Advertisement

p-Adic Analogue of the Wave Equation

  • Bo WuEmail author
  • Andrei Khrennikov
Article

Abstract

In this paper, a p-adic analogue of the wave equation with Lipschitz source is considered. Since it is hard to arrive the solution of the problem, we propose a regularized method to solve the problem from a modified p-adic integral equation. Moreover, we give an iterative scheme for numerical computation of the regularlized solution.

Keywords

Cauchy problem p-Adic elliptic equation Regularized solution 

Mathematics Subject Classification

11F85 46S10 47G30 

Notes

Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No.11701270), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 17KJB110003) and the Jiangsu Government Scholarship for Overseas Studies. We are grateful to Prof. Weiyi Su for fruitful discussions and valuable comments.

References

  1. 1.
    Albeverio, S., Khrennikov, A.Yu., Shelkovich, V.M.: Harmonic analysis in the \(p\)-adic Lizorkin spaces: fractional operators, pseudo-differential equations, \(p\)-adic wavelets. Tauberian theorems. J. Fourier Anal. Appl. 12(4), 393–425 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albeverio, S., Khrennikov, A.Yu., Shelkovich, V.M.: Theory of \(p\)-adic distributions. In: Linear and Nonlinear Models. Cambridge University Press, Cambridge (2010)Google Scholar
  3. 3.
    Albeverio, S., Khrennikov, A.Yu., Shelkovich, V.M.: The Cauchy problem for evolutionary pseudo-differential equations and the wavelet theory. J. Math. Anal. Appl. 375, 82–98 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aref’eva, I.Ya., Dragovich, B.G., Volovich, I.V.: On the \(p\)-adic summability of the an harmonic oscillator. Phys. Lett. B 200, 512–514 (1988)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Butzer, P.L., Wagner, H.J.: Walsh-Fourier series and the concept of a derivative. Appl. Anal. 3, 29–46 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chuong, N.M., Nguyen, V.C.: The Cauchy problem for a class of pseudo-differential equations over \(p\)-adic field. J. Math. Anal. Appl. 340, 629–645 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dragovich, B.G.: Adelic harmonic oscillator. Int. J. Modern Phys. A 10, 2349–2359 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gibbs, J.E., Millard, M.J.: Walsh functions as a solution of logical differential equations. NPL DES Rept. (1969)Google Scholar
  9. 9.
    Khoa, V.A., Hung, T.T.: Regularity bounds for a Gevrey criterion in a kernel-based regularization of the Cauchy problem of elliptic equations. Appl. Math. Lett. 69, 75–81 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khoa, V.A., Truong, M.T.N., Duy, N.H.M., Tuan, N.H.: The Cauchy problem of coupled elliptic sine-Gordon equations with noise: analysis of a general kernel-based regularization and reliable tools of computing. Comput. Math. Appl. 73, 141–162 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Khrennikov, A.Yu., Kochubei, A.N.: \(p\)-Adic analogue of the porous medium equation. arXiv:1611.08863
  12. 12.
    Khrennikov, A.Yu., Shelkovich, V.M.: Non-haar \(p\)-adic wavelets and their application to pseudo-differential operators and equations. Appl. Comp. Harmon. Anal. 28, 1–23 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Khrennikov, A.Yu., Shelkovich, V.M., Van Der Walt, J.H.: Adelic multiresolution analysis, construction of wavelet bases and pseudo-differential operators. J. Fourier Anal. Appl. 19, 1323–1358 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khrennikov, A.Yu., Oleschko, K., Correa Lopez, M.J.: Application of \(p\)-adic wavelets to model reaction-diffusion dynamics in random porous media. J. Fourier Anal. Appl. 22, 809–822 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Khrennikov, A.Yu., Oleschko, K., Correa Lopez, M.J.: Modeling fluid’s dynamics with master equations in ultrametric spaces representing the treelike structure of capillary networks. Entropy 18, 249 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Khrennikov, A.Yu., Kozyrev, S.V., Zuniga-Galindo, W.A.: Ultrametric Pseudodifferential Equations and Applications. Cambridge University Press, Cambridge (2018)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kochubei, A.N.: Pseudo-Differential Equations and Stochastics Over Non-archimedean Field. CRC Press, Boca Raton (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kochubei, A.N.: Pseudo-Differential Equations and Stochastics Over Non-archimedean Fields. Marcel Dekker, New York (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kochubei, A.N.: A non-archimedean wave equation. Pac. J. Math. 235, 245–261 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kozyrev, S.V.: Wavelet theory as \(p\)-adic spectral analysis. Izv. Math. 66, 367–376 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kozyrev, S.V.: \(P\)-adic pseudodifferential operators and \(p\)-adic wavelets. Theor. Math. Phys. 138, 322–332 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kozyrev, S.V.: Wavelets and spectral analysis of ultrametric pseudodifferential operators. Sb. Math. 198, 97–116 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Onneweer, C.W.: Differentiation on a p-adic or p-series field. In: Linear Spaces and Approximation, pp. 187–198. Birkhauser Verlag, Basel (1978)Google Scholar
  24. 24.
    Qian, Z., Fu, C.L., Li, Z.P.: Two regularization methods for a Cauchy problem for the Laplace equation. J. Math. Anal. Appl. 338, 479–489 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Qiu, H., Su, W.Y.: Pseudo-differential operators over \(p\)-adic fields. Sci. China Ser. A 41(4), 323–336 (2011)Google Scholar
  26. 26.
    Stankovic, R.S.: A note on differential operators on finite non-Abelian groups. Appl. Anal. 21, 31–41 (1986)CrossRefzbMATHGoogle Scholar
  27. 27.
    Su, W.Y.: Psuedo-differential operators and derivatives on locally compact Vilenkin groups. Sci. China Ser. A 35(7), 826–836 (1992)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Su, W.Y.: Harmonic Analysis and Fractal Analysis Over Local Fields and Applications. World Scientific, Singapore (2017)CrossRefGoogle Scholar
  29. 29.
    Taibleson, M.H.: Fourier Analysis on Local Fields. Princeton University Press, Princeton (1975)zbMATHGoogle Scholar
  30. 30.
    Torba, S.M., Zuniga-Galindo, W.A.: Parabolic type equations and Markov stochastic processes on Adeles. J. Fourier Anal. Appl. 19(4), 792–835 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tuan, N.H., Trong, D.D., Quan, P.H.: A note on a Cauchy problem for the Laplace equation: regularization and error estimates. Appl. Math. Comput. 217, 2913–2922 (2010)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Vladimirov, V.S.: Generalized functions over \(p\)-adic number field. Uspekhi Mat. Nauk. 43, 17–53 (1988)Google Scholar
  33. 33.
    Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: \(p\)-Adic Analysis and Mathematical Physics. WSP, Singapore (1994)CrossRefzbMATHGoogle Scholar
  34. 34.
    Volovich, I.V.: \(p\)-Adic string. Class. Quant. Grav. 4, 83–87 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Volovich, I.V.: \(p\)-Adic space-time and string theory. Theor. Math. Phys. 71, 574–576 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zuniga-Galindo, W.A.: Fundamental solutions of pseudo-differential operators over \(p\)-adic fields. Rend. Sem. Mat. Univ. Padova 109, 241–245 (2003)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zuniga-Galindo, W.A.: Parabolic equations and Markov processes over \(p\)-adic fields. Potential Anal. 28, 185–200 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zheng, W.X.: Derivatives and approximation theorems on local fields. Rocky Mt. J. Math. 15, 803–817 (1985)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Finance & EconomicsNanjingChina
  2. 2.International Center for Mathematical Modelling in Physics and Cognitive Science, Mathematical InstituteLinnnaeus University, VaxjoVaxjoSweden

Personalised recommendations