×

A local domain boundary element method for solving the nonlinear Fisher KPP diffusion-reaction equation. (English) Zbl 1521.74319


MSC:

74S15 Boundary element methods applied to problems in solid mechanics
65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs

Software:

PolyMesher
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fisher, RA, The wave of advance of advantageous genes, Ann Hum Genet, 7, 355-369 (1937) · JFM 63.1111.04
[2] Kolmogorov, A.; Petrovsky, I.; Piscounov, N., Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Bull. Math., 1, 1-25 (1937) · Zbl 0018.32106
[3] Murray, JD, Mathematical Biology I: An Introduction (2002), Springer-Verlag: Springer-Verlag Berlin Heidelberg
[4] Murray, JD, Mathematical Biology II: Spatial Models and Biomedical Applications (2003), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 1006.92002
[5] Britton, NF, Essential Mathematical Biology (2003), Springer-Verlag London · Zbl 1037.92001
[6] Wazwaz, A-M, The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations, Appl Math Comput, 169, 321-338 (2005) · Zbl 1121.65359
[7] Rosa, M.; Bruzón, MS; Gandarias, ML, A conservation law for a generalized chemical Fisher equation, J Math Chem, 53, 941-948 (2015) · Zbl 1331.92180
[8] Nadeem, M.; He, J-H, He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics, J Math Chem, 59, 1234-1245 (2021) · Zbl 1472.35009
[9] Habbal, A.; Barelli, H.; Malandain, G., Assessing the ability of the 2D Fisher- KPP equation to model cell-sheet wound closure, Math Biosci, 252, 45-59 (2014) · Zbl 1354.92018
[10] Grivas, KN; Vavva, MG; Sellountos, EJ; Fotiadis, DI; Polyzos, D., A Meshless LBIE/LRBF Method for Solving the Nonlinear Fisher Equation: Application to Bone Healing, Computer Methods in Engineering & Sciences, 105, 2, 87-122 (2015)
[11] Macías-Díaz, JE, Conciliating efficiency and dynamical consistency in the simulation of the effects of proliferation and motility of transforming growth factor β on cancer cells, Commun Nonlinear Sci Numer Simul, 40, 173-188 (2016) · Zbl 1510.92081
[12] Baabdulla, AA; Now, H.; Park, JA; Kim, W-J; Jung, S.; Yoo, J-Y; Hillen, T., Homogenization of a reaction diffusion equation can explain influenza A virus load data, J Theor Biol, 527, Article 110816 pp. (2021) · Zbl 1470.92280
[13] Browning, AP; Maclaren, OJ; Buenzli, PR; Lanaro, M.; Allenby, MC; Woodruff, MA; Simpson, MJ, Model-based data analysis of tissue growth in thin 3D printed scaffolds, J Theor Biol, 528, Article 110852 pp. (2021) · Zbl 1470.92132
[14] Chulián, S.; Rosa, M.; Gandarias, ML, Symmetries and solutions for a Fisher equation with a proliferation term involving tumor development, Mathematical Methods in Applied Sciences, 43, 2076-2084 (2020) · Zbl 1446.35219
[15] Chulián, S.; Martinez-Rubio, A.; Gandarias, ML; Rosa, M., Lie point symmetries for generalised Fisher’s equations describing tumor Dynamics, Mathematical Biosciences and Engineering, 18, 4, 3291-3312 (2021) · Zbl 1471.92093
[16] El-Hachem, M.; McCue, SW; JinW, Du Y.; Simpson, MJ, Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy, Proc. R. Soc. A, 475, Article 20190378 pp. (2019) · Zbl 1472.35398
[17] El-Hachem, M.; McCue, SW; Simpson, MJ, Invading and Receding Sharp-Fronted Travelling Waves, Bull Math Biol, 83, 35 (2021) · Zbl 1460.92232
[18] Engwer, C.; Wenske, M., Approximate stationalization of anisotropic advection-diffusion-reaction equations in the context of glioblastoma invasion, J Math Biol, 82, 10 (2021) · Zbl 1459.35360
[19] Schafer, A.; Weickenmeier, J.; Kuhl, E., The interplay of biochemical and biomechanical degeneration in Alzheimer’s disease, Comput Meth Appl Mech Eng, 352, 1, 369-388 (2019) · Zbl 1441.74132
[20] Mickens, RE, A Best Finite-Difference Scheme for the Fisher Equation, Numerical Methods for Partial Differential Equations, 10, 581-585 (1994) · Zbl 0810.65131
[21] Carey, GF; Shen, Y., Least-Squares Finite Element Approximation of Fisher’s Reaction-Diffusion Equation, Numerical Methods for Partial Differential Equations, 11, 175-186 (1995) · Zbl 0819.65124
[22] Kenkre, VM, Results from variants of the Fisher equation in the study of epidemics and bacteria, Physica A, 342, 242-248 (2004)
[23] Sari, M.; Gurarslan, G.; Zeytinoglu, A., High-order finite difference schemes for the solution of the generalized Burgers-Fisher equation, International Journal for Numerical Methods in Biomedical Engineering, 27, 1296-1308 (2011) · Zbl 1231.65146
[24] Lin R, Zhou H (2013) A discontinuous Galerkin least-square Finite Element Method for solving Fisher’s equation. Discrete and Continuous Dynamical Systems 2013 (special):489-497. doi: 10.3934/proc.2013.2013.489. · Zbl 1307.65135
[25] Bhalekar, S.; Patade, J., An Analytical Solution of Fisher’s Equation Using Decomposition Method, American Journal of Computational and Applied Mathematics, 6, 3, 123-127 (2016)
[26] Agbavon, KM; Appadu, AR; Khumalo, M., On the numerical solution of Fisher’s equation with coefficient of diffusion term much smaller than coefficient of reaction term, Advances in Difference Equations 2019:46 (2019) · Zbl 1459.65177
[27] Zhang, X.; Yao, L.; Liu, J., Numerical study of Fisher’s equation by the RBF-FD method, Appl Math Lett, 120, Article 107195 pp. (2021) · Zbl 1524.65695
[28] Defreitas, CL; Kane, SJ, A Laplace transform finite difference scheme for the Fisher-KPP equation, Journal of Algorithms & Computational Technology, 15, 1-11 (2021)
[29] Roessler, J.; Hussner, H., Numerical Solution of the 1 + 2 Dimensional Fisher’s Equation by Finite Elements and the Galerkin Method, Mathl. Comput. Modelling, 25, 3, 57-67 (1997) · Zbl 0885.65105
[30] Khiari, N.; Omra, K., Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions, Computers and Mathematics with Applications, 62, 4151-4160 (2014) · Zbl 1236.65113
[31] Yao, Y.; Yuan, G., Enforcing positivity with conservation for nine-point scheme of nonlinear diffusion equations, Comput Meth Appl Mech Eng, 223-224, 161-172 (2012) · Zbl 1253.65134
[32] Ilati, M.; Dehghan, M., Direct local boundary integral equation method for numerical solution of extended Fisher-Kolmogorov equation, Engineering with Computers, 34, 1, 203-213 (2018)
[33] Wrobel, LC, The Boundary Element Method Vol. 1: Application in Thermo-Fluids and Acoustics (2002), Wiley: Wiley West Sussex, UK · Zbl 0994.74002
[34] Rodopoulos, DC; Gortsas, TV; Tsinopoulos, SV; Polyzos, D., ACA/BEM for solving large-scale cathodic protection problems, Eng Anal Boundary Elem, 106, 139-148 (2019) · Zbl 1464.78023
[35] Kalovelonis, DT; Rodopoulos, DC; Gortsas, TV; Polyzos, D.; Tsinopoulos, SV, Cathodic Protection of a Container Ship Using a Detailed BEM Model, J Mar Sci Eng, 8, 359 (2020)
[36] Nardini, D.; Brebbia, CA, A new approach to free vibration analysis using boundary elements, (Brebbia, CA, Boundary Element Methods in Engineering (1982), Springer: Springer Berlin), 313-326 · Zbl 0541.73104
[37] Ahmad, S.; Banerjee, PK, Free vibration analysis by BEM using particular integrals, J Eng Mech, 112, 7, 682-695 (1986)
[38] Polyzos, D.; Dassios, G.; Beskos, DE, On the equivalence of dual reciprocity and particular integrals approaches in the BEM, Boundary Elements Communications, 5, 285-288 (1994)
[39] Kontoni DP, Beskos DE (1993) Transient dynamic elastoplastic analysis by the dual reciprocity BEM.
[40] Meral, G., Solution of the nonlinear diffusion equation using the dual reciprocity boundary element method and the relaxation type time integration scheme, (Kassab, A.; Brebbia, C. A.; Divo, E.; Poljak, D., Boundary Elements (2005)), 133-140, XXVII
[41] Gao, XW, The radial integration method for evaluation of domain integrals with boundary-only discretization, Eng Anal Boundary Elem, 26, 10, 905-916 (2002) · Zbl 1130.74461
[42] Sellountos, EJ, A single domain velocity - vorticity Fast Multipole Boundary Domain Element Method for three-dimensional incompressible fluid flow problems; part II, Eng Anal Boundary Elem, 114, 74-93 (2020) · Zbl 1465.76065
[43] Gortsas, TV; Tsinopoulos, SV; Polyzos, D., An accelerated boundary element method via cross approximation of integral kernels for large-scale cathodic protection problems. Computer-Aided Civil and Infrastructure Engineering, Early View (2021)
[44] Dargush, GF; Grigoriev, MM, A poly-region boundary element method for two-dimensional Boussinesq flows, Comput Meth Appl Mech Eng, 190, 1261-1287 (2000) · Zbl 0997.76056
[45] Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations (1999), Oxford Science Publications · Zbl 0931.65118
[46] Quarteroni, A.; Valli, A., Theory and Application of Steklov-Poincaré Operators for Boundary-Value Problems, (Spigler, R., Applied and Industrial Mathematics, Mathematics and Its Applications 56 (1991), Springer: Springer Dordrecht), 179-203 · Zbl 0723.65098
[47] Zhang, Y.; Varun, J.; Palha, A.; Gerritsma, M., The Discrete Steklov-Poincaré Operator Using Algebraic Dual Polynomial, Computational Method in Applied Mathematics, 19, 3, 645-661 (2019) · Zbl 1420.65129
[48] Liao, S-J, Boundary element method for general nonlinear differential operators, Eng Anal Boundary Elem, 20, 2, 91-99 (1997)
[49] Guiggiani, M.; Gigante, A., A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, J. Appl. Mech., 57, 906-915 (1990) · Zbl 0735.73084
[50] Tsepoura, KG; Tsinopoulos, SV; Polyzos, D.; Beskos, DE, A boundary element method for solving 2-D and 3-D static gradient elastic problems; Part II: Numerical implementation, Comput. Methods Appl. Mech. Eng., 192, 1875-2907 (2003) · Zbl 1054.74742
[51] Xie, G.; Zhong, Y.; Zhou, F.; Du, W.; Li, H.; Zhang, D., Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach, Appl Math Modell, 80, 647-667 (2020) · Zbl 1481.74709
[52] Zhong, Y.; Xie, G.; Hou, JJ; He, W.; Wang, L.; Wang, S., A boundary weak singularity elimination method for multilayer structures, Eng Anal Boundary Elem, 130, 2, 69-78 (2021) · Zbl 1521.74381
[53] Fata, S. N., Explicit expressions for 3D boundary integrals in potential theory, Int J Numer Methods Eng, 78, 32-47 (2009) · Zbl 1183.65155
[54] Henderson, HV; Searle, SR, On Deriving the Inverse of a Sum of Matrices, SIAM Rev, 23, 1, 53-60 (1981) · Zbl 0451.15005
[55] Arora, G.; Bhatia, GS, A Meshfree Numerical Technique Based on Radial Basis Function Pseudospectral Method for Fisher’s Equation, International Journal of Nonlinear Sciences and Numerical Simulation, 21, 1, 37-49 (2020) · Zbl 07168434
[56] Talischi, C.; Paulino, GH; Pereira, A.; Menezed, IFM, Polymesher: a general -purpose generator for polygonal elements written in Matlab, Struct Multidiscip Optim, 45, 309-328 (2012) · Zbl 1274.74401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.