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Method of Lines for Quasilinear Functional Differential Equations

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Ukrainian Mathematical Journal Aims and scope

We give a theorem on the estimation of error for approximate solutions to ordinary functional differential equations. The error is estimated by a solution of an initial problem for a nonlinear functional differential equation. We apply this general result to the investigation of convergence of the numerical method of lines for evolution functional differential equations. The initial boundary-value problems for quasilinear equations are transformed (by means of discretization in spatial variables) into systems of ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for given operators are assumed. Numerical examples are given.

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References

  1. A. Baranowska and Z. Kamont, “Numerical method of lines for first order partial differential functional equations,” Z. Anal. Anwend., 21, 949–962 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Brandi, Z. Kamont, and A. Salvadori, “Approximate solutions of mixed problems for first order partial differential functional equations,” Atti Semin. Mat. Fis. Univ. Modena, 39, 277–302 (1991).

    MATH  MathSciNet  Google Scholar 

  3. D. Bahuguna and J. Dabas, “Existence and uniqueness of a solution to a partial integro-differential equation by the method of lines,” Electron. J. Qual. Theor. Different. Equat., 4, 12 p. (2008).

  4. M. Bahuguna, J. Dabas, and R. K. Shukla, “Method of lines to hyperbolic integro-differential equations in ℝn,” Nonlin. Dynam. Syst. Theory, 8, 317–328 (2008).

    MATH  MathSciNet  Google Scholar 

  5. S. Brzychczy, “Existence solution for nonlinear systems of differential functional equations of parabolic-type in an arbitrary domain,” Ann. Pol. Math., 47, 309–317 (1987).

    MATH  MathSciNet  Google Scholar 

  6. W. Czernous, “Generalized solutions of mixed problems for first order partial functional differential equations,” Ukr. Mat. Zh., 58, No. 6, 813–828 (2006).

    Article  MathSciNet  Google Scholar 

  7. I. Gőri, “On the method of lines for the solutions of nonlinear partial differential equations,” Akad. Nauk SSSR, 166 (1987).

  8. I. Gőori, “The method of lines for the solution of some nonlinear partial differential equations,” Comput. Math. Appl., 15, 635–648 (1988).

    Article  MathSciNet  Google Scholar 

  9. W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, Berlin (2003).

    Book  MATH  Google Scholar 

  10. W. Jáger and L. Simon, “On a system of quasilinear parabolic functional differential equations,” Acta Math. Hung., 112, 39–55 (2006).

    Article  Google Scholar 

  11. Z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Kluwer Acad. Publ., Dordrecht, etc. (1999).

    Book  MATH  Google Scholar 

  12. Z. Kamont and K. Kropielnicka, “Differential difference inequalities related to hyperbolic functional differential systems and applications,” Different. Inequal. Appl., 8, 655–674 (2005).

    MATH  MathSciNet  Google Scholar 

  13. J. P. Kaughten, “The method of lines for parabolic partial integro-differential equations,” J. Integral Equat. Appl., 4, 69–81 (1992).

    Article  Google Scholar 

  14. S. Łojasiewicz, “Sur le probléme de Cauchy pour systémes d’ équations aux dérivées partielles du premier ordre dans le cas hyperbolique de deux variables indépendantes,” Ann. Pol. Math., 3, 87–117 (1956).

    MATH  Google Scholar 

  15. M. Netka, “Differential difference inequalities related to parabolic functional differential equations and applications,” Opusc. Math., 30, 95–115 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Redlinger, “Existence theorems for semilinear parabolic systems with functionals,” Nonlin. Anal., TMA, 8, 667–682 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  17. F. Shakeri and M. Dehghan, “The method of lines for solution of the m-dimensional wave equation subject to an integral conservation conditions,” Comput. Math. Appl., 56, 2175–2188 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  18. L. Simon, “Application of monotone-type operators to parabolic and functional parabolic PDE’s,” in: Evolutionary Equations, C. M. Dafermos and M. Pokorny (Eds), Elsevier, Amsterdam (2008), pp. 267–321.

    Google Scholar 

  19. K. Schmitt, R. C. Thompson, and W. Walter, “Existence of solutions of a nonlinear boundary-values problem via method of lines,” Nonlin. Anal., 2, 519–535 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  20. K. Topolski, “On the existence of classical solutions for differential-functional IBVP,” Abstr. Appl. Anal., 3, 363–375 (1998).

    Article  MathSciNet  Google Scholar 

  21. A.Vande Wouwer, Ph. Saucez, and W. E. Schiesser, Adaptative Method of Lines, Chapman & Hall-CRC, Roca Baton (2001).

    Book  Google Scholar 

  22. A.Voigt, “Line method approximations to the Cauchy problem for nonlinear parabolic differential equations,” Numer. Math., 23, 23–36 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  23. A.Voigt, “The method of lines for nonlinear parabolic differential equations with mixed derivatives,” Numer. Math., 32, 197–207 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  24. W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin (1970).

    Book  MATH  Google Scholar 

  25. W. Walter and A. Acker, “On the global existence of solutions of parabolic differential equations with a singular nonlinear term,” Nonlin. Anal., 2, 499–504 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  26. Dongming Wei, “Existence, uniqueness and numerical analysis of solutions of a quasilinear parabolic problem,” SIAM J. Numer. Anal., 29, 484–497 (1992).

  27. J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, Berlin (1996).

    Book  MATH  Google Scholar 

  28. S. Yuan, The Finite Method of Lines, Sci. Press, Beijing (1993).

    Google Scholar 

  29. B. Zubik-Kowal, “The method of lines for first order partial differential functional equations,” Stud. Sci. Math. Hung., 34, 413–428 (1998).

    MATH  MathSciNet  Google Scholar 

  30. B. Zubik-Kowal, “The method of lines for parabolic differential functional equations,” J. Numer. Anal., 17, 103–123 (1997).

    Article  MATH  MathSciNet  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 10, pp. 1363–1387, October, 2013.

Deceased (Z. Kamont)

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Czernous, W., Kamont, Z. Method of Lines for Quasilinear Functional Differential Equations. Ukr Math J 65, 1514–1541 (2014). https://doi.org/10.1007/s11253-014-0876-0

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  • DOI: https://doi.org/10.1007/s11253-014-0876-0

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