Numerical analysis of Galerkin meshless method for parabolic equations of tumor angiogenesis problem
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Numerical analysis of Galerkin meshless method for parabolic equations of tumor angiogenesis problem Hadi Jahanshahi1, Kamal Shanazari2 , Mehdi Mesrizadeh3 , Samaneh Soradi-Zeid4 , J. F. Gómez-Aguilar5,a 1 2 3 4
Department of Mechanical Engineering, University of Manitoba, Winnipeg R3T 5V6, Canada Department of Mathematics, University of Kurdistan, Sanandaj, Iran Department of Mathematics, Imam Khomeini International University, Qazvin, Iran 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 Received: 12 July 2020 / Accepted: 26 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 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.
1 Introduction Numerous phenomena are modeled by partial differential equations (PDEs) which depend on the state of the physical system [1,2] or specifically, cancer included [3,4]. Over the last few years, so many mathematical models of tumor growth have appeared in the research literature, for example, see [5–11]. The mathematical model for capillary formation in tumor angiogenesis is originally presented in [12]. In this model, Levine et al. used the cell transport (chemotactic) equations and developed the model using the theory of reinforced random walk derived by David [13] and performed a primary mathematical model of governing the motion of endothelial cells. Also, this model was used by Stevens and Othmer [14] to model fruiting bodies. Some work has been done on developing numerical methods to get the approximate solutions of the capillary formation in tumor angiogenesis. Saadatmandi and Dehghan [15] studied a numerical scheme based on the modified Legendre tau method to solve the problem of this model. Their approach was based on reducing the tumor angiogenesis problem to
a e-mail: [email protected] (corresponding author)
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