Effective particle methods for Fisher–Kolmogorov equations: Theory and applications to brain tumor dynamics |
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| Institution: | 1. Departamento de Matemáticas, E.T.S.I. Industriales and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain;2. Departamento de Matemáticas, E.T.S.I. Caminos, Canales y Puertos and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain;1. Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang 212013, Jiangsu, China;2. Key Laboratory of Measurement and Control for Complex System of Ministry of Education, Research Institute of Automation, Southeast University, Nanjing 210096, Jiangsu, China;1. Faculty of Physics, University of Belgrade, PO Box 44, 11001 Belgrade, Serbia;2. Department of Physics and Mathematics, Faculty of Pharmacy, University of Belgrade, Vojvode Stepe 450, Belgrade, Serbia;3. Department of Applied Mathematics, Faculty of Mining and Geology, University of Belgrade, PO Box 162, Belgrade, Serbia;4. Scientific Computing Lab., Institute of Physics, University of Beograd, PO Box 68, 11080 Beograd-Zemun, Serbia;1. Department of Computer Engineering, Munzur University, Tunceli, Turkey;2. Department of Mathematics, Federal University, Dutse, Jigawa, Nigeria;3. Department of Mathematics, Firat University, Elazig, Turkey;4. Department of Mathematics, Ordu University, Ordu, Turkey;1. Complex Systems and Cybernetic Control Lab., Faculty of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, P.O. Box 1591634311, Iran;2. Digestive Disease Research Center, Shariati Hospital, Tehran University of Medical Sciences, Tehran, Iran |
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| Abstract: | Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher–Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modeling scenarios. |
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| Keywords: | Fisher–Kolmogorov equations Brain tumors Effective particle methods |
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