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Asymptotic Solutions of Burgers Equation and Modified Burgers Equation Satisfying Flux Type Conditions

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Abstract

In this article, we solve exactly an initial boundary value problem (IBVP) for Burgers equation satisfying flux type conditions using Cole-Hopf transformation. Asymptotic expansions of the exact solution are obtained for different regions of the quarter plane \(x\ge 0,~t\ge 0\). We observe that travelling wave solution or stationary solution describes the large time behaviour of the solutions of the initial boundary value problem in different parameter ranges. A numerical study of the large time behaviour of solutions of Burgers equation and modified Burgers equation is also presented.

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References

  1. Burgers, J.M.: Application of a model system to illustrate some points of the statistical theory of turbulence. Proc. Roy Neth. Acad. Sci. Amst 43, 2–12 (1940)

    MathSciNet  MATH  Google Scholar 

  2. Joseph, K.T., Sachdev, P.L.: Exact analysis of Burgers equation on semiline with flux condition at the origin. Int. J. Non-Linear mechanics 28, 627–639 (1993)

    Article  MathSciNet  Google Scholar 

  3. Sachdev, P.L., Rao, C.S.: Large time asymptotics for solutions of nonlinear partial differential equations. Springer Science and Business Media, New York (2010)

    Book  Google Scholar 

  4. Biondini, G., De Lillo, S.: Semiline solutions of the Burgers equation with time dependent flux at the origin. Phys. Lett. A 220, 201–204 (1996)

    Article  MathSciNet  Google Scholar 

  5. Basha, H.A.: Burgers’ equation: A general nonlinear solution of infiltration and redistribution. Water Resour. Res. 38(11), 1–9 (2002)

    Article  Google Scholar 

  6. Calogero, F., De Lillo, S.: The Burgers equation on the semiline with general boundary conditions at the origin. J. Math. Phys. 32, 99–105 (1991)

    Article  MathSciNet  Google Scholar 

  7. Calogero, F., De Lillo, S.: The nonlinear diffusion-convection equation on the semiline with time-dependent flux at the origin. Theor. Math. Phys. 99, 531–537 (1994)

    Article  MathSciNet  Google Scholar 

  8. Calogero, F., De Lillo, S.: Flux infiltration into soils: analytic solutions. J. Phys. A: Math. Gen. 27, L137–L142 (1994)

    Article  MathSciNet  Google Scholar 

  9. Polyanin, A.D., Nazaikinskii, V.E.: Handbook of linear partial differential equations for engineers and scientists. CRC Press, Taylor and Francis Group (2016)

    MATH  Google Scholar 

  10. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications Inc, New York (1972)

    MATH  Google Scholar 

  11. Asmar, N.H., Grafakos, L.: Complex analysis with applications. Springer Nature Switzerland AG (2018)

    Book  Google Scholar 

  12. Schiesser, W.E., Griffiths, G.W.: A compendium of partial differential equation models: method of lines analysis with Matlab. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  13. Griffiths, G.W., Schiesser, W.E.: Traveling wave analysis of partial differential equations: numerical and analytical methods with Matlab and Maple. Academic Press, Burlington, USA (2012)

    MATH  Google Scholar 

  14. Mukundan, V., Awasthi, A.: Linearized implicit numerical method for Burgers’ equation. Nonlinear Eng. 5(4), 219–234 (2016)

    Article  Google Scholar 

  15. Samokhin, A.V.: The Burgers equation with periodic boundary conditions on an interval. Theor. Math. Phys. 188(3), 1371–1376 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Authors thank the referees for their suggestions which have improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Magadh University, Bihar, 824234, India

    Pintu Samanta

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    Ch. Srinivasa Rao

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  1. Pintu Samanta
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  2. Ch. Srinivasa Rao
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Both the authors contributed equally. All authors read and approved the final manuscript.

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Correspondence to Ch. Srinivasa Rao.

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Samanta, P., Rao, C.S. Asymptotic Solutions of Burgers Equation and Modified Burgers Equation Satisfying Flux Type Conditions. Int. J. Appl. Comput. Math 8, 205 (2022). https://doi.org/10.1007/s40819-022-01413-2

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  • DOI: https://doi.org/10.1007/s40819-022-01413-2

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