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Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations

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Abstract

The main purpose of this paper is to utilize the collocation method based on fractional Genocchi functions to approximate the solution of variable-order fractional partial integro-differential equations. In the beginning, the pseudo-operational matrix of integration and derivative has been presented. Then, using these matrices, the proposed equation has been reduced to an algebraic system. Error estimate for the presented technique is discussed and has been implemented the error algorithm on an example. At last, several examples have been illustrated to justify the accuracy and efficiency of the method.

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References

  1. Jerri AJ (1999) Introduction to integral equations with applications. Wiley, NewYork

    MATH  Google Scholar 

  2. Brunner H (2004) Collocation methods for volterra integral and related functional equations. Cambridge University Press, Cambridge

    Book  Google Scholar 

  3. Aghazadeh N, Khajehnasiri AA (2013) Solving nonlinear two-dimensional volterra integro-differential equations by block-pulse functions. Math Sci 7:1–7

    Article  MathSciNet  Google Scholar 

  4. Samko SG, Ross B (1993) Integration and differentiation to a variable fractional order. Integr Transf Spec Funct 1(4):277–300

    Article  MathSciNet  Google Scholar 

  5. Coimbra CFM, Soon CM, Kobayashi MH (2005) The variable viscoelasticity operator. Ann Phys 14:378–389

    Article  Google Scholar 

  6. Ingman D, Suzdalnitsky J (2004) Control of damping oscillations by fractional differential operator with time-dependent order. Comput Methods Appl Mech Eng 193(52):5585–5595

    Article  MathSciNet  Google Scholar 

  7. Ostalczyk PW, Duch P, Brzezinski DW, Sankowski D (2015) Order Functions selection in the variable-, fractional-order PID controller, advances in modelling and control of non-integer-order systems. Lecture Notes Electr Eng 320:159–170

    Article  Google Scholar 

  8. Li Z, Wang H, Xiao R, Yang S (2017) A variable-order fractional differential equation model of shape memory polymers. Chaos Solitons Fract 102:473–485

    Article  MathSciNet  Google Scholar 

  9. Heydari MH (2016) A new approach of the Chebyshev wavelets for the variable-order time fractional mobile-immobile advection-dispersion model. http://arXiv preprint arXiv:1605.06332

  10. Xu Y, He Z (2013) Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations. J Appl Math Comput 43(1–2):295–306

    Article  MathSciNet  Google Scholar 

  11. Doha EH, Abdelkawy MA, Amin AZM, Baleanu D (2018) Spectral technique for solving variable-order fractional volterra integro-differential equations. Numer Methods Partial Differ Equ 34(5):1659–1677

    Article  MathSciNet  Google Scholar 

  12. Moghaddam BP, Machado JAT (2017) A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fract Calc Appl Anal 20(4):1023–1042

    Article  MathSciNet  Google Scholar 

  13. Hosseinini M, Heydari MH, Avazzadeh Z, Maalek Ghaini FM (2018) two-dimensional Legendre wavelets for solving variable-order fractional nonlinear advection-diffusion equation with variable coefficients. Int J Nonlinear Sci Num 19(7–8):793–802

    Article  MathSciNet  Google Scholar 

  14. Araci S (2012) Novel identities for q-Genocchi numbers and polynomials. J Funct Sp Appl

  15. Araci S (2014) Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl Math Comput 233:599–607

    MathSciNet  MATH  Google Scholar 

  16. Roshan S, Jafari H, Baleanu D (2018) Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on Genocchi polynomials. Math Methods Appl Sci 41:1–8

    Article  MathSciNet  Google Scholar 

  17. Rahimkhani P, Ordokhani Y, Babolian E (2017) Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations. Appl Numer Math 122:66–81

    Article  MathSciNet  Google Scholar 

  18. Rabiei K, Parand K (2019) Collocation method to solve inequality-constrained optimal control problems of arbitrary order. Eng Comput 1:1–11

    Google Scholar 

  19. Kazem S, Abbasbandy S, Kumar S (2013) Fractional-order Legendre functions for solving fractional-order differential equations. Appl Math Model 37:5498–5510

    Article  MathSciNet  Google Scholar 

  20. Dehestani H, Ordokhani Y, Razzaghi M (2018) Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations. Appl Math Comput 336:433–453

    MathSciNet  MATH  Google Scholar 

  21. Dehestani H, Ordokhani Y, Razzaghi M (2019) Hybrid functions for numerical solution of fractional Fredholm‐Volterra functional integro‐differential equations with proportional delays. Int J Numer Model 32:e2606

    Article  Google Scholar 

  22. Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New York

    MATH  Google Scholar 

  23. Rivlin TJ (1981) An introduction to the approximation of functions. Dover, New York

    MATH  Google Scholar 

  24. Doha EH, Abdelkawy MA, Amin AZM, Baleanu D (2017) Spectral technique for solving variable-order fractional Volterra integro-differential equations. Numer Methods Partial Differ Equ 34(5):1659–1677

    Article  MathSciNet  Google Scholar 

  25. Danfu H, Xufeng S (2007) Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration. Appl Math Comput 194:460–466

    MathSciNet  MATH  Google Scholar 

  26. Darani P, Ebadian A (2007) A method for the numerical solution of the integro-differential equations. Appl Math Comput 188:657–668

    MathSciNet  MATH  Google Scholar 

  27. Yusufoglu E (2009) Improved homotopy perturbation method for solving Fredholm type integro-differential equations. Chaos Solitons Fract 41:28–37

    Article  Google Scholar 

  28. Yuzbas S, Sahin N, Yildirim AN (2012) A collocation approach for solving high-order linear Fredholm-Volterra integro-differential equations. Math Comput Model 55:547–563

    Article  MathSciNet  Google Scholar 

  29. Yuzbasi S (2016) A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations. Appl Math Comput 273:142–154

    MathSciNet  MATH  Google Scholar 

  30. Babolian E, Masouri Z, Hatamzadeh-Varmazyar S (2009) Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions. Comput Math Appl 58(2):239–247

    Article  MathSciNet  Google Scholar 

  31. Isik OR, Guney Z, Sezer M (2012) Bernstein series solutions of pantograph equations using polynomial interpolation. J Differ Equ Appl 18:357–374

    Article  MathSciNet  Google Scholar 

  32. Maleknejad K, Mahmoudi Y (2003) Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations. Appl Math Comput 145(2–3):641–653

    MathSciNet  MATH  Google Scholar 

  33. Rohaninasab N, Maleknejad K, Ezzati R (2018) Numerical solution of high-order Volterra-Fredholm integro-differential equations by using Legendre collocation method. Appl Math Comput 328:171–188

    MathSciNet  MATH  Google Scholar 

  34. Meng Z, Wang L, Li H, Wei Z (2015) Legendre wavelets method for solving fractional integro-differential equations. Int J Comput Math 92(6):1275–1291

    Article  MathSciNet  Google Scholar 

  35. Chen Y, Wu Y, Cui Y, Wang Z, Jin D (2010) Wavelet method for a class of fractional convection-diffusion equation with variable coefficients. J Comput Sci 1(3):146–149

    Article  Google Scholar 

  36. Saadatmandi A, Dehghan M, Azizi MR (2012) The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun Nonlinear Sci Numer Simul 17(11):4125–4136

    Article  MathSciNet  Google Scholar 

  37. Zhou F, Xu X (2016) The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients. Appl Math Comput 280:11–29

    MathSciNet  MATH  Google Scholar 

  38. Avazzadeh Z, Beygi Rizi Z, Maalek Ghaini FM, Loghmani GB (2012) A numerical solution of nonlinear parabolic-type Volterra partial integro-differential equations using radial basis functions. Eng Anal Bound Elem 36:881–893

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Alzahra university, Tehran, Iran

    Haniye Dehestani & Yadollah Ordokhani

  2. Department of Mathematics and Statistics, Mississippi State University, Starkville, MS, 39762, USA

    Mohsen Razzaghi

Authors
  1. Haniye Dehestani
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  2. Yadollah Ordokhani
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  3. Mohsen Razzaghi
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Correspondence to Yadollah Ordokhani.

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Dehestani, H., Ordokhani, Y. & Razzaghi, M. Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations. Engineering with Computers 37, 1791–1806 (2021). https://doi.org/10.1007/s00366-019-00912-z

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  • DOI: https://doi.org/10.1007/s00366-019-00912-z

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