Complex Analysis and Operator Theory

, Volume 6, Issue 1, pp 307–324 | Cite as

General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces

  • L. P. CastroEmail author
  • S. Saitoh
  • Y. Sawano
  • A. M. Simões


In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.


Linear partial differential equation with constant coefficients Discrete differential equation Approximation of functions Inverse source problem Reproducing kernel Tikhonov regularization Sobolev space Generalized inverse Approximate inverse Error estimate Noise 

Mathematics Subject Classification (2000)

Primary 35A35 Secondary 30C40 35A22 35C15 35E99 46E22 47A52 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asaduzzaman, M., Matsuura, T., Saitoh, S.: Constructions of approximate solutions for linear differential equations by reproducing kernels and inverse problems. In: Advances in Analysis, Proceedings of the 4th International ISAAC Congress, pp. 335–344. World Scientific, Singapore (2005)Google Scholar
  2. 2.
    Castro L.P., Chen Q., Saitoh S.: Source inversion of heat conduction from a finite number of observation data. Appl. Anal. 89, 801–813 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Higgins, J.R.: A sampling principle associated with Saitoh’s fundamental theory of linear transformations. In: Analytic Extension Formulas and Their Applications. International Society of Analytical Applications and Computing, vol. 9, pp. 73–86. Kluwer, Dordrecht (2001)Google Scholar
  4. 4.
    Itou H., Saitoh S.: Analytical and numerical solutions of linear singular integral equations. Int. J. Appl. Math. Stat. 12, 76–89 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Iwamura, K., Matsuura, T., Saitoh, S.: A numerical construction of a natural inverse of any matrix by using the theory of reproducing kernels with the Tikhonov regularization. Far East J. Math. Edu. (to appear)Google Scholar
  6. 6.
    Matsuura T., Saitoh S.: Analytical and numerical solutions of linear ordinary differential equations with constant coefficients. J. Anal. Appl. 3, 1–17 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Matsuura T., Saitoh S.: Dirichlet’s principle using computers. Appl. Anal. 84, 989–1003 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Matsuura T., Saitoh S.: Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley–Wiener spaces. Appl. Anal. 85, 901–915 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Saitoh S.: Hilbert spaces induced by Hilbert space valued functions. Proc. Am. Math. Soc. 89, 74–78 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Saitoh, S.: Integral transforms, reproducing kernels and their applications. In: Pitman Research Notes in Mathematics Series, vol. 369. Longman, Harlow (1997)Google Scholar
  11. 11.
    Stenger, F.: Numerical methods based on sinc and analytic functions. In: Springer Series in Computational Mathematics, vol. 20. Springer, New York (1993)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • L. P. Castro
    • 1
    Email author
  • S. Saitoh
    • 1
  • Y. Sawano
    • 2
  • A. M. Simões
    • 3
  1. 1.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Department of MathematicsKyoto UniversityKyotoJapan
  3. 3.Department of MathematicsUniversity of Beira InteriorCovilhãPortugal

Personalised recommendations