Eur. Phys. J. Plus (2020) 135: 866
https://doi.org/10.1140/epjp/s13360-020-00716-x

Regular Article

Numerical analysis of Galerkin meshless method for parabolic equations of tumor angiogenesis problem

¹ Department of Mechanical Engineering, University of Manitoba, Winnipeg, R3T 5V6, Canada
² Department of Mathematics, University of Kurdistan, Sanandaj, Iran
³ Department of Mathematics, Imam Khomeini International University, Qazvin, Iran
⁴ Faculty of Industry and Mining (Khash), University of Sistan and Baluchestan, Zahedan, Iran
⁵ CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico

^* e-mail: jose.ga@cenidet.tecnm.mx

Received: 12 July 2020
Accepted: 26 August 2020
Published online: 30 October 2020

Abstract

The stability and convergence of the Galerkin method for differential equations with symmetric operators have been confirmed with numerical results, while this is not the case when dealing with unsymmetric operators. In the present study, a sort of transformation is used as a preconditioner to convert the unsymmetric operator to a symmetric one. This method is implemented on the capillary formation mathematical model of tumor angiogenesis problem. Then, a Galerkin meshfree method based on the radial basis functions is presented for the numerical solution of this problem. The proposed strategy is based on applying the Galerkin method and group preserving scheme for the spatial and time variables, respectively. Also, the stability and the convergence of proposed method is considered. In addition, some of the advantages of the proposed technique over existing methods are shown. Finally, some numerical results will be provided to validate the theoretical achievements.