Meshfree method for solving mathematical fractional order model of capillary formation in tumor angiogenesis

Document Type : Original Paper

Authors

Department of Mathematics, Lorestan University

Abstract

This paper was devoted to numerical solution of capillary formation in tumor angiogenesis with time fractional derivative. A time discretization approach based on the θ-weighted fractional finite difference scheme was employed for time fractional derivative and a mesh free process was applied by using radial basis functions (RBFs). Stability analysis of the method was also investigated and some numerical cases were studied.

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Main Subjects


[1] Kilbas, A.A., Strvastava, H.M. and Trujilloو J.J. (2006). Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Vol. 204, Elsevier, Amsterdam.
[2] Lakshmikantham, V. and Leela, J.D. (2009). Theory of fractional dynamics systems, Cambridge Scientific Publishers, Cambridge, UK.
[3] Podlubny, I., (2002). Geometric and physical interpretation of fractionalintegration and fractional differentiation, Fract. Calc. Appl. Anal. 5, 367-386.
[4] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo J.J. (2012). Fractional calculua models and numerical methods, World Scientific, Singapore.
[5] Pudlubny, I. (1999). Fractional differential equations, Academic Press, New York.
[6] Hilfr, R. (2000). Applications of fractional calculus in physics, Publishing Company Singapore, World scientific. 
[7] Carpinteri, A. and. Mainardi, F. (1997). Fractional calculus: "Some basic problems in continuum and statistical mechanics", Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, 291-348.
[8] He,J.H., (1999). Some applications Of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15, 86-90.
[9] Luchko,A. and Gorenflo, R. (1998). The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98.
[10] Miller,K.S. and Ross, B. (1993). An Introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc., New York.
[11] Caputo, M. (1969). Elasticita e dissipazione, Bologna, Italy, Zanichelli.
[12] Jumarie, G. (2006). Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl., 51, 1367–1376.
[13] Momani, S. and Odibat, A. (2007). Numerical comparison of methods for solving linear differential equations of fractional order, Appl. Math. Comput. 31, 1248-1255.
[14] Levine, H.A., Pamuk, S., Sleeman, B.D. and Nilsen-Hamilton, M (2001). Mathematical model of capillary formation and development in tumor angiogenesis: penetration into the stroma, Bull. Math. Biol., 63, 801-863.
[15] Pamuk, S. and Endern, A. (2007). The method of lines for the numerical solution of a mathematical model for capillary formation: the role of endothelial cells in the capillary, Appl. Math. Comput. 186, 831–835.
[16] Saadatmandi, A. and Dehghan, M. (2008). Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method,Commun. Number. Math. Eng. 24, 1467–1474.
[17] Abbasbandy, S., Roohani, H.G. and Hashim, I. (2012). Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a mesh free method based on the radial basis function, Engineering Analysis with Boundary Elements: 36, 1811-1818.
[18] Buhmann, M.D. (2004). Radial basis functions: theory and implementation, Cambridge University Press.
[19] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related  interpolators, Comput. Math. Appl., 24, 121-138.
[20] Micchelli, C.A. (1986). Interpolation of scattered  data: distance matrices and conditionally positive definite functions, Constr. Approx.
[21] Horn, R.A. and Johnson, C.R. (2013). Matrix Analysis, second edition, Cambridge University Press.
[22] Hardy, R.L. (1971). Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 176, 1905-1915.