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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Finite difference solution of a nonlinear Klein-Gordon equation with an external source
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by G. Berikelashvili, O. Jokhadze, S. Kharibegashvili and B. Midodashvili PDF
Math. Comp. 80 (2011), 847-862 Request permission


In this paper, we consider the Darboux problem for a (1+1)-dimensional cubic nonlinear Klein-Gordon equation with an external source. Stable finite difference scheme is constructed on a four-point stencil, which does not require additional iterations for passing from one level to another. It is proved, that the finite difference scheme converges with the rate $O(h^2)$, when the exact solution belongs to the Sobolev space $W_2^2$.
  • F.J. Alexander and S. Habib, Statistical mechanics of kinds in 1+1 dimensions, Phys. Rev. Lett. 71 (1993), 955–958.
  • G. K. Berikelashvili, O. M. Dzhokhadze, B. G. Midodashvili, and S. S. Kharibegashvili, On the existence and nonexistence of global solutions of the first Darboux problem for nonlinear wave equations, Differ. Uravn. 44 (2008), no. 3, 359–372, 430 (Russian, with Russian summary); English transl., Differ. Equ. 44 (2008), no. 3, 374–389. MR 2437059, DOI 10.1134/S0012266108030087
  • A. R. Bishop, J. A. Krumhansl, and S. E. Trullinger, Solitons in condensed matter: a paradigm, Phys. D 1 (1980), no. 1, 1–44. MR 573367, DOI 10.1016/0167-2789(80)90003-2
  • A. V. Bitsadze, Some classes of partial differential equations, Advanced Studies in Contemporary Mathematics, vol. 4, Gordon and Breach Science Publishers, New York, 1988. Translated from the Russian by H. Zahavi. MR 1052404
  • T. Cazenave, A. Haraux, L. Vazquez and F.B. Weissler, Nonlinear effects in the wave equation with a cubic restoring force, Comput. Mech. 5 (1989) 49–72.
  • G. Darboux, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal, Vol. III(French) (Gauthier-Villars, 1894).
  • Moshé Flato, Jacques C. H. Simon, and Erik Taflin, Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations, Mem. Amer. Math. Soc. 127 (1997), no. 606, x+311. MR 1407900, DOI 10.1090/memo/0606
  • S. Gellerstedt, Sur un Problème aux Limites pour une Équation Linéaire aux Dérivées Partielles du Second Ordre de Type Mixte, (French) (Uppsala University, 1935).
  • O.G. Goman, The equation of a reflected wave, Vestn. Mosk. Univ., Ser. I (Russian) 23 (1968), 84–87.
  • Nicolas Gonzalez, An example of pure stability for the wave equation with moving boundary, J. Math. Anal. Appl. 228 (1998), no. 1, 51–59. MR 1659940, DOI 10.1006/jmaa.1998.6113
  • Édouard Goursat, Cours d’analyse mathématique. Tome III, 3rd ed., Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1992 (French). Intégrales infiniment voisines. Équations aux dérivées du second ordre. Équations intégrales. Calcul des variations. [Infinitely near integrals. Second-order partial differential equations. Integral equations. Calculus of variations]. MR 1296666
  • J. Hadamard, Résolution d’un problème aux limites pour les équations linéaires du type hyperbolique, Bull. Soc. Math. France 32 (1904), 242–268 (French). MR 1504486
  • Atsushi Inoue, Sur $cmu+u^{3}=f$ dans un domaine noncylindrique, J. Math. Anal. Appl. 46 (1974), 777–819 (French). MR 346329, DOI 10.1016/0022-247X(74)90273-X
  • Konrad Jörgens, Über die nichtlinearen Wellengleichungen der mathematischen Physik, Math. Ann. 138 (1959), 179–202 (German). MR 144073, DOI 10.1007/BF01342943
  • Nickolai A. Lar′kin and Márcio H. Simões, Nonlinear wave equation with a nonlinear boundary damping in a noncylindrical domain, Mat. Contemp. 23 (2002), 19–34. Seventh Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 2001). MR 1965774
  • Hans Lindblad and Avy Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys. 73 (2005), no. 3, 249–258. MR 2188297, DOI 10.1007/s11005-005-0021-y
  • B.N. Lu and S.M. Fang, Convergence on finite difference solution of semilinear wave equation in one space variable, Chinese Q. J. Math. 12 (1997), 35–40.
  • J. K. Perring and T. H. R. Skyrme, A model unified field equation, Nuclear Phys. 31 (1962), 550–555. MR 0138393
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
  • Jerome Sather, The existence of a global classical solution of the initial-boundary value problem for $cmu+u^{3}=f$, Arch. Rational Mech. Anal. 22 (1966), 292–307. MR 197965, DOI 10.1007/BF00285421
  • F. Tricomi, Lectures on Partial Differential Equations (Izdat. Inost. Lit., 1957).
  • S. D. Troitskaya, On a boundary value problem for hyperbolic equations, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 2, 193–224 (Russian, with Russian summary); English transl., Izv. Math. 62 (1998), no. 2, 399–428. MR 1623842, DOI 10.1070/im1998v062n02ABEH000180
  • C. Eugene Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), no. 3, 479–528. MR 1040892
  • Yu Lin Zhou, Applications of discrete functional analysis to the finite difference method, International Academic Publishers, Beijing, 1991. MR 1133399
  • A.A. Samarski, R.D. Lazarov and V.L. Makarov. Difference schemes for differential equations with generalized solutions. (Russian) Visshaya Shkola, Moscow, 1987.
  • D. B. Duncan, Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal. 34 (1997), no. 5, 1742–1760. MR 1472194, DOI 10.1137/S0036142993243106
  • Kang Feng and Dao Liu Wang, A note on conservation laws of symplectic difference schemes for Hamiltonian systems, J. Comput. Math. 9 (1991), no. 3, 229–237. MR 1150184
  • Wei Sha, Zhixiang Huang, Xianliang Wu, and Mingsheng Chen, Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation, J. Comput. Phys. 225 (2007), no. 1, 33–50. MR 2346670, DOI 10.1016/
  • Yushun Wang and Bin Wang, High-order multi-symplectic schemes for the nonlinear Klein-Gordon equation, Appl. Math. Comput. 166 (2005), no. 3, 608–632. MR 2150493, DOI 10.1016/j.amc.2004.07.007
  • Zhongqing Wang and Benyu Guo, Legendre rational spectral method for nonlinear Klein-Gordon equation, Numer. Math. J. Chinese Univ. (Engl. Ser.) 15 (2006), no. 2, 143–149. MR 2254925
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Additional Information
  • G. Berikelashvili
  • Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia
  • Email:
  • O. Jokhadze
  • Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia
  • Email:
  • S. Kharibegashvili
  • Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia
  • Email:
  • B. Midodashvili
  • Affiliation: I. Javakhishvili Tbilisi State University, 2, University str., 0186, Tbilisi, Georgia
  • Email:
  • Received by editor(s): May 13, 2008
  • Received by editor(s) in revised form: February 4, 2010
  • Published electronically: August 25, 2010
  • © Copyright 2010 American Mathematical Society
  • Journal: Math. Comp. 80 (2011), 847-862
  • MSC (2010): Primary 65M06, 35L70
  • DOI:
  • MathSciNet review: 2772098