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An efficient iterative method for multi-order nonlinear fractional differential equations based on the integrated Bernoulli polynomials

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Abstract

We present an effective practical approach for solving multi-order nonlinear fractional differential equations. Our method uses integrated Bernoulli polynomials and comes with a comprehensive convergence analysis. The integrated Bernoulli polynomials are combined with the collocation and simple iteration methods to approximate the solutions. We have provided several numerical examples to demonstrate the effectiveness, strength, and flexibility of our method. The results obtained from implementing the method have been compared with exact solutions and results obtained from other methods mentioned in the articles.

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References

  • Abd-Elhameed WM, Alsuyuti MM (2023) Numerical treatment of multi-term fractional differential equations via new kind of generalized chebyshev polynomials. Fractal Fract 7(1):74

    Article  Google Scholar 

  • Alipour M, Agahi H (2018) New computational techniques for solving nonlinear problems using g-fractional differential operator. J Comput Appl Math 330:70–74

    Article  MathSciNet  Google Scholar 

  • Alqhtani M, Owolabi KM, Saad KM, Pindza E (2022) Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology. Chaos, Solitons & Fractals 161:112394

    Article  MathSciNet  Google Scholar 

  • Amin R, Shah K, Mlaiki N, Yüzbaşi Ş, Abdeljawad T, Hussain A (2022) Existence and numerical analysis using Haar wavelet for fourth-order multi-term fractional differential equations. Comput Appl Math 41(7):329

    Article  MathSciNet  Google Scholar 

  • Azarnavid B (2023) The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis. Comput Appl Math 42(1):1–17

    Article  MathSciNet  Google Scholar 

  • Azarnavid B, Emamjomeh M, Nabati M (2022) A shooting like method based on the shifted Chebyshev polynomials for solving nonlinear fractional multi-point boundary value problem. Chaos, Solitons & Fractals 159:112159

    Article  MathSciNet  Google Scholar 

  • Bakhshandeh-Chamazkoti R, Alipour M (2022) Lie symmetries reduction and spectral methods on the fractional two-dimensional heat equation. Math Comput Simul 200:97–107

    Article  MathSciNet  Google Scholar 

  • Bhrawy AH, Tohidi E, Soleymani F (2012) A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl Math Comput 219(2):482–497

    MathSciNet  Google Scholar 

  • Bhrawy AH, Taha TM, Machado JAT (2015) A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn 81:1023–1052

    Article  MathSciNet  Google Scholar 

  • Chen X, Lei S, Wang L (2020) Bernoulli Polynomials Collocation Method for Multi-term Fractional Differential Equations. IAENG Int J Appl Math 50(3):1–8

    CAS  Google Scholar 

  • Chen W, Sun H, Li X (2022) Fractional derivative modeling in mechanics and engineering. Springer, New York

    Book  Google Scholar 

  • Dadkhah E, Shiri B, Ghaffarzadeh H, Baleanu D (2020) Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods. J Appl Math Comput 63:29–57

    Article  MathSciNet  Google Scholar 

  • Diethelm K (2010) The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, vol 2004. Springer, Berlin

    Google Scholar 

  • Diethelm K, Ford NJ (2004) Multi-order fractional differential equations and their numerical solutions. Appl Math Comput 154:621–640

    MathSciNet  Google Scholar 

  • Eftekhari T, Rashidinia J (2023) A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions. Mathematical Methods in the Applied Sciences 46(1):388–407

    Article  MathSciNet  ADS  Google Scholar 

  • El-Sayed AMA, El-Kalla IL, Ziada EAA (2010) Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations. Appl Numer Math 60(8):788–797

    Article  MathSciNet  Google Scholar 

  • Erturk VS, Alomari AK, Kumar P, Murillo-Arcila M (2022) Analytic solution for the strongly nonlinear multi-order fractional version of a BVP occurring in chemical reactor theory, Discrete Dynamics in Nature and Society 2022

  • Gil A, Segura J, Temme NM (2007) Numerical methods for special functions. Society for Industrial and Applied Mathematics

  • Gould WH (1972) Explicit formulas for Bernoulli numbers. Am Math Mon 79(1):44–51

    Article  MathSciNet  Google Scholar 

  • Izadi M, Cattani C (2020) Generalized Bessel polynomial for multi-order fractional differential equations. Symmetry 12(8):1260

    Article  ADS  Google Scholar 

  • Joujehi AS, Derakhshan MH, Marasi HR (2022) An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis. Commun Nonlinear Sci Numer Simul 114:106620

    Article  MathSciNet  Google Scholar 

  • Khan NA, Ibrahim Khalaf O, Andres Tavera Romero C, Sulaiman M., Bakar M.A. (2022) Application of intelligent paradigm through neural networks for numerical solution of multiorder fractional differential equations, Computational Intelligence and Neuroscience 2022

  • Khodabandelo HR, Shivanian E, Abbasbandy S (2022) A novel shifted Jacobi operational matrix method for nonlinear multi-terms delay differential equations of fractional variable-order with periodic and anti-periodic conditions. Mathematical Methods in the Applied Sciences 45(16):10116–10135

    Article  MathSciNet  ADS  Google Scholar 

  • Lehmer DH (1988) A new approach to Bernoulli polynomials. Am Math Mon 95(10):905–911

    Article  MathSciNet  Google Scholar 

  • Nagy AM (2022) Numerical solutions for nonlinear multi-term fractional differential equations via Dickson operational matrix. Int J Comput Math 99(7):1505–1515

    Article  MathSciNet  Google Scholar 

  • Napoli A (2016) Solutions of linear second order initial value problems by using Bernoulli polynomials. Appl Numer Math 99:109–120

    Article  MathSciNet  Google Scholar 

  • Odibat Z, Baleanu D (2022) Nonlinear dynamics and chaos in fractional differential equations with a new generalized Caputo fractional derivative. Chin J Phys 77:1003–1014

    Article  MathSciNet  Google Scholar 

  • Postavaru O (2022) Generalized fractional-order hybrid of block-pulse functions and Bernoulli polynomials approach for solving fractional delay differential equations, Soft Computing: 1-13

  • Postavaru O, Toma A (2022) A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems. Math Comput Simul 194:269–284

    Article  MathSciNet  Google Scholar 

  • B. Prakash, A. Setia and S. Bose, Numerical solution for a system of fractional differential equations with applications in fluid dynamics and chemical engineering, International Journal of Chemical Reactor Engineering 15(5) (2017)

  • Rezabeyk S, Abbasbandy S, Shivanian E, Derili H (2023) A new approach to solve weakly singular fractional-order delay integro-differential equations using operational matrices, Journal of Mathematical Modeling

  • Shi J, He K, Fang H (2022) Chaos, Hopf bifurcation and control of a fractional-order delay financial system. Math Comput Simul 194:348–364

    Article  MathSciNet  Google Scholar 

  • Shiralashetti SC, Deshi AB (2016) An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dyn 83:293–303

    Article  MathSciNet  Google Scholar 

  • Talib I, Raza A, Atangana A, Riaz MB (2022) Numerical study of multi-order fractional differential equations with constant and variable coefficients. Journal of Taibah University for Science 16(1):608–620

    Article  Google Scholar 

  • Verma P, Kumar M (2022) Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative, Engineering with Computers: 1-18

  • Zhang B, Tang Y, Zhang X (2021) Numerical solution of fractional differential equations using hybrid Bernoulli polynomials and block pulse functions. Mathematical Sciences 15:293–304

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran

    Babak Azarnavid

  2. Basic Sciences Group, Golpayegan College of Engineering, Isfahan University of Technology, Golpayegan, 87717-67498, Iran

    Mahdi Emamjomeh

  3. Department of Basic Sciences, Abadan Faculty of Petroleum Engineering, Petroleum University of Technology, Abadan, Iran

    Mohammad Nabati

  4. Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran

    Abdollah Dinmohammadi

Authors
  1. Babak Azarnavid
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  2. Mahdi Emamjomeh
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  3. Mohammad Nabati
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  4. Abdollah Dinmohammadi
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Correspondence to Babak Azarnavid.

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Communicated by Agnieszka Malinowska.

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Azarnavid, B., Emamjomeh, M., Nabati, M. et al. An efficient iterative method for multi-order nonlinear fractional differential equations based on the integrated Bernoulli polynomials. Comp. Appl. Math. 43, 68 (2024). https://doi.org/10.1007/s40314-023-02573-7

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  • DOI: https://doi.org/10.1007/s40314-023-02573-7

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