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Lie symmetries of generalized Burgers equations: application to boundary-value problems

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Abstract

There exist several approaches exploiting Lie symmetries in the reduction of boundary-value problems for partial differential equations modelling real-world phenomena to those problems for ordinary differential equations. Using an example of generalized Burgers equations appearing in non-linear acoustics we show that the direct procedure of solving boundary-value problems using Lie symmetries first described by Bluman is more general and straightforward than the method suggested by Moran and Gaggioli [J Eng Math 3:151–162, 1969]. After performing group classification of a class of generalized Burgers equations with time-dependent viscosity we solve an associated boundary-value problem using the symmetries obtained.

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Notes

  1. Note that if \(a=1/n\), the equivalence group \(\hat{G}^\sim _2\) was found previously in [36] (see also [37, 38]) in the course of studying form-preserving (admissible) transformations of the class of generalized Burgers equations, \(u_t+uu_x+f(t,x)u_{xx}=0.\)

References

  1. Bluman GW, Anco SC (2002) Symmetry and integration methods for differential equations. Springer, New York

    MATH  Google Scholar 

  2. Moran M, Gaggioli R (1968) Reduction of the number of variables in systems of partial differential equations, with auxiliary conditions. SIAM J Appl Math 16:202–215

    Article  MathSciNet  MATH  Google Scholar 

  3. Moran M, Gaggioli R (1969) A new systematic formalism for similarity analysis. J Eng Math 3:151–162

    Article  MathSciNet  MATH  Google Scholar 

  4. Bluman GW, Cole JD (1969) The general similarity solution of the heat equation. J Math Mech 18:1025–1042

    MathSciNet  MATH  Google Scholar 

  5. Bluman GW (1974) Application of the general similarity solution of the heat equation to boundary-value problems. Quart Appl Math 31:403–415

    MathSciNet  MATH  Google Scholar 

  6. Burde GI (1996) New similarity reductions of steady-state boundary layer equations. J Phys A 29:1665–1683

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Sophocleous C, O’Hara JG, Leach PGL (2011) Symmetry analysis of a model of stochastic volatility with time-dependent parameters. J Comput Appl Math 235:4158–4164

    Article  MathSciNet  MATH  Google Scholar 

  8. Sophocleous C, O’Hara JG, Leach PGL (2011) Algebraic solution of the Stein–Stein model for stochastic volatility. Commun Nonlinear Sci Numer Simul 16:1752–1759

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. O’Hara JG, Sophocleous C, Leach PGL (2013) Application of Lie point symmetries to the resolution of certain problems in financial mathematics with a terminal condition. J Eng Math 82:67–75

    Article  MathSciNet  Google Scholar 

  10. Abd-el-Malek MB, Helal MM (2011) Group method solutions of the generalized forms of Burgers, Burgers-KdV and KdV equations with time-dependent variable coefficients. Acta Mech 221:281–296

    Article  MATH  Google Scholar 

  11. Kovalenko SS (2011) Symmetry analysis and exact solutions of one class of (1+3)-dimensional boundary-value problems of the Stefan type. Ukr Math J 63:1534–1542

    Article  MathSciNet  Google Scholar 

  12. Kovalenko SS (2012) Lie symmetries and exact solutions of certain boundary-value problems with free boundaries. PhD thesis, Institute of Mathematics of NAS of Ukraine, Kiev (in Ukrainian)

  13. Bihlo A, Popovych RO (2012) Invariant discretization schemes for the shallow-water equations. SIAM J Sci Comput 34(6):B810–B839

    Article  MathSciNet  MATH  Google Scholar 

  14. Popovych RO, Bihlo A (2012) Symmetry preserving parameterization schemes. J Math Phys 53:073102

    Article  ADS  MathSciNet  Google Scholar 

  15. Goard J, Al-Nassar S (2014) Symmetries for initial value problems. Appl Math Lett 28:56–59

    Article  MathSciNet  Google Scholar 

  16. Forsyth AR (1959) Theory of differential equations. Part 4. Partial differential equations, vol 5–6. Dover, New York

  17. Bateman H (1915) Some recent researches on the motion of fluids. Mon Weather Rev 43(4):163–170

    Article  ADS  Google Scholar 

  18. Burgers JM (1948) A mathematical model illustrating the theory of turbulence. Adv Appl Mech 1:171–199

    Article  MathSciNet  Google Scholar 

  19. Hopf E (1950) The partial differential equation \(u_t + u u_x = \mu u_{xx}\). Commun Pure Appl Math 33:201–230

    Article  MathSciNet  Google Scholar 

  20. Cole JD (1951) On a quasilinear parabolic equation occurring in aerodynamics. Quart Appl Math 9:225–236

    MathSciNet  MATH  Google Scholar 

  21. Calogero F (1991) Why are certain nonlinear PDEs both widely applicable and integrable? In: What is integrability?. Springer Ser Nonlinear Dynam, Springer, Berlin, pp 1–62

  22. Hammerton PW, Crighton DG (1989) Approximate solution methods for nonlinear acoustic propagation over long ranges. Proc R Soc Lond A 426:125–152

  23. Doyle J, Englefield MJ (1990) Similarity solutions of a generalized Burgers equation. IMA J Appl Math 44:145–153

    Article  MathSciNet  MATH  Google Scholar 

  24. Wafo Soh C (2004) Symmetry reductions and new exact invariant solutions of the generalized Burgers equation arising in nonlinear acoustics. Int J Eng Sci 42:1169–1191

    Article  MathSciNet  MATH  Google Scholar 

  25. Sachdev PL (1987) Nonlinear diffusive waves. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  26. Sachdev PL (2000) Self-Similarity and beyond: exact solutions of nonlinear problems. Chapman & Hall/CRC, Boca Raton

    Book  Google Scholar 

  27. Leach JA (2013) A quarter-plane problem for the modified Burgers’ equation. J Math Phys 54:091502

    Article  ADS  MathSciNet  Google Scholar 

  28. Ovsiannikov LV (1982) Group analysis of differential equations. Academic Press, New York

    MATH  Google Scholar 

  29. Gasizov R, Ibragimov NH (1998) Lie symmetry analysis of differential equations in finance. Nonlinear Dyn 17:387–407

    Article  Google Scholar 

  30. Popovych RO, Kunzinger M, Eshraghi H (2010) Admissible transformations and normalized classes of nonlinear Schrödinger equations. Acta Appl Math 109:315–359

    Article  MathSciNet  MATH  Google Scholar 

  31. Kingston JG, Sophocleous C (1998) On form-preserving point transformations of partial differential equations. J Phys A 31:1597–1619

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Winternitz P, Gazeau JP (1992) Allowed transformations and symmetry classes of variable coefficient Korteweg-de Vries equations. Phys Lett A 167:246–250

    Article  ADS  MathSciNet  Google Scholar 

  33. Vaneeva OO, Popovych RO, Sophocleous C (2014) Equivalence transformations in the study of integrability. Phys Scr 89:038003

    Article  ADS  Google Scholar 

  34. Vaneeva OO, Popovych RO, Sophocleous C (2012) Extended group analysis of variable coefficient reaction-diffusion equations with exponential nonlinearities. J Math Anal Appl 396:225–242

    Article  MathSciNet  MATH  Google Scholar 

  35. Popovych RO, Ivanova NM (2004) New results on group classification of nonlinear diffusion-convection equations. J Phys A 37:7547–7565

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Kingston JG, Sophocleous C (1991) On point transformations of a generalized Burgers equation. Phys Lett A 155:15–19

    Article  ADS  MathSciNet  Google Scholar 

  37. Pocheketa OA, Popovych RO (2012) Reduction operators and exact solutions of generalized Burgers equations. Phys Lett A 376:2847–2850

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Pocheketa OA (2013) Normalized classes of generalized Burgers equations. Proc 6th workshop group analysis of differential equations and integrable systems, Protaras, Cyprus, 2012. University of Cyprus, Nicosia, pp 170–178

  39. Olver PJ (1986) Applications of Lie groups to differential equations. Springer, New York

  40. Lie S (1881) Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung. Arch Math 6(3):328–368. (Translation: Lie S (1994) On integration of a class of linear partial differential equations by means of definite integrals. In: CRC Handbook of Lie group analysis of differential equations, vol 2. Boca Raton, pp 473–508)

  41. Vaneeva OO, Papanicolaou NC, Christou MA, Sophocleous C (2014) Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries. Commun Nonlinear Sci Numer Simulat 19:3074–3085

    Article  ADS  MathSciNet  Google Scholar 

  42. Kuriksha O, Pošta S, Vaneeva O (2014) Group classification of variable coefficient generalized Kawahara equations. J Phys A 47:045201

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgments

The authors thank the five referees for their constructive suggestions for the improvement of this paper. OV is grateful for the hospitality and financial support of the University of Cyprus and to Roman Popovych and Sergii Kovalenko for useful comments. PGLL thanks the University of Cyprus for its kind hospitality and the University of KwaZulu-Natal and the National Research Foundation of South Africa for their continued support.

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Authors and Affiliations

  1. Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kiev-4,  01601, Ukraine

    O. O. Vaneeva

  2. Department of Mathematics and Statistics, University of Cyprus, Nicosia,  1678, Cyprus

    C. Sophocleous & P. G. L. Leach

  3. School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban, 4000, South Africa

    P. G. L. Leach

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  1. O. O. Vaneeva
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  2. C. Sophocleous
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  3. P. G. L. Leach
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Correspondence to C. Sophocleous.

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Vaneeva, O.O., Sophocleous, C. & Leach, P.G.L. Lie symmetries of generalized Burgers equations: application to boundary-value problems. J Eng Math 91, 165–176 (2015). https://doi.org/10.1007/s10665-014-9741-2

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