THE NUMERICAL MESHLESS APPROACH FOR SOLVING THE 2D TIME NONLINEAR MULTI-TERM FRACTIONAL CABLE EQUATION IN COMPLEX GEOMETRIES

Abstract

The cable equation plays a prominent role in biological neuron models, for instance, in spiking neuron models and electrophysiology. Thus, the current investigation scrutinizes the 2D time nonlinear multi-term fractional cable equation. By adopting a valid meshfree technique, the nonlinear multi-term time-fractional cable equations (NM-TTFCEs) that consist of government equations and their boundary conditions are transformed into the boundary value problems. For this purpose, the finite difference method is derived for temporal discretization such that the considered NM-TTFCEs can be transformed into a sequence of boundary value problems in inhomogeneous Helmholtz-type equations. The dual reciprocity method (DRM) is implemented to obtain a particular solution and the improved singular boundary method (ISBM) is employed to evaluate the homogeneous solution. Moreover, we apply the meshless method for solving two-dimensional NM-TTFCEs on regular and irregular distribution points with several computational domains. The numerical results vouch for the accuracy and high efficiency of the proposed method. Finally, we will conclude that the ISBM/DRM method can be considered a potential alternative to existing meshless strong form approaches in solving multi-term fractional equation problems with complex geometries.