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. AggouneH. 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.  Google Scholar

[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.   Google Scholar

[3]

C. ChesterB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Gasca and T. Sauer, Polynomial interpolation in several variables, Computational Mathematics, 12 (2000), 377-410.  doi: 10.1023/A:1018981505752.  Google Scholar

[7]

M. GiesbrechtG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. L. LopezP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. L. LopezE. 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.  Google Scholar

[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.  Google Scholar

[17]

V. P. NguyenT. RabczukS. 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.  Google Scholar

[18]

A. Quarteroni, Méthodes Numériques Pour le Calcul Scientifique. Programmes en Matlab, Collection IRIS, 2000. Google Scholar

[19]

J. N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, Boca Raton, 2004. Google Scholar

[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.  Google Scholar

[21]

Y. TampangoM. Potier-FerryY. 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.  Google Scholar

[22]

Y. TampangoM. Potier-FerryY. 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.  Google Scholar

[23]

J. YangH. 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.  Google Scholar

[24]

D. S. ZezeM. 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.  Google Scholar

show all references

References:
[1]

W. AggouneH. 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.  Google Scholar

[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.   Google Scholar

[3]

C. ChesterB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Gasca and T. Sauer, Polynomial interpolation in several variables, Computational Mathematics, 12 (2000), 377-410.  doi: 10.1023/A:1018981505752.  Google Scholar

[7]

M. GiesbrechtG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. L. LopezP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. L. LopezE. 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.  Google Scholar

[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.  Google Scholar

[17]

V. P. NguyenT. RabczukS. 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.  Google Scholar

[18]

A. Quarteroni, Méthodes Numériques Pour le Calcul Scientifique. Programmes en Matlab, Collection IRIS, 2000. Google Scholar

[19]

J. N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, Boca Raton, 2004. Google Scholar

[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.  Google Scholar

[21]

Y. TampangoM. Potier-FerryY. 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.  Google Scholar

[22]

Y. TampangoM. Potier-FerryY. 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.  Google Scholar

[23]

J. YangH. 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.  Google Scholar

[24]

D. S. ZezeM. 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.  Google Scholar

Figure 1.  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)
Figure 2.  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
Figure 3.  Example 1, 2-point Taylor series (n = 2). Convergence with the degree p
Figure 4.  Example 1, degree p = 4. Convergence with the number expansion points
Figure 5.  Example 1, 2-point Taylor series, c $\pm \frac{10}{3}$, p = 6. Distribution of the residual and of the error in the interval
Figure 6.  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
Figure 7.  Example 2, 4-point Taylor series, expansion points located in ($\pm 0.8; \pm 1.8$). Convergence with the degree
Figure 8.  Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 1)$, $u_{max} = 4.718$
Figure 9.  Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 0.1)$, $u_{max} = 5.0788$
Figure 10.  Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 4$, $(L = 10, a = 0.1)$, $u_{max} = 5.109$
Figure 11.  Example 3, residual with 3 subdomains, $p = 5$, $n = 4$. Case of a smooth distribution of points
[1]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[2]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[3]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[4]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[5]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[6]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[7]

Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[8]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[9]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[10]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[11]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

[12]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[13]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[14]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010

[15]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

[16]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[17]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[18]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[19]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013

[20]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (388)
  • HTML views (474)
  • Cited by (1)

[Back to Top]