Abstract
In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.
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References
Ahlers, G., Cannell, D. S.: Vortex-front propagation in rotating Couette-Taylor flow. Phys. Rev. Lett. 50(20), 1583–1586 (1983)
Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)
Belmonte-Beitia, J., Calvo, G. F., Perez-Garcia, V. M.: Effective particle methods for Fisher–Kolmogorov equations: theory and applications to brain tumor dynamics. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3267–3283 (2014)
Belytschko, T., Lu, Y. Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Meth. Eng. 37(2), 229–256 (1994)
Cheng, R., Cheng, Y.: Error estimates for the finite point method. Appl. Numer. Math. 58(6), 884–898 (2008)
Cheng, Y., Bai, F., Liu, C., Peng, M.: Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method. Int. J. Comput. Mater. Sci. Eng. 5(04), 1650023 (2016)
Cheng, Y., Bai, F., Peng, M.: A novel interpolating element-free Galerkin (IEFG) method for twodimensional elastoplasticity. Appl. Math. Model. 38(21-22), 5187–5197 (2014)
Danumjaya, P., Pani, A. K.: Orthogonal cubic spline collocation method for the extended Fisher–Kolmogorov equation. J. Comput. Appl. Math. 174(1), 101–117 (2005)
Danumjaya, P., Pani, A. K.: Numerical methods for the extended Fisher–Kolmogorov (EFK) equation. Int. J. Numer. Anal. Model. 3(2), 186–210 (2006)
Dee, G., van Saarloos, W.: Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett. 60(25), 2641–2644 (1988)
Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math. 286, 211–231 (2015)
Elphick, C., Coullet, P., Repaux, D.: Nature of spatial chaos. Phys. Rev. Lett 58, 431–434 (1987)
Gerin, C., Pallud, J., Grammaticos, B., Mandonnet, E., Deroulers, C., Varlet, P., Capelle, L., Taillandier, L., Bauchet, L., Duffau, H., et al.: Improving the time-machine: estimating date of birth of grade II gliomas. Cell Proliferation 45(1), 76–90 (2012)
Guozhen, Z.: Experiments on director waves in nematic liquid crystals. Phys. Rev. Lett. 49(18), 1332–1335 (1982)
Hong-Ping, R., Yu-Min, C., Wu, Z.: An improved boundary element-free method (IBEFM) for two-dimensional potential problems. Chinese Physics B 18(10), 4065–4073 (2009)
Hornreich, R., Luban, M., Shtrikman, S.: Critical behavior at the onset of k-space instability on the λ line. Phys. Rev. Lett. 35(25), 1678–1681 (1975)
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)
Jbabdi, S., Mandonnet, E., Duffau, H., Capelle, L., Swanson, K. R., Pélégrini-Issac, M., Guillevin, R., Benali, H.: Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging. Magnetic Resonance in Medicine: an Official Journal of the International Society for Magnetic Resonance in Medicine 54(3), 616–624 (2005)
Kadri, T., Omrani, K.: A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation. Comput. Math. Appl. 61(2), 451–459 (2011)
Khiari, N., Omrani, K.: Finite difference discretization of the extended Fisher–Kolmogorov equation in two dimensions. Comput. Math. Appl. 62(11), 4151–4160 (2011)
Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37(155), 141–158 (1981)
Li, X.: Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces. Appl. Numer. Math. 99, 77–97 (2016)
Li, X., Zhang, S., Wang, Y., Chen, H.: Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations. Comput. Math. Appl. 71(8), 1655–1678 (2016)
Liu, F., Cheng, Y.: The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems. Int. J. Comput. Mater. Sci. Eng. 7(03), 1850023 (2018)
Liu, F., Cheng, Y.: The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels. Int. J. Appl. Mech. 10(04), 1850047 (2018)
Liu, F., Wu, Q., Cheng, Y.: A meshless method based on the nonsingular weight functions for elastoplastic large deformation problems. Int. J. Appl. Mech. 11(01), 1950006 (2019)
Liu, F., Zhao, X., Liu, B.: Fourier pseudo-spectral method for the extended Fisher-Kolmogorov equation in two dimensions. Advances in Difference Equations 2017(1), 94 (2017)
Liu, G.-R., Gu, Y.-T.: An Introduction to Meshfree Methods and their Programming. Springer Science & Business Media, Berlin (2005)
Mittal, R., Arora, G.: Quintic B-spline collocation method for numerical solution of the extended Fisher–Kolmogorov equation. Int. J. Appl. Math. Mech. 6 (1), 74–85 (2010)
Mittal, R., Dahiya, S.: A study of quintic B-spline based differential quadrature method for a class of semi-linear Fisher–Kolmogorov equations. Alexandria Engineering Journal 55(3), 2893–2899 (2016)
Murray, J.: Mathematical Biology II. Springer, Spatial Models and Biomedical Applications (2003)
Pérez-García, V. M., Bogdanska, M., Martínez-González, A., Belmonte-Beitia, J., Schucht, P., Pérez-Romasanta, L. A.: Delay effects in the response of low-grade gliomas to radiotherapy: a mathematical model and its therapeutical implications. Mathematical medicine and biology: A journal of the IMA 32(3), 307–329 (2014)
Sun, F., Wang, J.: Interpolating element-free Galerkin method for the regularized long wave equation and its error analysis. Appl. Math. Comput. 315, 54–69 (2017)
Sun, F., Wang, J., Cheng, Y.: An improved interpolating element-free Galerkin method for elastoplasticity via nonsingular weight functions. Int. J. Appl. Mech. 8(08), 1650096 (2016)
Sun, F., Wang, J., Cheng, Y., Huang, A.: Error estimates for the interpolating moving least-squares method in n-dimensional space. Appl. Numer. Math. 98, 79–105 (2015)
Van Saarloos, W.: Dynamical velocity selection: marginal stability. Physical review letters 58(24), 2571–2574 (1987)
Van Saarloos, W.: Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection. Phys. Rev. A 37(1), 211–229 (1988)
Zhang, T., Li, X.: A variational multiscale interpolating element-free Galerkin method for convection-diffusion and Stokes problems. Engineering Analysis with Boundary Elements 82, 185–193 (2017)
Zhao, N., Ren, H.: The interpolating element-free Galerkin method for 2D transient heat conduction problems Mathematical Problems in Engineering (2014)
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Ilati, M. Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. Numer Algor 85, 485–502 (2020). https://doi.org/10.1007/s11075-019-00823-6
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DOI: https://doi.org/10.1007/s11075-019-00823-6
Keywords
- Extended Fisher–Kolmogorov equation
- Interpolating element-free Galerkin method
- Interpolating moving least squares method
- Error estimate
- Brain tumor dynamics modeling