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.
Similar content being viewed by others
Data Availability
Not applicable.
References
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)
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)
Sachdev, P.L., Rao, C.S.: Large time asymptotics for solutions of nonlinear partial differential equations. Springer Science and Business Media, New York (2010)
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)
Basha, H.A.: Burgers’ equation: A general nonlinear solution of infiltration and redistribution. Water Resour. Res. 38(11), 1–9 (2002)
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)
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)
Calogero, F., De Lillo, S.: Flux infiltration into soils: analytic solutions. J. Phys. A: Math. Gen. 27, L137–L142 (1994)
Polyanin, A.D., Nazaikinskii, V.E.: Handbook of linear partial differential equations for engineers and scientists. CRC Press, Taylor and Francis Group (2016)
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications Inc, New York (1972)
Asmar, N.H., Grafakos, L.: Complex analysis with applications. Springer Nature Switzerland AG (2018)
Schiesser, W.E., Griffiths, G.W.: A compendium of partial differential equation models: method of lines analysis with Matlab. Cambridge University Press, Cambridge (2009)
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)
Mukundan, V., Awasthi, A.: Linearized implicit numerical method for Burgers’ equation. Nonlinear Eng. 5(4), 219–234 (2016)
Samokhin, A.V.: The Burgers equation with periodic boundary conditions on an interval. Theor. Math. Phys. 188(3), 1371–1376 (2016)
Acknowledgements
Authors thank the referees for their suggestions which have improved the presentation of the paper.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
Both the authors contributed equally. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
DOI: https://doi.org/10.1007/s40819-022-01413-2