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**Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method.**
*(English)*
Zbl 1151.92017

Summary: A numerical procedure is developed to obtain the solution of a mathematical model for capillary formation in tumor angiogenesis. The proposed method is based on the shifted Legendre tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre polynomial with unknown coefficients. The operational matrices of integrals and derivatives together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. An illustrative example is included to demonstrate the validity and applicability of the presented technique.

### MSC:

92C50 | Medical applications (general) |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35K15 | Initial value problems for second-order parabolic equations |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

### Keywords:

numerical solutions; shifted Legendre tau method; capillary formation; tumor angiogenesis; operational matrix
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\textit{A. Saadatmandi} and \textit{M. Dehghan}, Commun. Numer. Methods Eng. 24, No. 11, 1467--1474 (2008; Zbl 1151.92017)

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### References:

[1] | Pamuk, Qualitative analysis of a mathematical model for capillary formation in tumor angiogenesis, Mathematical Models and Methods in Applied Sciences 13 (1) pp 19– (2003) · Zbl 1043.92014 |

[2] | Pamuk, Steady-state analysis of a mathematical model for capillary network formation in the absence of tumor source, Mathematical Biosciences 189 pp 21– (2004) · Zbl 1072.92027 |

[3] | Pamuk, The method of lines for the numerical solution of a mathematical model for capillary formation: the role of endothelial cells in the capillary, Applied Mathematics and Computation 186 pp 831– (2007) · Zbl 1114.65111 |

[4] | Levine, Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma, Bulletin of Mathematical Biology 63 (5) pp 801– (2001) · Zbl 1323.92029 |

[5] | Angelis, Advection-diffusion models for solid tumour evolution in vivo and related free boundary problem, Mathematical Models and Methods in Applied Sciences 10 (3) pp 379– (2000) · Zbl 1008.92017 |

[6] | Anderson, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Applied Mathematics Letters 11 pp 109– (1998) · Zbl 0935.92024 |

[7] | Dehghan, Numerical solution of a parabolic equation subject to specification of energy, Applied Mathematics and Computation 149 pp 31– (2004) · Zbl 1038.65088 |

[8] | Lanczos, Applied Analysis (1956) |

[9] | Razzaghi, Tau method approximation for radiative transfer problems in a slab medium, Journal of Quantitative Spectroscopy and Radiative Transfer 72 (4) pp 439– (2002) |

[10] | Saadatmandi, A tau method approximation for the diffusion equation with nonlocal boundary conditions, International Journal of Computer Mathematics 81 pp 1427– (2004) |

[11] | Saadatmandi, Numerical solution of the one-dimensional wave equation with an integral condition, Numerical Methods for Partial Differential Equations 23 pp 282– (2007) · Zbl 1112.65097 |

[12] | Canuto, Spectral Methods in Fluid Dynamic (1988) · doi:10.1007/978-3-642-84108-8 |

[13] | Gottlieb, Theory and Applications of Spectral Methods for Partial Differential Equations (1984) |

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