Analytical solutions to a nonlinear diffusion–advection equation

  • Shiva P. PudasainiEmail author
  • Sayonita Ghosh Hajra
  • Santosh Kandel
  • Khim B. Khattri


In this paper, we construct some exact and analytical solutions to a nonlinear diffusion and advection model (Pudasaini in Eng Geol 202: 62–73, 2016) using the Lie symmetry, travelling wave, generalized separation of variables, and boundary layer methods. The model in consideration can be viewed as an extension of viscous Burgers equation, but it describes significantly different physical phenomenon. The nonlinearity in the model is associated with the quadratic diffusion and advection fluxes which are described by the sub-diffusive and sub-advective fluid flow in general porous media and debris material. We also observe that different methods consistently produce similar analytical solutions. This highlights the intrinsic characteristics of the flow of fluid in porous material. The nonlinear diffusion and advection is characterized by a gradually thinning tail that stretches to the rear of the fluid and the evolution of forward advecting frontal bore head, in contrast to the classical linear diffusion and advection. Additionally, we compare solutions for the linear and nonlinear diffusion and advection models highlighting the similarities and differences. The analytical solutions constructed in this paper and the existing high-resolution numerical solution presented previously for the nonlinear diffusion and advection model independently support each other. This implies that the exact and analytical solutions constructed here are physically meaningful and can potentially be applied to calculate the complex nonlinear re-distribution of fluid in porous landscape, and debris and porous materials.


Analytical solutions Exact solutions Nonlinear diffusion–advection Flow in porous media 

Mathematics Subject Classification

35K55 76R50 76S05 76T99 22E60 22E70 86A05 



This work has been financially supported by the German Research Foundation (DFG) through the research projects, PU 386/3-1: “Development of a GIS-based Open Source Simulation Tool for Modelling General Avalanche and Debris Flows over Natural Topography” within a transnational research project, D-A-CH, and PU 386/5-1: “A novel and unified solution to multi-phase mass flows”: U\(^\text {MultiSol}\). Santosh Kandel’s research is partially supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, by the SNF Grant No. 200020 172498/1, and by the COST Action MP1405 QSPACE, supported by COST(European Cooperation in Science and Technology).


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Shiva P. Pudasaini
    • 1
    Email author
  • Sayonita Ghosh Hajra
    • 2
  • Santosh Kandel
    • 3
  • Khim B. Khattri
    • 4
  1. 1.Department of Geophysics, Steinmann InstituteUniversity of BonnBonnGermany
  2. 2.Department of Mathematics and StatisticsCalifornia State University SacramentoSacramentoUSA
  3. 3.Institute for MathematicsUniversity of ZurichZurichSwitzerland
  4. 4.School of ScienceKathmandu UniversityDhulikhel, KavreNepal

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