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A Note on the Multiplier Approach for Derivation of Conservation Laws for Partial Differential Equations in the Complex Domain

  • R. NazEmail author
  • F. M. Mahomed
Conference paper
  • 424 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)

Abstract

We study the conservation laws of scalar partial differential equations (PDEs) with two real independent variables in the complex plane. The complex PDE is split into a system of two real coupled or uncoupled PDEs. We invoke the multiplier method for the derivation of conserved quantities for the complex PDEs and their split systems. The approach is applied to both variational and non-variational complex PDEs. The decomposed complex multipliers of the complex PDE yields two real multipliers for the related split system in the real plane. The multipliers of the split system are derived by utilizing the multiplier method. The multipliers of the split system are compared with the multipliers of the complex PDE after decomposition of the complex multipliers. It is demonstrated that the split multipliers of the complex PDE are not in general the same as the multipliers of the decomposed system of real PDEs. They are shown to be identical when all the multipliers of the complex PDE have either pure real or imaginary parts. We moreover look at the number of conserved vectors that arise by a complex split and from the real system by use of the multipliers.

Keywords

Multiplier approach Complex domain Conservation laws 

Notes

Acknowledgements

F.M. is grateful to the N.R.F. of South Africa for a research grant which has facilitated this work.

References

  1. 1.
    Naz, R., Mahomed, F.M., Mason, D.P.: Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics. Appl. Math. Comput. 205(1), 212–230 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anco, S.C., Bluman, G.: Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78(15), 2869–2873 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anco, S.C., Bluman, G.: Direct construction method for conservation laws of partial differential equations part II: general treatment. Eur. J. Appl. Math. 13(05), 567–585 (2002)zbMATHGoogle Scholar
  4. 4.
    Ali, S., Mahomed, F.M., Qadir, A.: Complex Lie symmetries for variational problems. J. Nonlinear Math. Phys. 15(sup1), 25–35 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Farooq, M.U., Ali, S., Mahomed, F.M.: Two-dimensional systems that arise from the Noether classification of Lagrangians on the line. Appl. Math. Comput. 217(16), 6959–6973 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ali, S., Mahomed, F.M., Qadir, A.: Complex Lie symmetries for scalar second-order ordinary differential equations. Nonlinear Anal.: Real. World Appl. 10(6), 3335–3344 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Soh, C.W., Mahomed, F.M.: Hypercomplex analysis and integration of systems of ordinary differential equations. Math. Methods Appl. Sci. (2016)Google Scholar
  8. 8.
    Mahomed, F.M., Naz, R.: A note on the Lie symmetries of complex partial differential equations and their split real systems. Pramana 77(3), 483–491 (2011)CrossRefGoogle Scholar
  9. 9.
    Naz, R., Mahomed, F.M.: A complex Noether approach for variational partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 27(1), 120–135 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cheviakov, A.F.: GeM software package for computation of symmetries and conservation laws of differential equations. Comput. Phys. Commun. 176(1), 48–61 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cheviakov, A.F.: Computation of fluxes of conservation laws. J. Eng. Math. 66(1–3), 153–173 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Lahore School of EconomicsCentre for Mathematics and Statistical SciencesLahorePakistan
  2. 2.School of Computer Science and Applied Mathematics, DST-NRF Centre of Excellence in Mathematical and Statistical SciencesUniversity of the WitwatersrandJohannesburg, WitsSouth Africa

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