10.1007/s40096-022-00490-0

A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations

  • Mo Faheem1,
  • Arshad Khan*1,
  • Ömer Oruç2,
  1. Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, IN
  2. Department of Mathematics, Faculty of Science, Dicle University, Diyarbakır, TR

Published in Issue 2022-11-15

How to Cite

Faheem, M., Khan, A., & Oruç, Ömer. (2022). A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations. Mathematical Sciences, 18(2 (June 2024). https://doi.org/10.1007/s40096-022-00490-0
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

Abstract In this work, we have investigated p -type fractional neutral delay differential equations ( p -FNDDE) and p -type fractional neutral delay partial differential equations ( p -FNDPDE) via generalized Gegenbauer wavelet. Generalized Gegenbauer scaling function fractional integral operator (GGSFIO) is constructed using the Riemann–Liouville definition of fractional integral to handle the fractional derivatives present in p -FNDDE and p -FNDPDE. The operation of Gegenbauer wavelet basis and GGSFIO to p -FNDDE and p -FNDPDE returns a system of equations which is later solved by Newton’s method for unknown wavelet coefficients. With the help of these coefficients, we get the approximate solution. We have established the convergence analysis to assure the theoretical authenticity of the present method. The developed scheme is tested on several examples of p -FNDDE and p -FNDPDE to ensure computational convergence which validated the theoretical findings. The comparison of the numerical results of our method with the existing methods concludes the superiority of the proposed method.

Keywords

  • Gegenbauer wavelet,
  • Collocation grids,
  • Fractional neutral delay differential equations,
  • Fractional neutral delay partial differential equations,
  • Convergence

References

  1. Abdulaziz et al. (2008) Solving systems of fractional differential equations by homotopy-perturbation method 372(4) (pp. 451-459)
  2. Adomian and Rach (1983) Nonlinear stochastic differential delay equations 91(1) (pp. 94-101)
  3. Antoine et al. (2008) Cambridge University Press
  4. Asl and Ulsoy (2003) Analysis of a system of linear delay differential equations 125(2) (pp. 215-223)
  5. Aziz and Amin (2016) Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet (pp. 10286-10299)
  6. Bohannan (2008) Analog fractional order controller in temperature and motor control applications 14(9–10) (pp. 1487-1498)
  7. Chen et al. (2010) Fractional diffusion equations by the Kansa method 59(5) (pp. 1614-1620)
  8. Darvishi et al. (2021) Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions
  9. Darvishi et al. (2021) Some optical soliton solutions of space-time conformable fractional Schrödinger-type models 96(6)
  10. De Oliveira and Tenreiro Machado (2014) A review of definitions for fractional derivatives and integral
  11. Dehghan and Abbaszadeh (2018) A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation 75(8) (pp. 2903-2914)
  12. Do et al. (2021) A generalized fractional-order Chebyshev wavelet method for two-dimensional distributed-order fractional differential equations
  13. Dumrongpokaphan et al. (2007) An intracellular delay-differential equation model of the HIV infection and immune control 2(1) (pp. 84-112)
  14. Faheem et al. (2020) On some wavelet solutions of singular differential equations arising in the modeling of chemical and biochemical phenomena 2020(1) (pp. 1-23)
  15. Faheem et al. (2020) Collocation methods based on Gegenbauer and Bernoulli wavelets for solving neutral delay differential equations (pp. 72-92)
  16. Ghosh, U., Chowdhury, S., Khan, D.K.: Mathematical Modelling of Epidemiology in Presence of Vaccination and Delay, pp. 91–98. Computer Science and Information Technology (CS and IT) (2013)
  17. Glowinski, R.: Wavelet Solution of Linear and Nonlinear Elliptic, Parabolic and Hyperbolic Problems in One Space Dimension. Aware, Inc. (1989)
  18. Gopalsamy (2013) Springer
  19. He (1999) Some applications of nonlinear fractional differential equations and their approximations 15(2) (pp. 86-90)
  20. Hörmander (2015) Springer
  21. Hu and Zhang (2012) Implicit compact difference schemes for the fractional cable equation 36(9) (pp. 4027-4043)
  22. Insperger (2015) On the approximation of delayed systems by Taylor series expansion 10(2)
  23. Khan, A., Faheem, M., Raza, A.: Solution of third-order Emden–Fowler-type equations using wavelet methods. Eng. Comput. (2021).
  24. https://doi.org/10.1108/EC-04-2020-0218
  25. Kheybari et al. (2020) A semi-analytical approach to Caputo type time-fractional modified anomalous sub-diffusion equations (pp. 103-122)
  26. Krasznai et al. (2010) The modified chain method for a class of delay differential equations arising in neural networks 51(5–6) (pp. 452-460)
  27. Lakestani et al. (2012) The construction of operational matrix of fractional derivatives using b-spline functions 17(3) (pp. 1149-1162)
  28. Lakshmanan and Senthilkumar (2011) Springer
  29. Larsson et al. (2015) Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity (pp. 196-209)
  30. Mainardi (2010) World Scientific
  31. Mehra (2018) Springer
  32. Miller and Ross (1993) Wiley
  33. Morgado et al. (2013) Analysis and numerical methods for fractional differential equations with delay (pp. 159-168)
  34. Ngo et al. (2022) An effective method for solving nonlinear fractional differential equations (pp. 207-218)
  35. Niu et al. (2020) A reproducing kernel method for solving heat conduction equations with delay
  36. Odibat and Shawagfeh (2007) Generalized Taylor’s formula 186(1) (pp. 286-293)
  37. Oruç (2018) A numerical procedure based on Hermite wavelets for two-dimensional hyperbolic telegraph equation 34(4) (pp. 741-755)
  38. Qian and Weiss (1993) Wavelets and the numerical solution of boundary value problems 6(1) (pp. 47-52)
  39. Rahimkhani et al. (2016) Fractional-order Bernoulli wavelets and their applications 40(17–18) (pp. 8087-8107)
  40. Rahimkhani et al. (2019) A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations (pp. 1-27)
  41. Rahimkhani et al. (2019) An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets (pp. 1-27)
  42. Raza and Khan (2019) Haar wavelet series solution for solving neutral delay differential equations 31(4) (pp. 1070-1076)
  43. Raza et al. (2021) Solution of singularly perturbed differential difference equations and convection delayed dominated diffusion equations using Haar wavelet 15(2) (pp. 123-136)
  44. Rossikhin and Shitikova (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids (pp. 15-67)
  45. Salehi and Darvishi (2016) An investigation of fractional Riccati differential equation 127(23) (pp. 11505-11521)
  46. Salehi et al. (2018) Numerical solution of space fractional diffusion equation by the method of lines and splines (pp. 465-480)
  47. Solodushkin et al. (2015) First order partial differential equations with time delay and retardation of a state variable (pp. 322-330)
  48. Vo et al. (2021) A numerical method for solving variable-order fractional diffusion equations using fractional-order Taylor wavelets 37(3) (pp. 2668-2686)
  49. Yuttanan et al. (2021) A fractional-order generalized Taylor wavelet method for nonlinear fractional delay and nonlinear fractional pantograph differential equations 44(5) (pp. 4156-4175)
  50. Zhang and Zhang (2013) A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equations 26(2) (pp. 306-312)
  51. Zhou (2008) Existence and uniqueness of fractional functional differential equations with unbounded delay 1(4) (pp. 239-244)
  52. Zhou et al. (2009) Existence and uniqueness for p-type fractional neutral differential equations 71(7–8) (pp. 2724-2733)