Computation of Green's functions for boundary value problems with Mathematica

Applied Mathematics and Computation

Volumn 219, Issue 4, 2012, Pages 1919-1936

  Cabada, Alberto ^a   Cid, José Ángel ^b   Máquez Villamarín, Beatriz ^a  

^a UNIVERSITY OF SANTIAGO DE COMPOSTELA   (Spain)

^b UNIVERSITY OF VIGO   (Spain)

Author keywords

Boundary value problem; Green's function; Mathematica package

Indexed keywords

CONSTANT COEFFICIENTS; LINEAR ORDINARY DIFFERENTIAL EQUATIONS; MATHEMATICA; TWO-POINT;

ALGORITHMS; BOUNDARY VALUE PROBLEMS; ORDINARY DIFFERENTIAL EQUATIONS;

GREEN'S FUNCTION;

EID: 84867582938     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2012.08.035     Document Type: Article

References (39)

1. M.L. Abell, and J.P. Braselton Differential Equations with Mathematica 2004 Elsevier
2. R.P. Agarwal, and D. O'Regan Ordinary and Partial Differential Equations. With Special Functions, Fourier Series, and Boundary Value Problems Universitext 2009 Springer New York
3. I.V. Barteneva, A. Cabada, and A.O. Ignatyev Maximum and anti-maximum principles for the general operator of second order with variable coefficients Appl. Math. Comput. 134 2003 173 184
4. S.R. Bernfeld, and V. Lakshmikantham An Introduction to Nonlinear Boundary Value Problems Mathematics in Science and Engineering vol. 109 1974 Academic Press New York
5. G. Birkhoff, and G.C. Rota Ordinary Differential Equations fourth ed. 1989 John Wiley & Sons, Inc. New York
6. A. Cabada The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems J. Math. Anal. Appl. 185 1994 302 320
7. A. Cabada The method of lower and upper solutions for third-order periodic boundary value problems J. Math. Anal. Appl. 195 1995 568 589
8. A. Cabada Maximum principles for third-order initial and terminal value problems Differential & Difference Equations and Applications New York 2006 Hindawi Publ. Corp. 247 255
9. A. Cabada, and J.A. Cid On the sign of the Green's function associated to Hill's equation with an indefinite potential Appl. Math. Comput. 205 2008 303 308
10. Cabada A., Cid, J.A., Máquez-Villamarín, B. (2011) Green's Function, available at , Wolfram demonstrations project, Published: October 3.
11. A. Cabada, J.A. Cid, and L. Sanchez Positivity and lower and upper solutions for fourth order boundary value problems Nonlinear Anal. 67 2007 1599 1612
12. J.A. Cid, D. Franco, and F. Minhós Positive fixed points and fourth-order equations Bull. Lond. Math. Soc. 41 2009 72 78
13. E.A. Coddington, and N. Levinson Theory of Ordinary Differential Equations 1955 McGraw-Hill Book Company, Inc. New York-Toronto-London
14. C. De Coster, and P. Habets Two-Point Boundary Value Problems: Lower and Upper Solutions Mathematics in Science and Engineering vol. 205 2006 Elsevier B.V. Amsterdam
15. B. Davies Integral Transforms and their Applications third ed. Texts in Applied Mathematics vol. 41 2002 Springer-Verlag New York
16. D.G. Duffy Green's Functions with Applications Studies in Advanced Mathematics 2001 Chapman & Hall/CRC Boca Raton, FL
17. H.M. Fried Green's Functions and Ordered Exponentials 2002 Cambridge University Press Cambridge
18. B. Friedman Principles and Techniques of Applied Mathematics Reprint of the 1956 Original 1990 Dover Publications, Inc. New York
19. A. Gray, M. Mezzino, and M.A. Pinsky Introduction to Ordinary Differential Equations with Mathematica 1997 Springer-Verlag New York
20. R. Hakl, and P. Torres Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign Appl. Math. Comput. 217 2011 7599 7611
21. S. Heikkilä, and V. Lakshmikantham Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations Monographs and Textbooks in Pure and Applied Mathematics vol. 181 1994 Marcel Dekker, Inc. New York
22. P.K. Kythe, P. Puri, and M.R. Schäferkotter Partial Differential Equations and Boundary Value Problems with Mathematica second revised ed. 2003 Chapman and Hall/CRC Boca Raton
23. G.S. Ladde, V. Lakshmikantham, and A.S. Vatsala Monotone Iterative Techniques for Nonlinear Differential Equations 1985 Pitman Boston, M.A.
24. S. Lynch Dynamical Systems with Applications using Mathematica 2007 Birkhäuser Boston
25. J. Mawhin Maximum principle for bounded solutions of the telegraph equation: the case of high dimensions Prog. Nonlinear Differ. Equat. Appl. 63 2005 343 351
26. C.W. Maynard, and M.R. Scott Some relationships between Green's functions and invariant imbedding J. Optim. Theory Appl. 12 1973 6 15
27. S. Novo, R. Obaya, and J. Rojo Equations and Differential Systems 1995 McGraw-Hill (in Spanish)
28. P. Omari, and M. Trombetta Remarks on the lower and upper solutions method for second and third-order periodic boundary value problems Appl. Math. Comput. 50 1992 1 21
29. A. Peterson On the sign of Green's functions J. Differ. Equat. 21 1976 167 178
30. M.H. Protter, and H.F. Weinberger Maximum Principles in Differential Equations 1967 Prentice-Hall
31. G.F. Roach Green's Functions second ed. 1982 Cambridge University Press Cambridge-New York
32. M. Scott Invariant imbedding and the calculation of internal values J. Math. Anal. Appl. 28 1969 112 119
33. I. Stakgold Green's Functions and Boundary Value Problems second ed. Pure and Applied Mathematics 1998 John Wiley & Sons, Inc. New York
34. M. Struwe Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems 2000 Springer-Verlag
35. P.J. Torres Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem J. Differ. Equat. 190 2003 643 662
36. J.R.L. Webb Solutions of nonlinear equations in cones and positive linear operators J. Lond. Math. Soc. 82 2 2010 420 436
37. J.R.L. Webb, G. Infante, and D. Franco Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions Proc. Roy. Soc. Edinburgh Sec. A 138 2008 427 446
38. M. Zhang, Optimal conditions for maximum and anti-maximum principles of the periodic solution problem, Bound. Value Probl. 2010, (Article ID 410986), 26 p, .
39. Lα norms Proc. Amer. Math. Soc. 130 2002 3325 3333