Fractional differential equations and inverse problems.
Diagramas
- Autores:
-
Echeverry Franco, Manuel Danilo
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2020
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/80051
- Palabra clave:
- 510 - Matemáticas
Problemas inversos (Ecuaciones diferenciales)
Fractional Derivatives
Mollification
Inverse Problems
Differential Equations
Derivadas Fraccionarias
Molificación
Problemas Inversos
Ecuaciones Diferenciales
- Rights
- openAccess
- License
- Atribución-NoComercial-SinDerivadas 4.0 Internacional
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| dc.title.eng.fl_str_mv |
Fractional differential equations and inverse problems. |
| dc.title.translated.spa.fl_str_mv |
Ecuaciones diferenciales fraccionarias y problemas inversos. |
| title |
Fractional differential equations and inverse problems. |
| spellingShingle |
Fractional differential equations and inverse problems. 510 - Matemáticas Problemas inversos (Ecuaciones diferenciales) Fractional Derivatives Mollification Inverse Problems Differential Equations Derivadas Fraccionarias Molificación Problemas Inversos Ecuaciones Diferenciales |
| title_short |
Fractional differential equations and inverse problems. |
| title_full |
Fractional differential equations and inverse problems. |
| title_fullStr |
Fractional differential equations and inverse problems. |
| title_full_unstemmed |
Fractional differential equations and inverse problems. |
| title_sort |
Fractional differential equations and inverse problems. |
| dc.creator.fl_str_mv |
Echeverry Franco, Manuel Danilo |
| dc.contributor.advisor.none.fl_str_mv |
Mejía-Salazar, Carlos Enrique |
| dc.contributor.author.none.fl_str_mv |
Echeverry Franco, Manuel Danilo |
| dc.contributor.researchgroup.spa.fl_str_mv |
Computación Científica |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas |
| topic |
510 - Matemáticas Problemas inversos (Ecuaciones diferenciales) Fractional Derivatives Mollification Inverse Problems Differential Equations Derivadas Fraccionarias Molificación Problemas Inversos Ecuaciones Diferenciales |
| dc.subject.lemb.none.fl_str_mv |
Problemas inversos (Ecuaciones diferenciales) |
| dc.subject.proposal.eng.fl_str_mv |
Fractional Derivatives Mollification Inverse Problems Differential Equations |
| dc.subject.proposal.spa.fl_str_mv |
Derivadas Fraccionarias Molificación Problemas Inversos Ecuaciones Diferenciales |
| description |
Diagramas |
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2020 |
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2020 |
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2021-08-30T14:45:39Z |
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2021-08-30T14:45:39Z |
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Trabajo de grado - Doctorado |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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eng |
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C. 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. C. 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. C. 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. C. D. Acosta and C. E. Mejía. Stabilization of explicit methods for convection diffusion equations by discrete mollification. Comput. Math. Appl., 55:368–380, 2008. C.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. C.D. Acosta and C.E. Mejía. Stable computations by discrete mollification. Universidad Nacional de Colombia, 2014. A. 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. E.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. Isaac Asimov. Asimov’s Biographical Encyclopedia Of Science And Technology. Dou- bleday, 2nd edition, 1982. D. Benson, M. Meerschaert, and J. Revielle. Fractional calculus in hydrologic modeling: A numerical perspective. Advances in Water Resources, 51:479–497, 2013. D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert. Application of a fractional advection-dispersion equation. Water Resources Research, 36(6):1403–1412, 2000. H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer New York, 2010. R. 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. T. 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. K. 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. M.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. S. 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. D. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer Berlin Heidelberg, 2001. R. Gorenflo and F. Mainardi. Fractional calculus: integral and differential equations of fractional order. arXiv:0885.3823v1, 2008. C. Groetsch. Inverse Problems in the Mathematical Sciences. Vieweg, 1993. P. Hansen. Rank-Deficient and Discrete Ill-Posed Problems. Society for Industrial and Applied Mathematics, 1998. P. Hansen. Discrete inverse problems, insight and algorithms. SIAM, 2010. G. 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. V. Isakov. Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences. Springer International Publishing, 2017. J.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. P. Linz. Analytical and Numerical Methods for Volterra Equations. Society for Indus- trial and Applied Mathematics, 1985. F. Liu, Anh V.V., I. Turner, and P. Zhuang. Time fractional advection-dispersion equation. J. Appl. Math. and Computing, 13:233–245, 2003. Y. 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. Agnieszka B. Malinowska, Tatiana Odzijewicz, and Delfim F. M. Torres. Advanced Methods in the Fractional Calculus of Variations. Springer Publishing Company, Incor- porated, 2015. M.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. C. 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. C. 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. K.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. Wiley, 1993. D.A. Murio. The Mollification Method and the Numerical Solution of Ill-Posed Prob- lems. Wiley, 1993. P.G. Nutting. A new general law of deformation. Journal of the Franklin Institute, 191(5):679 – 685, 1921. P.G. Nutting. A general stress-strain-time formula. Journal of the Franklin Institute, 235(5):513 – 524, 1943. K.B. Oldham and J. Spanier. The fractional calculus: theory and applications of dif- ferentiation and integration to arbitrary order. Dover, Mineola, 2006. I. Podlubny. Fractional differential equations. Academic Press, 1999. A. 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. K. 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. S. Salsa. Partial Differential Equations in Action: From Modelling to Theory. UNI- TEXT. Springer International Publishing, 2016. G. W. Scott Blair and M. Reiner. The rheological law underlying the nutting equation. Applied Scientific Research, 2(1):225, Jan 1951. R. Shikrani and M.S. Hashmi et al. An efficient numerical approach for space fractional partial differential equations. Alexandria Engineering Journal, Article in press, 2020. Martin. 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. H. 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. H. 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. C. Vogel and M. Oman. Iterative methods for total variation denoising. SIAM Journal on Scientific Computing, 17(1):227–238, 1996. H. 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. R. 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. X. 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. K. Yosida. Functional Analysis. Classics in mathematics / Springer. World Publishing Company, 1980. D. 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. P. 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. |
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Medellín - Ciencias - Doctorado en Ciencias - Matemáticas |
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Escuela de matemáticas |
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Facultad de Ciencias |
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Atribución-NoComercial-SinDerivadas 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Mejía-Salazar, Carlos Enrique857d01016c4a3b5adaea406d23f10665600Echeverry Franco, Manuel Daniloac9c5e6f11058eaef7a1c7200ce1ee46Computación Científica2021-08-30T14:45:39Z2021-08-30T14:45:39Z2020https://repositorio.unal.edu.co/handle/unal/80051Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/DiagramasOur 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.Nuestro 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)Convocatoria 647 de ColcienciasDoctoradoDoctor en Ciencias - MatemáticasAnálisis Numéricoix, 80 páginasapplication/pdfengUniversidad Nacional de ColombiaMedellín - Ciencias - Doctorado en Ciencias - MatemáticasEscuela de matemáticasFacultad de CienciasMedellínUniversidad Nacional de Colombia - Sede Medellín510 - MatemáticasProblemas inversos (Ecuaciones diferenciales)Fractional DerivativesMollificationInverse ProblemsDifferential EquationsDerivadas FraccionariasMolificaciónProblemas InversosEcuaciones DiferencialesFractional differential equations and inverse problems.Ecuaciones diferenciales fraccionarias y problemas inversos.Trabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDC. 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.C. 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.C. 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.C. D. Acosta and C. E. Mejía. Stabilization of explicit methods for convection diffusion equations by discrete mollification. Comput. Math. Appl., 55:368–380, 2008.C.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.C.D. Acosta and C.E. Mejía. Stable computations by discrete mollification. Universidad Nacional de Colombia, 2014.A. 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.E.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.Isaac Asimov. Asimov’s Biographical Encyclopedia Of Science And Technology. Dou- bleday, 2nd edition, 1982.D. Benson, M. Meerschaert, and J. Revielle. Fractional calculus in hydrologic modeling: A numerical perspective. Advances in Water Resources, 51:479–497, 2013.D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert. Application of a fractional advection-dispersion equation. Water Resources Research, 36(6):1403–1412, 2000.H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer New York, 2010.R. 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.T. 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.K. 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.M.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.S. 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.D. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer Berlin Heidelberg, 2001.R. Gorenflo and F. Mainardi. Fractional calculus: integral and differential equations of fractional order. arXiv:0885.3823v1, 2008.C. Groetsch. Inverse Problems in the Mathematical Sciences. Vieweg, 1993.P. Hansen. Rank-Deficient and Discrete Ill-Posed Problems. Society for Industrial and Applied Mathematics, 1998.P. Hansen. Discrete inverse problems, insight and algorithms. SIAM, 2010.G. 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.V. Isakov. Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences. Springer International Publishing, 2017.J.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.P. Linz. Analytical and Numerical Methods for Volterra Equations. Society for Indus- trial and Applied Mathematics, 1985.F. Liu, Anh V.V., I. Turner, and P. Zhuang. Time fractional advection-dispersion equation. J. Appl. Math. and Computing, 13:233–245, 2003.Y. 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.Agnieszka B. Malinowska, Tatiana Odzijewicz, and Delfim F. M. Torres. Advanced Methods in the Fractional Calculus of Variations. Springer Publishing Company, Incor- porated, 2015.M.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.C. 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.C. 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.K.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.D.A. Murio. The Mollification Method and the Numerical Solution of Ill-Posed Prob- lems. Wiley, 1993.P.G. Nutting. A new general law of deformation. Journal of the Franklin Institute, 191(5):679 – 685, 1921.P.G. Nutting. A general stress-strain-time formula. Journal of the Franklin Institute, 235(5):513 – 524, 1943.K.B. Oldham and J. Spanier. The fractional calculus: theory and applications of dif- ferentiation and integration to arbitrary order. Dover, Mineola, 2006.I. Podlubny. Fractional differential equations. Academic Press, 1999.A. 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.K. 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.S. Salsa. Partial Differential Equations in Action: From Modelling to Theory. UNI- TEXT. Springer International Publishing, 2016.G. W. Scott Blair and M. Reiner. The rheological law underlying the nutting equation. Applied Scientific Research, 2(1):225, Jan 1951.R. Shikrani and M.S. Hashmi et al. An efficient numerical approach for space fractional partial differential equations. Alexandria Engineering Journal, Article in press, 2020.Martin. 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.H. 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.H. 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.C. Vogel and M. Oman. Iterative methods for total variation denoising. SIAM Journal on Scientific Computing, 17(1):227–238, 1996.H. 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.R. 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.X. 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.K. Yosida. Functional Analysis. Classics in mathematics / Springer. World Publishing Company, 1980.D. 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.P. 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.EspecializadaColcienciasORIGINAL1128470198.2020.pdf1128470198.2020.pdfTesis Doctorado en Ciencias-Matemáticasapplication/pdf889370https://repositorio.unal.edu.co/bitstream/unal/80051/3/1128470198.2020.pdf6f4b6605cbfc737e8b3b74277cf50051MD53LICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/80051/4/license.txtcccfe52f796b7c63423298c2d3365fc6MD54THUMBNAIL1128470198.2020.pdf.jpg1128470198.2020.pdf.jpgGenerated Thumbnailimage/jpeg4632https://repositorio.unal.edu.co/bitstream/unal/80051/5/1128470198.2020.pdf.jpgcfd94942aa72ae1d36aaf4b24c4edd78MD55unal/80051oai:repositorio.unal.edu.co:unal/800512023-07-21 23:03:57.987Repositorio Institucional Universidad Nacional de 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