Advertisement

On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems

  • Mahmoud A. ZakyEmail author
  • Ibrahem G. Ameen
Article
  • 298 Downloads

Abstract

Multi-order fractional differential equations are motivated by their flexibility to describe complex multi-rate physical processes. This paper is concerned with the convergence behavior of a spectral collocation method when used to approximate solutions of nonlinear multi-order fractional initial value problems. The collocation scheme and its convergence analysis are developed based on the novel spectral collocation method which has been recently presented by Wang et al. (J Sci Comput 76(1):166–188, 2018) for single fractional-order boundary value problems. The proposed method is an indirect approach since we act on the equivalent Volterra integral equation of the second kind. More precisely, the spectral rate of convergence for the proposed method is established in the \(L^2 \)- and \( L^{\infty } \)-norms. The method enjoys high accuracy for problems with smooth solutions. Exponentially rapid convergence is observed with a small number of degree of freedoms and for all samples of fractional orders. Numerical examples are presented to support the theoretical finding.

Keywords

Multi-term fractional differential equations Volterra integral equations Collocation method Convergence analysis 

Mathematics Subject Classification

58C40 65N35 34L10 34K28 41A25 

Notes

Acknowledgements

We thank the five reviewers for their valuable comments and for bringing to our attention some important references, which helped improve our paper greatly.

References

  1. Abdelkawy MA, Lopes MA, Zaky MA (2016) Shifted fractional Jacobi spectral algorithm for solving distributed order time-fractional reaction-diffusion equations. Comput Appl Math 38:81MathSciNetzbMATHCrossRefGoogle Scholar
  2. Aboelenen T, Bakr SA, El-Hawary HM (2017) Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain. Int J Comput Math 94:570–596MathSciNetzbMATHCrossRefGoogle Scholar
  3. Alsuyuti MM, Doha EH, Ezz-Eldien SS, Bayoumi BI, Baleanu D (2019) Modified Galerkin algorithm for solving multitype fractional differential equations. Math Method Appl Sci 42(5):1389–1412 Journal of Computational and Applied MathematicMathSciNetzbMATHCrossRefGoogle Scholar
  4. Atanackovic TM, Budincevic M, Pilipovic S (2005) On a fractional distributed-order oscillator. J Phys A 38:6703–6713MathSciNetzbMATHCrossRefGoogle Scholar
  5. Bhrawy AH, Zaky MA (2015) A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J Comput Phys 281:876–895MathSciNetzbMATHCrossRefGoogle Scholar
  6. Bhrawy AH, Zaky MA (2016) Shifted fractional-order jacobi orthogonal functions: application to a system of fractional differential equations. Appl Math Model 40:832–845MathSciNetCrossRefGoogle Scholar
  7. Bhrawy AH, Zaky MA (2016) A fractional-order Jacobi tau method for a class of time-fractional PDEs with variable coefficients. Math Method Appl Sci 39:1765–1779MathSciNetzbMATHCrossRefGoogle Scholar
  8. Bhrawy AH, Zaky MA (2017) Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput Math Appl 73:1100–1117MathSciNetzbMATHCrossRefGoogle Scholar
  9. Bhrawy AH, Zaky MA (2018) Numerical simulation of multi-dimensional distributed-order generalized Schrödinger equations. Nonlinear Dyn 89:1415–1432zbMATHCrossRefGoogle Scholar
  10. Bhrawy AH, Doha EH, Ezz-Eldien SS, Abdelkawy MA (2016) A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations. Calcolo 53(1):1–17MathSciNetzbMATHCrossRefGoogle Scholar
  11. Caputo M (2001) Distributed order differential equations modelling dielectric induction and diffusion. Fract Calc Appl Anal 4:421–442MathSciNetzbMATHGoogle Scholar
  12. Chen S, Shen J (2018) Enriched spectral methods and applications to problems with weakly singular solutions. J Sci Comput 77:1468–1489MathSciNetzbMATHCrossRefGoogle Scholar
  13. Chen S, Shen J, Wang L-L (2016) Generalized Jacobi functions and their applications to fractional differential equations. Math Comput 85:1603–1638MathSciNetzbMATHCrossRefGoogle Scholar
  14. Dabiri A, Butcher EA (2017) Stable fractional Chebyshev differentiation matrix for numerical solution of fractional differential equations. Nonlinear Dyn 90(1):185–201MathSciNetzbMATHCrossRefGoogle Scholar
  15. Dabiri A, Butcher EA (2017) Efficient modified Chebyshev differentiation matrices for fractional differential equations. Commun Nonlinear Sci Numer Simul 50:284–310MathSciNetCrossRefGoogle Scholar
  16. Dabiri A, Butcher EA (2018) Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods. Appl Math Model 56:424–448MathSciNetCrossRefGoogle Scholar
  17. Dabiri A, Moghaddam BP, Machado JA (2018) Optimal variable-order fractional PID controllers for dynamical systems. J Comput Appl Math 339:40–48MathSciNetzbMATHCrossRefGoogle Scholar
  18. Diethelm K (2010) The analysis of fractional differential equations. Springer, BerlinzbMATHCrossRefGoogle Scholar
  19. Doha EH, Zaky MA, Abdelkawy M (2019) Spectral methods within fractional calculus, applications in engineering, life and socialsciences, vol 8, part B. De Gruyter, Berlin, pp 207–232CrossRefGoogle Scholar
  20. Erfani S, Babolian E, Javadi S, Shamsi M (2019) Stable evaluations of fractional derivative of the Müntz–Legendre polynomials and application to fractional differential equations. J Comput Appl Math 348:70–88MathSciNetzbMATHCrossRefGoogle Scholar
  21. Ezz-Eldien SS (2016) New quadrature approach based on operational matrix for solving a class of fractional variational problems. J Comput Phys 317:362–381MathSciNetzbMATHCrossRefGoogle Scholar
  22. Ezz-Eldien SS, Doha EH (2019) Fast and precise spectral method for solving pantograph type Volterra integro-differential equations. Numer Algorithms 81(1):57–77MathSciNetzbMATHCrossRefGoogle Scholar
  23. Ford NJ, Morgado ML, Rebelo M (2013) Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract Calc Appl Anal 16(4):874–891MathSciNetzbMATHCrossRefGoogle Scholar
  24. Hafez RM (2018) Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method. Comput Appl Math 37(4):5253–5273MathSciNetzbMATHCrossRefGoogle Scholar
  25. Keshi FK, Moghaddam BP, Aghili A (2019) A numerical technique for variable-order fractional functional nonlinear dynamic systems. Int J Dyn Control.  https://doi.org/10.1007/s40435-019-00521-0 MathSciNetCrossRefGoogle Scholar
  26. Kopteva N, Stynes M (2015) An efficient collocatio nmethod for a Caputo two-point boundary value problem. BIT 55:1105–1123MathSciNetzbMATHCrossRefGoogle Scholar
  27. Liang H, Stynes M (2018) Collocation methods for general Caputo two-point boundary value problems. J Sci Comput 76:390–425MathSciNetzbMATHCrossRefGoogle Scholar
  28. Lischke A, Zayernouri M, Karniadakis GE (2017) A Petrov–Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. SIAM J Sci Comput 39(3):A922–A946MathSciNetzbMATHCrossRefGoogle Scholar
  29. Liu W, Wang L-L, Xiang S (2019) A new spectral method using nonstandard singular basis functions for time-fractional differential equations. Commun Appl Math Comput.  https://doi.org/10.1007/s42967-019-00012-1 MathSciNetCrossRefGoogle Scholar
  30. Mastroianni G, Occorsto D (2001) Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey. J Comput Appl Math 134:325–341MathSciNetzbMATHCrossRefGoogle Scholar
  31. Moghaddam BP, Dabiri A, Lopes AM, Machado JA (2019) Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials. Comput Appl Math 38(2):46MathSciNetzbMATHCrossRefGoogle Scholar
  32. Moghaddam BP, Machado JA, Morgado ML (2019) Numerical approach for a class of distributed order time fractional partial differential equations. Appl Numer Math 136:152–162MathSciNetzbMATHCrossRefGoogle Scholar
  33. Mokhtary P, Ghoreishi F, Srivastava HM (2016) The müntz-legendre tau method for fractional differential equations. Appl Math Model 40:671–684MathSciNetCrossRefGoogle Scholar
  34. Mokhtary P, Moghaddam BP, Lopes AM, Machado JA (2019) A computational approach for the non-smooth solution of nonlinear weakly singular Volterra integral equation with proportional delay. Numer Algorithms.  https://doi.org/10.1007/s11075-019-00712-y CrossRefGoogle Scholar
  35. Pedas A, Tamme E (2011) Spline collocation methods for linear multi-term fractional differential equations. J Comput Appl Math 236(2):167–176MathSciNetzbMATHCrossRefGoogle Scholar
  36. Pezza L, Pitolli F (2018) A multiscale collocation method for fractional differential problems. Math Comput Simul 147:210–219MathSciNetzbMATHCrossRefGoogle Scholar
  37. Shen J, Wang Y (2016) Müntz–Galerkin methods and applications to mixed Dirichlet–Neumann boundary value problems. SIAM J Sci Comput 38:A2357–A2381zbMATHCrossRefGoogle Scholar
  38. Shen J, Tang T, Wang LL (2011) Spectral methods: algorithms, analysis and applications, springer series in computational mathematics, vol 41. Springer, BerlinCrossRefGoogle Scholar
  39. Sun HG, Zhang Y, Baleanu D, Chen W, Chen YQ (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231CrossRefGoogle Scholar
  40. Teodoro GS, Machado JA, Oliveira EC (2019) A review of definitions of fractional derivatives and other operators. J Comput Phys 388(1):195–208MathSciNetCrossRefGoogle Scholar
  41. Wang C, Wang Z, Wang L (2018) A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative. J Sci Comput 76(1):166–188MathSciNetzbMATHCrossRefGoogle Scholar
  42. Zaky MA (2018a) A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput Appl Math 37:3525–3538MathSciNetzbMATHCrossRefGoogle Scholar
  43. Zaky MA (2018b) An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid. Comput Math Appl 75:2243–2258MathSciNetzbMATHCrossRefGoogle Scholar
  44. Zaky MA (2019a) Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Appl Numer Math.  https://doi.org/10.1016/j.apnum.2019.05.008 MathSciNetzbMATHCrossRefGoogle Scholar
  45. Zaky MA (2019b) Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J Comput Appl Math 357:103–122MathSciNetzbMATHCrossRefGoogle Scholar
  46. Zaky MA, Ezz-Eldien SS, Doha EH, Machado JA, Bhrawy AH (2016) An efficient operational matrix technique for multi-dimensional variable-order time fractional diffusion equations. J Comput Nonlinear Dyn 11:1–8 061002Google Scholar
  47. Zaky MA, Doha EH, Machado JA (2018a) A spectral framework for fractional variational problems based on fractional Jacobi functions. Appl Numer Math 132:51–72MathSciNetzbMATHCrossRefGoogle Scholar
  48. Zaky M, Doha E, Machado JA (2018b) A spectral numerical method for solving distributed-order fractional Initial value problems. J Comput Nonlinear Dyn 3(10):1–8 101007. rGoogle Scholar
  49. Zeng F, Zhang Z, Karniadakis GE (2017) Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput Methods Appl Mech Eng 327:478–502MathSciNetCrossRefGoogle Scholar
  50. Zhang J, Liu F, Lin Z, Anh V (2019) Analytical and numerical solutions of a multi-term time-fractional Burgers’ fluid model. Appl Math Comput 356:1–12MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Research CentreGizaEgypt
  2. 2.Department of Mathematics, Faculty of ScienceAl-Azhar UniversityCairoEgypt

Personalised recommendations