Research Article
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Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain

Year 2024, Volume: 73 Issue: 3, 833 - 844, 27.09.2024
https://doi.org/10.31801/cfsuasmas.1434079

Abstract

In this study, the finite propogation speed properties investigated for a two dimensional exterior problem defined by nonlinear Klein-Gordon equation. Under some assumptions on the initial data and the nonlinearity, the solution is shown to have a finite propogation speed. Furthermore, it is demonstrated that the problem has a unique solution, and accurate numerical solutions have been produced by the use of the dual reciprocity boundary element approach with linear radial basis functions.

References

  • Segel, L. A., Mathematics Applied to Continuum Mechanics, Macmillan Publication, New York, 1977.
  • Whitham, G. B., Linear and Nonlinear Waves, Wiley Interscience Publication, New York, 1974.
  • Axelsson, O., Finite Difference Methods, Encyclopedia of Computational Mechanics, Stein E., de Borst R., Hughes T., eds., Vol. 1, chap. 2, John Wiley & Sons. Ltd., West Sussex, 2004.
  • Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, United States of America, 1998.
  • Macias-Diaz, J. E., Medina-Guavera, M. G., Vargas-Rodriguez, H., Exact solutions of non-linear Klein-Gordon equation with non-constant coefficients through the trial equation method, J. Math. Chem., 59 (2021) 827-839. https://doi.org/10.1007/s10910-021-01220-y
  • Nakao, M., Energy decay for a nonlinear generalized Klein-Gordon equation in exterior domains with a nonlinear localized dissipative term, J. Math. Soc. Japan., 64(3), (2012) 851-883. https://doi.org/10.2969/jmsj/06430851
  • Mohamad, H., Energy asymptotics for the strongly damped Klein-Gordon equation, Partial Differ. Equ., 3(71) 2022, 1-12. https://doi.org/10.1007/s42985-022-00207-x
  • Datti, P. S., Nonlinear wave equations in exterior domains, Nonlinear Anal. Theory Methods Appl., 15(4), (1990) 312-331. https://doi.org/10.1016/0362-546X(90)90140-C
  • Taskesen, H., Global existence and nonexistence of solutions for a Klein-Gordon equation with exponential type nonlinear term, TWMS J. App. and Eng. Math., 10(3), (2020) 669-676.
  • Hörmander, L., Remarks on the Klein-Gordon equation, J. Equations aux Derivees Partielles, (1987), 1-9.
  • Malloug, M., Local energy decay for the damped Klein-Gordon equation in exterior domain, Appl. Anal., 96(2)(2017), 349-362. https://doi.org/10.1080/00036811.2015.1136821
  • Dehghan, M., Shokri, A., Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, J. Comput. Appl. Math., 230 (2009), 400-410. https://doi.org/10.1016/j.cam.2008.12.011
  • Bülbül, B., Sezer, M., A new approach to numerical solution of nonlinear Klein-Gordon equation, Math. Probl. Eng., 2013(2013), 1-7. http://dx.doi.org/10.1155/2013/869749
  • Macias-Diaz, J. E., Puri, A., A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein Gordon equation, Numer. Methods Partial Differ. Equ., 21(5), (2005),998-1015. https://doi.org/10.1002/num.20094
  • Macias-Diaz, J. E., On the bifurcation of energy in media governed by (2+1)-dimensional modified Klein-Gordon equations, Appl. Math. Comput., 206, (2008), 221-235. https://doi.org/10.1016/j.amc.2008.09.013
  • Pekmen, B., Tezer-Sezgin, M., Differential quadrature solution of nonlinear Klein-Gordon and sine-Gordon equations, Comput. Phys. Commun., 183 (2012), 1702-1713. https://doi.org/10.1016/j.cpc.2012.03.010
  • Tai, Y., Zhou, Z., Jiang, Z., Numerical solution of coupled nonlinear Klein-Gordon equations on unbounded domains, Phys. Rev. E., 106(2) (2022), 025317-1-10. https://doi.org/10.1103/PhysRevE.106.025317
  • Givoli, D., Recent Advances in the DtN FE Method, Arch. Comput. Methods Eng., 6(2), 71-116, 1999. https://doi.org/10.1007/BF02736182
  • Meral, G., Tezer-Sezgin, M., DRBEM solution of exterior wave problem using FDM and LSM time integrations. Eng. Anal. Bound. Elem., 34 (2010) 574-580. https://doi.org/10.1016/j.enganabound.2010.01.006
  • Givoli, D., Patlashenko, I., Finite element solution of nonlinear time dependent exterior wave problems, Jour. of Comput. Phys., 143 (1998) 241-258. https://doi.org/10.1006/jcph.1998.9999
  • Brebbia, C. A., Dominguez, J., Boundary Elements an Introductory Course, 2nd edn. Comput. Mech. Publications, Southampton, Boston, 1992.
  • Senel, P., Comparison study on the numerical stability of dual reciprocity boundary element method for the MHD slip flow problem, Eng. Anal. Bound. Elem., 151(2023) 370-386. https://doi.org/10.1016/j.enganabound.2023.03.010
  • Meral, G., DRBEM-FDM solution of a chemotaxis–haptotaxis model for cancer invasion, J. Comput. Appl. Math., 354 (2019) 299-309. https://doi.org/10.1016/j.cam.2018.04.047
Year 2024, Volume: 73 Issue: 3, 833 - 844, 27.09.2024
https://doi.org/10.31801/cfsuasmas.1434079

Abstract

References

  • Segel, L. A., Mathematics Applied to Continuum Mechanics, Macmillan Publication, New York, 1977.
  • Whitham, G. B., Linear and Nonlinear Waves, Wiley Interscience Publication, New York, 1974.
  • Axelsson, O., Finite Difference Methods, Encyclopedia of Computational Mechanics, Stein E., de Borst R., Hughes T., eds., Vol. 1, chap. 2, John Wiley & Sons. Ltd., West Sussex, 2004.
  • Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, United States of America, 1998.
  • Macias-Diaz, J. E., Medina-Guavera, M. G., Vargas-Rodriguez, H., Exact solutions of non-linear Klein-Gordon equation with non-constant coefficients through the trial equation method, J. Math. Chem., 59 (2021) 827-839. https://doi.org/10.1007/s10910-021-01220-y
  • Nakao, M., Energy decay for a nonlinear generalized Klein-Gordon equation in exterior domains with a nonlinear localized dissipative term, J. Math. Soc. Japan., 64(3), (2012) 851-883. https://doi.org/10.2969/jmsj/06430851
  • Mohamad, H., Energy asymptotics for the strongly damped Klein-Gordon equation, Partial Differ. Equ., 3(71) 2022, 1-12. https://doi.org/10.1007/s42985-022-00207-x
  • Datti, P. S., Nonlinear wave equations in exterior domains, Nonlinear Anal. Theory Methods Appl., 15(4), (1990) 312-331. https://doi.org/10.1016/0362-546X(90)90140-C
  • Taskesen, H., Global existence and nonexistence of solutions for a Klein-Gordon equation with exponential type nonlinear term, TWMS J. App. and Eng. Math., 10(3), (2020) 669-676.
  • Hörmander, L., Remarks on the Klein-Gordon equation, J. Equations aux Derivees Partielles, (1987), 1-9.
  • Malloug, M., Local energy decay for the damped Klein-Gordon equation in exterior domain, Appl. Anal., 96(2)(2017), 349-362. https://doi.org/10.1080/00036811.2015.1136821
  • Dehghan, M., Shokri, A., Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, J. Comput. Appl. Math., 230 (2009), 400-410. https://doi.org/10.1016/j.cam.2008.12.011
  • Bülbül, B., Sezer, M., A new approach to numerical solution of nonlinear Klein-Gordon equation, Math. Probl. Eng., 2013(2013), 1-7. http://dx.doi.org/10.1155/2013/869749
  • Macias-Diaz, J. E., Puri, A., A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein Gordon equation, Numer. Methods Partial Differ. Equ., 21(5), (2005),998-1015. https://doi.org/10.1002/num.20094
  • Macias-Diaz, J. E., On the bifurcation of energy in media governed by (2+1)-dimensional modified Klein-Gordon equations, Appl. Math. Comput., 206, (2008), 221-235. https://doi.org/10.1016/j.amc.2008.09.013
  • Pekmen, B., Tezer-Sezgin, M., Differential quadrature solution of nonlinear Klein-Gordon and sine-Gordon equations, Comput. Phys. Commun., 183 (2012), 1702-1713. https://doi.org/10.1016/j.cpc.2012.03.010
  • Tai, Y., Zhou, Z., Jiang, Z., Numerical solution of coupled nonlinear Klein-Gordon equations on unbounded domains, Phys. Rev. E., 106(2) (2022), 025317-1-10. https://doi.org/10.1103/PhysRevE.106.025317
  • Givoli, D., Recent Advances in the DtN FE Method, Arch. Comput. Methods Eng., 6(2), 71-116, 1999. https://doi.org/10.1007/BF02736182
  • Meral, G., Tezer-Sezgin, M., DRBEM solution of exterior wave problem using FDM and LSM time integrations. Eng. Anal. Bound. Elem., 34 (2010) 574-580. https://doi.org/10.1016/j.enganabound.2010.01.006
  • Givoli, D., Patlashenko, I., Finite element solution of nonlinear time dependent exterior wave problems, Jour. of Comput. Phys., 143 (1998) 241-258. https://doi.org/10.1006/jcph.1998.9999
  • Brebbia, C. A., Dominguez, J., Boundary Elements an Introductory Course, 2nd edn. Comput. Mech. Publications, Southampton, Boston, 1992.
  • Senel, P., Comparison study on the numerical stability of dual reciprocity boundary element method for the MHD slip flow problem, Eng. Anal. Bound. Elem., 151(2023) 370-386. https://doi.org/10.1016/j.enganabound.2023.03.010
  • Meral, G., DRBEM-FDM solution of a chemotaxis–haptotaxis model for cancer invasion, J. Comput. Appl. Math., 354 (2019) 299-309. https://doi.org/10.1016/j.cam.2018.04.047
There are 23 citations in total.

Details

Primary Language English
Subjects Numerical Analysis
Journal Section Research Articles
Authors

Gülnihal Meral 0000-0003-0072-0609

Publication Date September 27, 2024
Submission Date February 8, 2024
Acceptance Date May 10, 2024
Published in Issue Year 2024 Volume: 73 Issue: 3

Cite

APA Meral, G. (2024). Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(3), 833-844. https://doi.org/10.31801/cfsuasmas.1434079
AMA Meral G. Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2024;73(3):833-844. doi:10.31801/cfsuasmas.1434079
Chicago Meral, Gülnihal. “Mathematical Analysis and Numerical Simulations for a Nonlinear Klein Gordon Equation in an Exterior Domain”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, no. 3 (September 2024): 833-44. https://doi.org/10.31801/cfsuasmas.1434079.
EndNote Meral G (September 1, 2024) Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 3 833–844.
IEEE G. Meral, “Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 73, no. 3, pp. 833–844, 2024, doi: 10.31801/cfsuasmas.1434079.
ISNAD Meral, Gülnihal. “Mathematical Analysis and Numerical Simulations for a Nonlinear Klein Gordon Equation in an Exterior Domain”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/3 (September 2024), 833-844. https://doi.org/10.31801/cfsuasmas.1434079.
JAMA Meral G. Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:833–844.
MLA Meral, Gülnihal. “Mathematical Analysis and Numerical Simulations for a Nonlinear Klein Gordon Equation in an Exterior Domain”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 73, no. 3, 2024, pp. 833-44, doi:10.31801/cfsuasmas.1434079.
Vancouver Meral G. Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(3):833-44.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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