A Meshless LBIE/LRBF Method for Solving the Nonlinear Fisher Equation: Application to Bone Healing
K. N. Grivas, M. G. Vavva, E. J. Sellountos, D. I. Fotiadis and D. Polyzos

doi:10.3970/cmes.2015.105.087
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 105, No. 2, pp. 87-122, 2015
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Keywords Fisher equation, Local Boundary Integral Equation Method, LBIE, Local Radial Basis Functions, Bone healing
Abstract

A simple Local Boundary Integral Equation (LBIE) method for solving the Fisher nonlinear transient diffusion equation in two dimensions (2D) is reported. The method utilizes, for its meshless implementation, randomly distributed nodal points in the interior domain and nodal points corresponding to a Boundary Element Method (BEM) mesh, at the global boundary. The interpolation of the interior and boundary potentials is accomplished using a Local Radial Basis Functions (LRBF) scheme. At the nodes of global boundary the potentials and their fluxes are treated as independent variables. On the local boundaries, potential fluxes are avoided by using the Laplacian companion solution. Potential gradients are accurately evaluated without RBFs via a LBIE, valid for gradient of potentials. Nonlinearity is treated using the Newton-Raphson scheme. The accuracy of the proposed methodology is demonstrated through representative numerical examples. Fisher equation is solved here via the LBIE/LRBF method in order to predict cell proliferation during bone healing. Cell concentrations and their gradients are numerically evaluated in a 2D model of fractured bone. The results are demonstrated and discussed.

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