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Mathematical Biosciences and Engineering

2015, Volume 12, Issue 1: 41-69. doi: 10.3934/mbe.2015.12.41
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Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach

  • 1.
    Department of Mathematics, University College London, Gower Street, London WC1E 6BT
  • 2.
    Department of Computing and Information Management, Hong Kong Institute of Vocational Education, Chai Wan, Hong Kong
  • Received: 01 May 2014 Accepted: 29 June 2018 Published: 01 December 2014

  • MSC : Primary: 35C07, 34A05; Secondary: 92C37.

  • We consider quasi-stationary (travelling wave type) solutions to a nonlinearreaction-diffusion equation with arbitrary, autonomous coefficients,describing the evolution of glioblastomas, aggressive primary brain tumorsthat are characterized by extensive infiltration into the brain and arehighly resistant to treatment. The second order nonlinear equationdescribing the glioblastoma growth through travelling waves can be reduced to a first order Abel typeequation. By using the integrability conditions for the Abel equationseveral classes of exact travelling wave solutions of the general reaction-diffusion equation that describes glioblastoma growth are obtained, corresponding to different forms of the product of the diffusion and reaction functions. The solutions are obtained by using the Chiellini lemma and the Lemke transformation, respectively, and the corresponding equations represent generalizations of the classical Fisher--Kolmogorov equation. The biological implications of two classes of solutions are also investigated by using both numerical and semi-analytical methods for realistic values of the biological parameters.

    Citation: Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach[J]. Mathematical Biosciences and Engineering, 2015, 12(1): 41-69. doi: 10.3934/mbe.2015.12.41

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  • We consider quasi-stationary (travelling wave type) solutions to a nonlinearreaction-diffusion equation with arbitrary, autonomous coefficients,describing the evolution of glioblastomas, aggressive primary brain tumorsthat are characterized by extensive infiltration into the brain and arehighly resistant to treatment. The second order nonlinear equationdescribing the glioblastoma growth through travelling waves can be reduced to a first order Abel typeequation. By using the integrability conditions for the Abel equationseveral classes of exact travelling wave solutions of the general reaction-diffusion equation that describes glioblastoma growth are obtained, corresponding to different forms of the product of the diffusion and reaction functions. The solutions are obtained by using the Chiellini lemma and the Lemke transformation, respectively, and the corresponding equations represent generalizations of the classical Fisher--Kolmogorov equation. The biological implications of two classes of solutions are also investigated by using both numerical and semi-analytical methods for realistic values of the biological parameters.


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