. Chaos, Solitons and Fractals, 108:39–48, 03 2018.spa
dc.relation.referencesAgnieszka B. Malinowska, Tatiana Odzijewicz, and Delfim F. M. Torres. Advanced Methods in the Fractional Calculus of Variations. Springer Publishing Company, Incor- porated, 2015.spa
dc.relation.referencesM.M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathemat- ics, 172:65–77, 2004.spa
dc.relation.referencesC. E. Mejía and A. Piedrahita. Solution of a time fractional inverse advection-dispersion problem by discrete mollification. Revista Colombiana de Matemáticas, 51(1):83–102, 2017.spa
dc.relation.referencesC. E. Mejía and A. Piedrahita. A finite difference approximation of a two dimensional time fractional advection-dispersion problem. https://arxiv.org/abs/1807.07393, 2018.spa
dc.relation.referencesK.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.spa
dc.relation.referencesD.A. Murio. The Mollification Method and the Numerical Solution of Ill-Posed Prob- lems. Wiley, 1993.spa
dc.relation.referencesP.G. Nutting. A new general law of deformation. Journal of the Franklin Institute, 191(5):679 – 685, 1921.spa
dc.relation.referencesP.G. Nutting. A general stress-strain-time formula. Journal of the Franklin Institute, 235(5):513 – 524, 1943.spa
dc.relation.referencesK.B. Oldham and J. Spanier. The fractional calculus: theory and applications of dif- ferentiation and integration to arbitrary order. Dover, Mineola, 2006.spa
dc.relation.referencesI. Podlubny. Fractional differential equations. Academic Press, 1999.spa
dc.relation.referencesA. Saadatmandi and M. Dehghan. A tau approach for solution of the space fractional diffusion equation. Computers and Mathematics with Applications, 62(3):1135 – 1142, 2011.spa
dc.relation.referencesK. Sakamoto and M. Yamamoto. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. Journal of Mathe- matical Analysis and Applications, 382(1):426 – 447, 2011.spa
dc.relation.referencesS. Salsa. Partial Differential Equations in Action: From Modelling to Theory. UNI- TEXT. Springer International Publishing, 2016.spa
dc.relation.referencesG. W. Scott Blair and M. Reiner. The rheological law underlying the nutting equation. Applied Scientific Research, 2(1):225, Jan 1951.spa
dc.relation.referencesR. Shikrani and M.S. Hashmi et al. An efficient numerical approach for space fractional partial differential equations. Alexandria Engineering Journal, Article in press, 2020.spa
dc.relation.referencesMartin. Stynes, Eugene. O’Riordan, and José Luis. Gracia. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM Journal on Numerical Analysis, 55(2):1057–1079, 2017.spa
dc.relation.referencesH. Sun, Y. Zhang, D. Baleanu, and W. Chen. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simulat, 64:213–231, 2018.spa
dc.relation.referencesH. Sun, Y. Zhang, D. Baleanu, Chen W., and Chen Y. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simulat, 64:213–231, 2018.spa
dc.relation.referencesC. Vogel and M. Oman. Iterative methods for total variation denoising. SIAM Journal on Scientific Computing, 17(1):227–238, 1996.spa
dc.relation.referencesH. Wei, W. Dang, and S. Wei. Parameter identification of solute transport with spatial fractional advection-dispersion equation via tikhonov regularization. Optik, 129:8–14, 2017.spa
dc.relation.referencesR. Weiss and R. S. Anderssen. A product integration method for a class of singular first kind volterra equations. Numer. Math., 18(5):442–456, October 1971.spa
dc.relation.referencesX. Xiong and X. Xue. Fractional tikhonov method for an inverse time-fractional diffu- sion problem in 2-dimensional space. Bulletin of the Malaysian Mathematical Sciences Society, 2018.spa
dc.relation.referencesK. Yosida. Functional Analysis. Classics in mathematics / Springer. World Publishing Company, 1980.spa
dc.relation.referencesD. Zhang and G. Li et al. Numerical identification of multiparameters in the space fractional advection dispersion equation by final observations. Journal of Applied Math- ematics, 2012:740385, 2012.spa
dc.relation.referencesP. Zhuang and F. Liu. Finite difference approximation for two-dimensional time frac- tional diffusion equation. Journal of Algorithms & Computational Technology, 1(1):1– 16, 2007.spa
dc.rights.accessrightsinfo:eu-repo/semantics/openAccessspa
dc.rights.licenseAtribución-NoComercial-SinDerivadas 4.0 Internacionalspa
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/spa
dc.subject.ddc510 - Matemáticasspa
dc.subject.lembProblemas inversos (Ecuaciones diferenciales)
dc.subject.proposalFractional Derivativeseng
dc.subject.proposalDerivadas Fraccionariasspa
dc.subject.proposalMollificationeng
dc.subject.proposalMolificaciónspa
dc.subject.proposalInverse Problemseng
dc.subject.proposalProblemas Inversosspa
dc.subject.proposalDifferential Equationseng
dc.subject.proposalEcuaciones Diferencialesspa
dc.titleFractional differential equations and inverse problems.eng
dc.title.translatedEcuaciones diferenciales fraccionarias y problemas inversos.spa
dc.typeTrabajo de grado - Doctoradospa
dc.type.coarhttp://purl.org/coar/resource_type/c_db06spa
dc.type.coarversionhttp://purl.org/coar/version/c_ab4af688f83e57aaspa
dc.type.contentTextspa
dc.type.driverinfo:eu-repo/semantics/doctoralThesisspa
dc.type.redcolhttp://purl.org/redcol/resource_type/TDspa
dc.type.versioninfo:eu-repo/semantics/acceptedVersionspa
dcterms.audienceEspecializadaspa
oaire.accessrightshttp://purl.org/coar/access_right/c_abf2spa
oaire.fundernameColcienciasspa

Archivos

Bloque original

Mostrando 1 - 1 de 1
Miniatura
Nombre:
1128470198.2020.pdf
Tamaño:
868.53 KB
Formato:
Adobe Portable Document Format
Descripción:
Tesis Doctorado en Ciencias-Matemáticas
Descargar

Bloque de licencias

Mostrando 1 - 1 de 1
No hay miniatura disponible
Nombre:
license.txt
Tamaño:
3.87 KB
Formato:
Item-specific license agreed upon to submission
Descripción:
Descargar

Colecciones

Doctorado en Ciencias - Matemáticas