# American Institute of Mathematical Sciences

October 2019, 12(6): 1791-1806. doi: 10.3934/dcdss.2019118

## Multi-point Taylor series to solve differential equations

 1 Laboratoire de Mathématiques-Informatique, Université Nangui Abrogoua, Unité de Formation et de Recherche en Sciences Fondamentales et Appliquées, 02 B.P. V 102 Abidjan, Côte d'Ivoire 2 Université de Lorraine, CNRS, Arts et Métiers ParisTech, LEM3, F-57000 Metz, France 3 EDF R & D Saclay, 7 boulevard Gaspard Monge 91120 Palaiseau, France

* Corresponding author: Zézé

Received  November 2017 Revised  February 2018 Published  November 2018

The use of Taylor series is an effective numerical method to solve ordinary differential equations but this fails when the sought function is not analytic or when it has singularities close to the domain. These drawbacks can be partially removed by considering multi-point Taylor series, but up to now there are only few applications of the latter method in the literature and not for problems with very localized solutions. In this respect, a new numerical procedure is presented that works for an arbitrary cloud of expansion points and it is assessed from several numerical experiments.

Citation: Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118
##### References:
 [1] W. Aggoune, H. Zahrouni and M. Potier-Ferry, Asymptotic numerical methods for unilateral contact, International Journal for Numerical Methods in Engineering, 68 (2006), 605-631. doi: 10.1002/nme.1714. [2] H. Ben Dhia, Multiscale mechanical problems: The arlequin method, Comptes Rendus de l'Academie des Sciences Series IIB Mechanics Physics Astronomy, 12326 (1998), 899-904. [3] C. Chester, B. Friedman and F. Ursell, An extension of the method of steepest descents, Mathematical Proceedings of the Cambridge Philosophical Society, 53 (1957), 599-611. doi: 10.1017/S0305004100032655. [4] G. Corliss and Y. F. Chang, Solving ordinary differential equations using taylor series, ACM Transact Math Software, 8 (1982), 114-144. doi: 10.1145/355993.355995. [5] F. Costabile and A. Napoli, Solving bvps using two-point taylor formula by a symbolic software, Journal of Computational and Applied Mathematics, 210 (2007), 136-148. doi: 10.1016/j.cam.2006.10.081. [6] M. Gasca and T. Sauer, Polynomial interpolation in several variables, Computational Mathematics, 12 (2000), 377-410. doi: 10.1023/A:1018981505752. [7] M. Giesbrecht, G. Labahn and W. S. Lee, Symbolic-numeric sparse interpolation of multivariate polynomials, Journal of Symbolic Computation, 44 (2009), 943-959. doi: 10.1016/j.jsc.2008.11.003. [8] U. Haussler-Combe and C. Korn, An adaptive approach with the element-free-galerkin method, Computer Methods in Applied Mechanics and Engineering, 162 (1998), 203-222. doi: 10.1016/S0045-7825(97)00344-7. [9] E. J. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics-ii solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & Mathematics with Applications, 19 (1990), 147-161. doi: 10.1016/0898-1221(90)90271-K. [10] J. L. Lopez and N. M. Temme, New series expansions of the gauss hypergeometric function, Advances in Computational Mathematics, 39 (2013), 349-365. doi: 10.1007/s10444-012-9283-y. [11] J. L. Lopez and E. Perez Sinusia, New series expansions for the confluent hypergeometric function m (a, b, z), Applied Mathematics and Computation, 235 (2014), 26-31. doi: 10.1016/j.amc.2014.02.099. [12] J. L. Lopez, P. Pagola and E. Perez Sinusia, New series expansions of the 3f2 function, Journal of Mathematical Analysis and Applications, 421 (2015), 982-995. doi: 10.1016/j.jmaa.2014.07.065. [13] J. L. Lopez and N. M. Temme, Two-point taylor expansions of analytic functions, Studies in Applied Mathematics, 109 (2002), 297-311. doi: 10.1111/1467-9590.00225. [14] J. L. Lopez and N. M. Temme, Multi-point taylor expansions of analytic functions, Transactions of the American Mathematical Society, 356 (2004), 4323-4342. doi: 10.1090/S0002-9947-04-03619-0. [15] J. L. Lopez, E. Perez Sinusia and N. M. Temme, Multi-point taylor approximations in one- dimensional linear boundary value problems, Applied Mathematics and Computation, 207 (2009), 519-527. doi: 10.1016/j.amc.2008.11.015. [16] J. L. Lopez and E. Perez Sinusia, Two-point taylor approximations of the solutions of two-dimensional boundary value problems, Applied Mathematics and Computation, 218 (2012), 9107-9115. doi: 10.1016/j.amc.2012.02.060. [17] V. P. Nguyen, T. Rabczuk, S. P. A. Bordas and M. Duflot, Meshless methods: A review and computer implementation aspects, Mathematics and Computers in Simulation, 79 (2008), 763-813. doi: 10.1016/j.matcom.2008.01.003. [18] A. Quarteroni, Méthodes Numériques Pour le Calcul Scientifique. Programmes en Matlab, Collection IRIS, 2000. [19] J. N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, Boca Raton, 2004. [20] P. Rentrop, A taylor series method for the numerical solution of two-point boundary value problems, Numerische Mathematik, 31 (1978), 359-375. doi: 10.1007/BF01404566. [21] Y. Tampango, M. Potier-Ferry, Y. Koutsawa and S. Belouettar, Convergence analysis and detection of singularities within a boundary meshless method based on taylor series, Engineering Analysis with Boundary Elements, 36 (2012), 1465-1472. doi: 10.1016/j.enganabound.2012.03.014. [22] Y. Tampango, M. Potier-Ferry, Y. Koutsawa and S. Tiem, Coupling of polynomial approximations with application to a boundary meshless method, International Journal for Numerical Methods in Engineering, 95 (2013), 1094-1112. doi: 10.1002/nme.4549. [23] J. Yang, H. Hu and M. Potier-Ferry, Solving large scale problems by Taylor Meshless method, International Journal for Numerical Methods in Engineering, 112 (2017), 103-124. doi: 10.1002/nme.5508. [24] D. S. Zeze, M. Potier-Ferry and N. Damil, A boundary meshless method with shape functions computed from the pde, Engineering Analysis with Boundary Elements, 34 (2010), 747-754. doi: 10.1016/j.enganabound.2010.03.008.

show all references

##### References:
 [1] W. Aggoune, H. Zahrouni and M. Potier-Ferry, Asymptotic numerical methods for unilateral contact, International Journal for Numerical Methods in Engineering, 68 (2006), 605-631. doi: 10.1002/nme.1714. [2] H. Ben Dhia, Multiscale mechanical problems: The arlequin method, Comptes Rendus de l'Academie des Sciences Series IIB Mechanics Physics Astronomy, 12326 (1998), 899-904. [3] C. Chester, B. Friedman and F. Ursell, An extension of the method of steepest descents, Mathematical Proceedings of the Cambridge Philosophical Society, 53 (1957), 599-611. doi: 10.1017/S0305004100032655. [4] G. Corliss and Y. F. Chang, Solving ordinary differential equations using taylor series, ACM Transact Math Software, 8 (1982), 114-144. doi: 10.1145/355993.355995. [5] F. Costabile and A. Napoli, Solving bvps using two-point taylor formula by a symbolic software, Journal of Computational and Applied Mathematics, 210 (2007), 136-148. doi: 10.1016/j.cam.2006.10.081. [6] M. Gasca and T. Sauer, Polynomial interpolation in several variables, Computational Mathematics, 12 (2000), 377-410. doi: 10.1023/A:1018981505752. [7] M. Giesbrecht, G. Labahn and W. S. Lee, Symbolic-numeric sparse interpolation of multivariate polynomials, Journal of Symbolic Computation, 44 (2009), 943-959. doi: 10.1016/j.jsc.2008.11.003. [8] U. Haussler-Combe and C. Korn, An adaptive approach with the element-free-galerkin method, Computer Methods in Applied Mechanics and Engineering, 162 (1998), 203-222. doi: 10.1016/S0045-7825(97)00344-7. [9] E. J. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics-ii solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & Mathematics with Applications, 19 (1990), 147-161. doi: 10.1016/0898-1221(90)90271-K. [10] J. L. Lopez and N. M. Temme, New series expansions of the gauss hypergeometric function, Advances in Computational Mathematics, 39 (2013), 349-365. doi: 10.1007/s10444-012-9283-y. [11] J. L. Lopez and E. Perez Sinusia, New series expansions for the confluent hypergeometric function m (a, b, z), Applied Mathematics and Computation, 235 (2014), 26-31. doi: 10.1016/j.amc.2014.02.099. [12] J. L. Lopez, P. Pagola and E. Perez Sinusia, New series expansions of the 3f2 function, Journal of Mathematical Analysis and Applications, 421 (2015), 982-995. doi: 10.1016/j.jmaa.2014.07.065. [13] J. L. Lopez and N. M. Temme, Two-point taylor expansions of analytic functions, Studies in Applied Mathematics, 109 (2002), 297-311. doi: 10.1111/1467-9590.00225. [14] J. L. Lopez and N. M. Temme, Multi-point taylor expansions of analytic functions, Transactions of the American Mathematical Society, 356 (2004), 4323-4342. doi: 10.1090/S0002-9947-04-03619-0. [15] J. L. Lopez, E. Perez Sinusia and N. M. Temme, Multi-point taylor approximations in one- dimensional linear boundary value problems, Applied Mathematics and Computation, 207 (2009), 519-527. doi: 10.1016/j.amc.2008.11.015. [16] J. L. Lopez and E. Perez Sinusia, Two-point taylor approximations of the solutions of two-dimensional boundary value problems, Applied Mathematics and Computation, 218 (2012), 9107-9115. doi: 10.1016/j.amc.2012.02.060. [17] V. P. Nguyen, T. Rabczuk, S. P. A. Bordas and M. Duflot, Meshless methods: A review and computer implementation aspects, Mathematics and Computers in Simulation, 79 (2008), 763-813. doi: 10.1016/j.matcom.2008.01.003. [18] A. Quarteroni, Méthodes Numériques Pour le Calcul Scientifique. Programmes en Matlab, Collection IRIS, 2000. [19] J. N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, Boca Raton, 2004. [20] P. Rentrop, A taylor series method for the numerical solution of two-point boundary value problems, Numerische Mathematik, 31 (1978), 359-375. doi: 10.1007/BF01404566. [21] Y. Tampango, M. Potier-Ferry, Y. Koutsawa and S. Belouettar, Convergence analysis and detection of singularities within a boundary meshless method based on taylor series, Engineering Analysis with Boundary Elements, 36 (2012), 1465-1472. doi: 10.1016/j.enganabound.2012.03.014. [22] Y. Tampango, M. Potier-Ferry, Y. Koutsawa and S. Tiem, Coupling of polynomial approximations with application to a boundary meshless method, International Journal for Numerical Methods in Engineering, 95 (2013), 1094-1112. doi: 10.1002/nme.4549. [23] J. Yang, H. Hu and M. Potier-Ferry, Solving large scale problems by Taylor Meshless method, International Journal for Numerical Methods in Engineering, 112 (2017), 103-124. doi: 10.1002/nme.5508. [24] D. S. Zeze, M. Potier-Ferry and N. Damil, A boundary meshless method with shape functions computed from the pde, Engineering Analysis with Boundary Elements, 34 (2010), 747-754. doi: 10.1016/j.enganabound.2010.03.008.
Discretization points in the case of three subdomains and two point-Taylor series. The expansion points are $x_i$ and the additional collocation points are $y_j$ (2 per subdomain)
Example 1: $f(x) = g(x) = 1, L = 10$. Comparison of a one-point Taylor series, one-point Taylor series in two subdomains and two-point Taylor series
Example 1, 2-point Taylor series (n = 2). Convergence with the degree p
Example 1, degree p = 4. Convergence with the number expansion points
Example 1, 2-point Taylor series, c $\pm \frac{10}{3}$, p = 6. Distribution of the residual and of the error in the interval
Example 1, 2-point Taylor series, p = 6. Maximal value of the residual and of the error according to the location of the expansion points
Example 2, 4-point Taylor series, expansion points located in ($\pm 0.8; \pm 1.8$). Convergence with the degree
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 1)$, $u_{max} = 4.718$
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 0.1)$, $u_{max} = 5.0788$
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 4$, $(L = 10, a = 0.1)$, $u_{max} = 5.109$
Example 3, residual with 3 subdomains, $p = 5$, $n = 4$. Case of a smooth distribution of points
 [1] Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090 [2] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [3] Richard D. Neidinger. Efficient recurrence relations for univariate and multivariate Taylor series coefficients. Conference Publications, 2013, 2013 (special) : 587-596. doi: 10.3934/proc.2013.2013.587 [4] Martin Oberlack, Andreas Rosteck. New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 451-471. doi: 10.3934/dcdss.2010.3.451 [5] Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759 [6] John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273 [7] Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457 [8] Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control & Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437 [9] Ricardo M. Martins, Otávio M. L. Gomide. Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142 [10] Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 [11] Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255 [12] Tae Gab Ha. On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. Evolution Equations & Control Theory, 2018, 7 (2) : 281-291. doi: 10.3934/eect.2018014 [13] Fei Jiang, Song Jiang, Weiwei Wang. Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1853-1898. doi: 10.3934/dcdss.2016076 [14] M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129 [15] Ricardo García López. A note on L-series and Hodge spectrum of polynomials. Electronic Research Announcements, 2009, 16: 56-62. doi: 10.3934/era.2009.16.56 [16] R. Wong, L. Zhang. Global asymptotics of Hermite polynomials via Riemann-Hilbert approach. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 661-682. doi: 10.3934/dcdsb.2007.7.661 [17] David W. Pravica, Michael J. Spurr. Unique summing of formal power series solutions to advanced and delayed differential equations. Conference Publications, 2005, 2005 (Special) : 730-737. doi: 10.3934/proc.2005.2005.730 [18] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 [19] Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178 [20] Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861

2017 Impact Factor: 0.561