Implicit difference methods for Hamilton-Jacobi differential functional equations

• Autorzy: Kępczyńska, A.
• Języki publikacji: EN

Abstrakty

EN

Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference functional equations. The numerical methods are difference schemes which are implicit with respect to time variable. A complete convergence analysis for the methods is given and it is shown that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type. Numerical examples are given.

Słowa kluczowe

PL

funkcjonalne równania różniczkowe; bezwarunkowe metody różniczkowe; operatory interpolacyjne

EN

functional differential equations; implicit difference methods; stability and convergence; interpolating operators

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2007
• Tom: Vol. 40, nr 1
• Strony: 125-150
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Kępczyńska, A.

• Gdańsk University of Technology Department of Applied Physics and Mathematics G. Narutowicz Street 11-12 80-952 Gdańsk, Poland

Bibliografia

• [1] S. Bal, Convergence of a difference method for a system of first order partial differential equations of hyperbolic type, Ann. Polon. Math. 30 (1974), 19–36.
• [2] P. Brandi, Z. Kamont and A. Salvadori, Approximate solutions of mixed problems for first order partial differential functional equations, Atti. Sem Mat. Fis. Univ. Modena 39 (1991), no. 1, 277–302.
• [3] D. Jaruszewska-Walczak and Z. Kamont, Difference methods for quasilinear hyperbolic differential functional systems on the Haar pyramid, Bull. Belg. Math. Soc. 10 (2003), no. 2, 267–290.
• [4] Z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Kluver Acad. Publ., Dordrecht, Boston, London (1999).
• [5] Z. Kamont and K. Prządka, Difference methods for first order partial differential functional equations with initial - boundary conditions, Zh. Vychisl. Mat. i Mat. Fiz. 31 (1991), no. 10, 1476–1488.
• [6] Z. Kamont and K. Prządka, Difference methods for nonlinear partial differential equations of the first order, Ann. Polon. Math. 48 (1988), no. 3, 227–246.
• [7] Z. Kowalski, A difference method for certain hyperbolic systems of nonlinear partial differential equations of the first order, Ann. Polon. Math. 19 (1967), 313–322.
• [8] Z. Kowalski, On the difference method for certain hyperbolic systems on nonlinear partial differential equations of the first order, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 297–302.
• [9] V. Lakshmikantham and S. Leela, Differential and integral inequalities: Theory and applications, Academic Press, New York, London, (1969).
• [10] H. Leszczyński, Convergence results for unbounded solutions of first order nonlinear differential-functional equations, Ann. Polon. Math. 64 (1996), no. 1, 1–16.
• [11] J. Szarski, Differential inequalities, PWN, Warsaw (1965), 256 pp.
• [12] J. Wu, Theory and Applications of Partial Functional Differential Equations, Berlin, Springer (1996).