A weighted combination of reproducing kernel particle shape functions with cardinal functions of scalable polyharmonic spline radial kernel utilized in Galerkin weak form of a mathematical model related to anti-angiogenic therapy

Abstract

In this manuscript, a new localized meshfree (meshless) approximation, i.e., a combination of the reproducing kernel particle (RKP) shape functions with the cardinal functions based on the scalable polyharmonic spline (PHS) radial function is introduced. It is called the RKP+PHS+poly approximation, and its convergence rate is of order O(hm+1), where m is the total degree of polynomials. We apply this method to construct the spaces of trial and test functions in a Galerkin scheme of an extended version of the biological mathematical model in two dimensions describing the interactions between endothelial cells, fibronectin, angiogenic growth factors, and fasentin concentrations. By considering the row-sum method, the eigenvalue stability is also numerically carried out for the discrete equations corresponding to the obtained weak formulation. Accordingly, a semi-implicit form of the backward difference method of order 1 (SBDF1) has been utilized to approximate the weak form in time. We complete our numerical algorithm by solving the obtained full-discretized problem overtime via the biconjugate gradient stabilized (BiCGSTAB) solver with a proper preconditioner. Some simulation results are investigated by estimating the maximum velocity parameter of an enzymatic reaction from the experimental dose–response curve of fasentin to demonstrate the effect of using fasentin drug.