10.1007/s40096-022-00507-8

Numerical solution of fractional pantograph equations via Müntz–Legendre polynomials

  • M. Tavassoli Kajani*1,
  1. Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, IR

Published in Issue 2023-01-25

How to Cite

Tavassoli Kajani, M. (2023). Numerical solution of fractional pantograph equations via Müntz–Legendre polynomials. Mathematical Sciences, 18(3 (September 2024). https://doi.org/10.1007/s40096-022-00507-8
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

Abstract In this paper, two new collocation methods to obtain a solution for fractional pantograph equations are developed. The primary one is a single-domain collocation scheme based on modified Müntz–Legendre polynomials which have high accuracy. The other one, modified shifted Müntz–Legendre polynomials are applied to derive a multi-domain collocation scheme equipped with domain decomposition. The use of the shifted Müntz–Legendre polynomials and the domain decomposition leads to an appropriate approximate solution with few collocation points. We illustrate the accuracy and efficiency of these methods with a few numerical examples.

Keywords

  • Shifted Müntz–Legendre polynomials,
  • Fractional pantograph equations,
  • Approximate solution,
  • Collocation method

References

  1. Bahmanpour et al. (2019) Solving Fredholm integral equations of the first kind using Müntz wavelets (pp. 159-171) https://doi.org/10.1016/j.apnum.2019.04.007
  2. Ghasemi et al. (2007) Numerical solution of linear Fredholm integral equations using sine-cosine wavelets 84(7) (pp. 979-987) https://doi.org/10.1080/00207160701242300
  3. Gokmen et al. (2018) A numerical technique for solving functional integro-differential equations having variable bounds (pp. 5609-5623) https://doi.org/10.1007/s40314-018-0653-z
  4. Kurkcu, O.K., Aslan, E., Sezer, M.: An inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function. Comp. Appl. Math.
  5. 38
  6. (34), (2019)
  7. Milovanovic, G.V.: Müntz orthogonal polynomials and their numerical evaluation, in: Applications and Computation of Orthogonal Polynomials, in: Internat. Ser. Numer. Math., Birkhäuser, Basel
  8. 131
  9. , 179–194 (1999)
  10. Saffarian and Mohebbi (2021) Numerical solution of two and three dimensional time fractional damped nonlinear Klein-Gordon equation using ADI spectral element method
  11. Mirzaee et al. (2021) Approximate solution of stochastic Volterra integro-differential equations by using moving least squares scheme and spectral collocation method
  12. Yuzbasi and Yildirim (2022) A collocation method to solve the parabolic-type partial integro-differential equations via Pell-Lucas polynomials
  13. Taghizadeh, E., Matinfar, M.: Modified numerical approaches for a class of Volterra integral equations with proportional delays. Comp. Appl. Math.
  14. 38
  15. (63), (2019)
  16. Xie et al. (2018) A new computational approach for the solutions of generalized pantograph-delay differential equations (pp. 1756-1783) https://doi.org/10.1007/s40314-017-0418-0
  17. Bahmani and Shokri (2022) Numerical study of the unsteady 2D coupled magneto-hydrodynamic equations on regular/irregular pipe using direct meshless local Petrov-Galerkin method
  18. Maleknejad et al. (2021) Numerical solutions of distributed order fractional differential equations in the time domain using the Muntz-Legendre wavelets approach (pp. 707-731) https://doi.org/10.1002/num.22548
  19. Mohammadi (2018) Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis (pp. 4122-4144) https://doi.org/10.1007/s40314-017-0550-x
  20. Keshi et al. (2018) A numerical approach for solving a class of variable-order fractional functional integral equations (pp. 4821-4834) https://doi.org/10.1007/s40314-018-0604-8
  21. Maleki and Kajani (2015) Numerical approximations for Volterra’s population growth model with fractional order via a multi-domain pseudospectral method 39(15) (pp. 4300-4308) https://doi.org/10.1016/j.apm.2014.12.045
  22. Yuzbasi et al. (2015) Müntz-Legendre Matrix Method to solve Delay Fredholm Integro-Differential Equations with constant coefficients 3(2) (pp. 159-167)
  23. Yuzbasi et al. (2013) Müntz-Legendre Polynomial Solutions of Linear Delay Fredholm Integro-Differential Equations and Residual Correction 18(3) (pp. 476-485)
  24. Rahimkhani et al. (2018) Müntz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations (pp. 1283-1305) https://doi.org/10.1007/s11075-017-0363-4
  25. Ockendon and Tayler (1971) The dynamics of a current collection system for an electric locomotive (pp. 447-468) https://doi.org/10.1098/rspa.1971.0078
  26. Ajello et al. (1992) A model of stage structured population growth with density depended time delay (pp. 855-869) https://doi.org/10.1137/0152048
  27. Cushing (1977) Springer-Verlag
  28. Tohidi et al. (2013) A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation (pp. 4283-4294) https://doi.org/10.1016/j.apm.2012.09.032
  29. Sedaghat et al. (2012) Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials (pp. 4815-4830) https://doi.org/10.1016/j.cnsns.2012.05.009
  30. Doha et al. (2014) A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations (pp. 43-54) https://doi.org/10.1016/j.apnum.2013.11.003
  31. Li and Wu (2011) Periodic boundary value problems for neutral multi-pantograph Equations (pp. 1983-1986) https://doi.org/10.1016/j.camwa.2010.08.045
  32. Raja (2014) Numerical treatment for boundary value problems of Pantograph functional differential equation using computational intelligence algorithms (pp. 806-821) https://doi.org/10.1016/j.asoc.2014.08.055
  33. Yusufoglu (2010) An efficient algorithm for solving generalized pantograph equations with linear functional argument (pp. 3591-3595)
  34. Li and Liu (2005) Runge-Kutta methods for the multi-pantograph delay equation (pp. 383-395)
  35. Balachandran et al. (2013) Existence of solutions of nonlinear fractional pantograph equations 33B(3) (pp. 712-720) https://doi.org/10.1016/S0252-9602(13)60032-6
  36. Borwein et al. (1994) Müntz systems and orthogonal Müntz-Legendre polynomials 342(2) (pp. 523-542)
  37. Esmaeili et al. (2011) Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials (pp. 918-929) https://doi.org/10.1016/j.camwa.2011.04.023
  38. Torelli (1989) Stability of numerical methods for delay differential equations 25(1) (pp. 15-26) https://doi.org/10.1016/0377-0427(89)90071-X
  39. Canuto et al. (2006) Fundamentals in Single Domains), Springer, Springer https://doi.org/10.1007/978-3-540-30726-6
  40. Li and Zhang (2020) Long time numerical behaviors of fractional pantograph equations (pp. 244-257) https://doi.org/10.1016/j.matcom.2019.12.004
  41. Alsuyuti et al. (2021) Spectral Galerkin schemes for a class of multi-order fractional pantograph equations https://doi.org/10.1016/j.cam.2020.113157
  42. Sezer et al. (2008) A taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term (pp. 1055-1063) https://doi.org/10.1080/00207160701466784