Métodos Computacionais para Problemas Singulares

Computational Methods for Singular Problems

 POCTI/MAT/45700/2002

(Project supported by Fundação para a Ciência e Tecnologia)

Principal Research Unity: Centro de Matemática e Aplicações

Research Team:

Pedro Lima (Principal Investigator)
Michel Carpentier,  Dep. Matemática, Instituto Superior Técnico, Lisboa (PhD)
Nikolai Chemetov, Dep. Matemática, Faculdade de Ciências,  Universidade de Lisboa (PhD)
Teresa Diogo,  Dep. Matemática, Instituto Superior Técnico, Lisboa (PhD)

António Oliveira, Dep. Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (PhD)

Ricardo Barros,Dep. Matemática, Faculdade de Ciências e Tecnologia, Universidade de Lisboa (PhD Student)
Madalena Consciência, Dep. Matemática, Faculdade de Ciências, Universidade de Lisboa  (MSc )
Madalena Menezes, Dep. Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (PhD Student)
Luisa Morgado, Dep. Matemática, Universidade de Trás-os-Montes e Alto Douro, Vila Real (PhD Student) 
Matilde Pato, Dep.Eng. Química, Instituto Superior de Engenharia de Lisboa (MSc)
Magda Rebelo, Dep. Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (PhD Student)
Filomena Teodoro, Dep. Matemática, Instituto Politécnico de Setúbal  (PhD Student)
Svilen Valtchev, Dep. Matemática, Instituto Superior Técnico, Lisboa (PhD Student)

Objectives of the project

The main objective of the present project is to develop new efficient  methods for the solution of mathematical problems  which have singularities and cannot be solved by means of classical methods. The mathematical models that we shall consider arise in such fields as fluid mechanics, heat conduction and wave propagation. The investigation of these models leads to boundary value problems for nonlinear differential equations and Volterra integral equations. The project has two components: 1) the theoretical investigation of the considered mathematical models and 2) the development of numerical methods and creation of computer algorithms for the numerical solution of the problems. The main characteristic of the  methods to be introduced is that they take into account the asymptotic behavior of the solutions near the singularities. In the field of Volterra integral equations,  we intend to improve the efficency of the known algorithms for integral equations by introducing such techniques as variable transformations and graded meshes.

References:  Article 1  Article 2  Article 3  Article 4  Article 5

 

Starting date: September 1st 2003

Activities

2003-2004

P.M.Lima participated in the XXIV  Iberian Latin-American Congress on Computational Methods in Engineering (CILAMCE 24),29-31 October 2003, Ouro Preto, Brazil.

N. B. Konyukhova, from the Computing Center of Russian Academy of Sciences (Moscow), has visited the Center of Mathematics and Applications, 5-22 November 2003.

P.M.Lima and T.Diogo participated in the Ninth International Conference on Enhancement and Promotion of Computational Methods in Engineering and Science (EPMESC IX), 25-28 November 2003, Macau.

T.Diogo visited Prof. Tao Tang, at Mathematics department of the Baptist University of Hong Kong, 19 November-4 December 2003.

Prof. A.I.Sukov, from the Moscow Technological University and Russian Academy of Sciences, visited IST from 28 February to 28 March 2004.

P.M.Lima visited Prof. Neville Ford at Mathematics department of the Chester University College, 19 April – 29 April 2004.

P.M. Lima and S.Valtchev participated in the congress “Métodos Computacionais em Engenharia”, held in Lisbon, 31May-2nd June 2004.

P.M. Lima participated in the International Congress on Computational and Applied Mathematics (ICCAM 2004), held in Leuven (Belgium),  July 26-30 2004.

2004-2005

P.M. Lima and N.V. Chemetov participated in the VI  International Congress on Mathematical Modelling, held in Nizhny Novgorod (Russia),  September 20-26 2004.

The International Workshop on Analysis and Numerical Approximation of Singular Problems (IWANASP) took place at IST, on November 10-12th  2004. The Organizing Committee was composed by P. M. Lima, M.T. Diogo, N.V. Chemetov and A . Oliveira.  The conference was attended by about 40 persons from 10 countries.

M. Rebelo, L. Morgado and S.Valtchev  presented  communications at IWANASP.

Prof. N. J Ford visited the Mathematics Department of IST, from January 30th to February 4th 2005.  

The MSc  Thesis of M. Consciência,  entitled “Formação de bolhas num fluido””,  was discussed and approved  on April 11 , at the Faculdade de Ciências of the of  the University of Lisbon.

Prof. U. Steffanelli  has visited the Faculdade de Ciências of  the University of Lisbon  on May  2005.

M.T. Diogo  and P.M. Lima  participated in the XXI Biennial Conference on Numerical Analysis , held in Dundee (Scotland), June 28th- July 1st.

P.M. Lima and L. Morgado participated in the conference  Equadiff  11,   held in Bratislava (Slovakia), 25-29 July 2005.

M.T. Diogo and M. Rebelo participated in the 7th Hellenic European Conference on Computer Mathematics and  its Applications (Hercma 20005) , held in Athens (Greece), September 22-24, 2005.

2005-2006

Prof. V. Goncharov (Universidade de Évora) visited Prof. N. Chemetov at Faculdade de Ciências da Universidade de Lisboa  on October 2005.

M.T.Diogo and P.M. Lima visited the Institute of Computational Mathematics and Scientific Engineering , Chinese Academy of Sciences  (ICMSEC) , in Beiging,  from 5 to 11 December 2005.

M.T. Diogo delivered the invited talk “Convergence Properties of Collocation Methods for a Volterra Integral Equation  with Weakly Singular Kernel”  at the seminar of  the Institute of Computational Mathematics and Scientific Engineering, on December 7 2005.

M.T. Diogo  and P.M. Lima  presented invited communications in the II International Conference on Scientific Computing and Partial Differential Equations (SCPDE 05),  held in Hong Kong , December 12-16 2005.

Prof.  S. Antontsev (Universidade da Beira Interior) visited Prof.  N. Chemetov in Faculdade de Ciências da Universidade de Lisboa  in January 2006.

Prof, Neville J.  Ford  (Chester University) visited the Mathematics Department of IST,  on  February  5-11 , 2006 , and delivered a lecture entitled “Super-exponential solutions: their numerical modelling and detection “.

Prof. F. Sadyrbaev (Univ. of Latvia, Univ. of Daugavpils) visited the Mathematics Department of IST, on April 2006, and delivered a lecture, entitled “Two-point nonlinear boundary value problems for ordinary differential equations: existence, multiple solutions, properties”.

Prof. Neville J. Ford (Chester University) visited the Mathematics Department of IST, on  April 26-28 , 2006.

M. Rebelo and  L.Morgado participated  and presented communications in the “Encontro Nacional da Sociedade Portuguesa de Matemática” (ENSPM06), Instituto Superior de Engenharia de Lisboa,  June 20-23 2006.

T. Diogo participated and presented a communication at the Twelfth International Congress on Computational and Applied Mathematics  (ICCAM 2006), at the Catholic University of Leuven (Belgium), July  10-14 2006.

T. Diogo and P. Lima participated as organizers at the Second International Workshop on Analysis and Numerical Approximation of Singular Problems (IWANASP06), held in Samos  (Greece), September 6-8 2006.

L. Morgado and M. Rebelo presented communications at the Second International Workshop on Analysis and Numerical Approximation of Singular Problems (IWANASP06), held in Samos  (Greece), September 6-8 2006.

T. Diogo and P. Lima participated and presented communications at the XXIX Congresso Nacional de Matemática Aplicada e Computacional (CNMAC), Campinas (Brasil),  September 18-21 2006.

 2006-March 2007 (extension)

 P. Lima participated and presented communication in the 4th WSEAS International Conference on Fluid Mechanics, Gold Coast (Australia),   17-19 January 2007.

Prof. A. Pedas  (Institute of Applied Mathematics, Tartu University, Estonia) visited the Mathematics Department of IST, on  4th -10th  February 2007.

Prof. A.I. Sukov (Moscow Technological University, Russia) visited the Mathematics Department of IST, on 11th February- 4th March 2007.

 

Publications

·            O. Antontsev and N. Chemetov, Flux of superconducting vortices through a domain,  SIAM Journal on Mathematical Analysis, 39  (2006) p. 263-280.

·            N.Chemetov and M. Marques, Non-convex quasi-variational differential inclusions, submitted.

  • N.Chemetov, The numerical analysis of a two-shape memory model, submitted.
  • N. Chemetov and F. Cipriano, The 2D-Euler Equations and the statistical transport equations,  Commun. Math. Phys 267 (2006), 543-558.
  • N. Chemetov, Well-posedness of two-shape memory model,  Mathematical Models of Applied Sciences, 29  (2006), 209-233.
  •  T.Diogo, P.M.Lima, S.Valtchev and N.Ford, Numerical methods for a nonuniquely solvable Volterra integral equation,  in proceedings of CILAMCE XXIV (CD), C.E.Mello et al. (eds), Ouro Preto, Brasil, 2003.
  • T.Diogo, S.Valtchev and P.Lima, Numerical solution of a singular Volterra integral equation by piecewise polynomial collocation, in Computational Methods in Engineering and Science,  Iu ,Lamas, Li and Mok. (eds),  Swets & Zellinger, Lisse, 2003, p. 195-200.
  • T. Diogo, N.B. Franco and P. Lima, High order product integration methods for a Volterra integral equation with logarithmic singular kernel , Comm  Pure Appl. Anal., v.3 (2004), n.2, p. 217-235. 
  • T.Diogo,  N.Ford J. Edwards and S.Thomas, Numerical analysis of a singular integral equation,  Appl. Math. Comp., v. 167 (2005),  p.372-382.
  • T. Diogo, P.M. Lima and M.S. Rebelo, Computational methods for a nonlinear Volterra integral equation,  Proceedings of the 7th Hellenic European Conference on Computer Mathematics and its Applications (HERCMA 2005),  Athens, September 22-24 2005.
  • T. Diogo, N.J. Ford, P.M.Lima and S.M.Thomas, Solution of a singular integral equation by a split-interval method,   Int. J.  Numer. Anal.  Model., 4 (2007), p. 63-73.
  • T. Diogo, P.M. Lima and M.S.Rebelo,  Numerical solution of a nonlinear Abel- type Volterra integral equation,   Comm.   Pure  Appl.  Anal. , 5 (2006), p.277-288.
  • T. Diogo, N.J. Ford, P.M.Lima and S. S.Valtchev, Numerical methods for a Volterra integral equations with non-smooth solutions,   J.Comput. Appl. Math. , 189 (2006) , p. 412-423.
  • T. Diogo and P.Lima, Superconvergence of collocation methods for a class of weakly singular Volterra integral equations, to appear in Journal of Computational and Applied Mathematics.
  • T. Diogo and P. Lima,  Collocation solutions of a weakly singular Volterra integral equation, submitted to TEMA.
  • T. Diogo, Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations, submitted to J. Comp. Appl. Math. 
  • N.J.Ford, T.Diogo, J.Ford and P.Lima, A note on numerical modelling of qualitative behaviour of solutions to convolution integral equations, to appear in Journal of Computational and Applied Mathematics.
  •  G. Kitzhofer, O. Koch, P.M. Lima and E. Weinmüller, Efficient numerical solution of the density profile equation in hydrodynamics,  accepted for publication in J. of Sci. Comput.
  • P.Lima, N.Chemetov, N.B.Konyukhova and A.I.Sukov, Analytical-numerical approach to a singular boundary value, proceedings of CILAMCE XXIV (CD), C.E.Mello et al. (eds), Ouro Preto, Brasil, 2003.
  • P.M.Lima and M.P.Pato, Computational methods for a singular boundary-value problem, in Computational Methods in Engineering and Science, Iu , Lamas, Li and Mok(eds),  Swets & Zellinger, Lisse, 2003, p. 211-217.
  • P.M.Lima, N.Chemetov, N.B.Konyukhova and A.I.Sukov,  Modelação matemática e simulação numérica da formação de bolhas de gás numa mistura gás-líquido, in proceedings of the Congress Métodos Computacionais em Engenharia (CD),  C.M.Soares et al.(eds), Lisboa, 2004.
  • P.M.Lima, N.B.Konyukhova, N.V.Chemetov and A.I.Sukov, Mathematical analysis and numerical solution of  a singular problem   i n  nonlinear  physics,  Vestnik Nizhegorodskogo Universiteta,  v. 1(28), ser. Mathematical Modeling and Optimal Control  (2005),  p.162-171.
  • P.M.Lima, N.B.Konyukhova, N.V.Chemetov and A.I.Sukov, Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems,  J.Comput. Appl. Math. ,189 (2006) , p. 260-273.
  • P.M.Lima and L. Morgado, Analysis of singular boundary value problems for an Emden-Fowler equation,  Comm. Pure  Appl.  Anal.  5 (2006),  p.321-336.
  • P.M.Lima and L. Morgado, Numerical approximation of singular boundary value problems for a nonlinear differential equation, accepted for publication in Proceedings of Equadiff 11, Bratislava,  2005.
  • P.M.Lima and L. Morgado, Numerical Approximation of singular boundary value problems on bounded and unbounded  domains, in Fundamental Physical-Mathematical Problems and Modeling of Technical Systems , vol. 9, ed. Janus-K, Moscow, 2006, 105-110.
  • P.M. Lima and L. Morgado, Asymptotic and numerical approximation of a boundary value problem for a quasi-linear differential equation, WSEAS Transactions on Mathematics, Issue 5, V. 6 (2007), 639-647.
  •  P.M. Lima and L. Morgado, Analysis and numerical approximation of a singular boundary value problem for the one-dimensional p-laplacian, Proceedings of the 4th WSEAS International Conference on Fluid Mechanics, Gold Coast, Australia, January 17-19, 2007, p. 42-47.
  • P.M. Lima and L. Morgado, Analysis and numerical approximation of a free boundary problem for a singular ordinary differential equation, submitted to TEMA (Tendencias em Matemática Aplicada), 2006.
  • P.M. Lima and L. Morgado, Analytical-numerical investigation of a singular boundary value problem for a generalized Emden-Fowler equation, submitted to J.Comput. Appl. Math, 2007.