On some difference analogues of PDEs with a delay

  • Henryk Leszczyński
  • Jerzy Pieniażek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 879)


  1. 1.

    No difference scheme for equation (2) seems to be definitely convergent

  2. 2.

    In our experiments the schemes of type A and B and mixed A-B behave less stable than C or D, especially close to the boundary errors for A and B are relatively big

  3. 3.

    The schemes C and D are comparable and the error expressions are reasonable

  4. 4.

    Our scheme of type D for equation (2) is easily adaptable to a parallel computing process. It is enough to divide the set E = [0, T] × [−b, b] into rectangles E = [0, T] × [b j , b j ,b j+1 ] (j = 0,..., m − 1) where −b = b0 < ... < b m = b, and then we can minimize functional F j defined on E like F on E. Using many processors one can achieve better results for meshes with smaller steps h t and h x . The exchange of data between neighbouring regions is minimal.

  5. 5.

    A proper division of the set E and the question of well-posedness of initialor initial-boundary-value problem for equation (3) is still open and demands some more extensive investigations with the use of multi-processor mashines.



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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Henryk Leszczyński
    • 1
  • Jerzy Pieniażek
    • 2
  1. 1.Institute of MathematicsUniversity of GdańskGdańskPoland
  2. 2.Institute of MathematicsUniversity of GdańskGdanskPoland

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