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Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative

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Abstract

The main focus of this study is to apply the two-step Adomian decomposition method (TSADM) for finding a solution of a fractional-order non-linear differential equation by using the Caputo derivative. We are interested in obtaining an analytical solution with two main constraints, that are, without converting the non-linear fractional differential equation to a system of linear algebraic fractional equation, and secondly, with less number of iterations. Moreover, we have investigated conditions for the existence and uniqueness of a solution with the help of some fixed point theorems. Furthermore, the method is demonstrated with the help of some examples. We also compare the results with the Adomian decomposition method (ADM), the modified Adomian decomposition method, and the combination of the ADM and a spectral method. It is concluded that the TSADM provides an analytical solution of fractional-order non-linear differential equation, while the other methods furnish an approximate solution.

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References

  1. Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  2. Lokenath D (2003) Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 54:3413–3442

    MathSciNet  MATH  Google Scholar 

  3. Abbaszadeh M, Dehghan M (2019) Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection-diffusion equation. Eng Comput. https://doi.org/10.1007/s00366-019-00861-7

    Article  Google Scholar 

  4. JinRong W, Yong Z (2011) A class of fractional evolution equations and optimal controls. Nonlinear Anal Real World Appl 12(1):262–272

    Article  MathSciNet  Google Scholar 

  5. Lakshmikantham V (2008) Theory of fractional functional differential equations. Nonlinear Anal Real World Appl 69(10):3337–3343

    Article  MathSciNet  Google Scholar 

  6. Yong Z, Feng J (2010) Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal Real World Appl 11(5):4465–4475

    Article  MathSciNet  Google Scholar 

  7. Fakhar-Izadi F (2020) Fully Petrov-Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation. Eng Comput. https://doi.org/10.1007/s00366-020-00968-2

    Article  Google Scholar 

  8. JinRong W, Yong Z (2011) Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal Real World Appl 12(6):3642–3653

    Article  MathSciNet  Google Scholar 

  9. Hengfei D (2019) A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation. Appl Numer Math 135:30–46

    Article  MathSciNet  Google Scholar 

  10. Jafari H, Seifi S (2009) Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Commun Nonlinear Sci Numer Simul 14(5):2006–2012

    Article  MathSciNet  Google Scholar 

  11. Mousa I, Jafar B, Zainab A (2018) General solution of second order fractional differential equations. Int J Appl Math Res 7(2):56–61

    Article  Google Scholar 

  12. Hendy AS (2020) Numerical treatment for after-effected multi-term time-space fractional advection-diffusion equations. Eng Comput. https://doi.org/10.1007/s00366-020-00975-3

    Article  Google Scholar 

  13. Albadarneh Ramzi B, Batiha Iqbal M, Mohammad Z (2016) Numerical solutions for linear fractional differential equations of order 1 \(\le \alpha \le 2\) using finite difference method (FFDM). J Math Comput Sci 16:103–111

  14. Galip O, Aydin K, Yildiray K (2008) A new analytical approximate method for the solution of fractional differential equations. Int J Comput Math 85(1):131–142

    Article  MathSciNet  Google Scholar 

  15. Oruc O, Esen A, Bulut F (2019) A haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation. Eng Comput 35:75–86

    Article  Google Scholar 

  16. Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135(2):501–544

    Article  MathSciNet  Google Scholar 

  17. Veyis T, Nuran G (2013) On solving partial differential equations of fractional order by using the variational iteration method and multivariate \(Pad\acute{e}\) approximations. Eur J Pure Appl Math 6(2):147–171

    MathSciNet  MATH  Google Scholar 

  18. Zaid O, Shaher M (2009) The variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput Math Appl 58(11–12):2199–2208

    MathSciNet  MATH  Google Scholar 

  19. Zaid O, Shaher M (2008) Numerical method for nonlinear partial differential equation of fractional order. Appl Math Model 32(1):28–39

    Article  Google Scholar 

  20. Ray Santanu S (2009) Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method. Commun Non-linear Sci Numer Simul 14(4):1295–1306

    Article  MathSciNet  Google Scholar 

  21. Bashir A, Nieto Juan J, Ahmed A (2011) Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions. Acta Mathematica Scientia 31(6):2122–2130

    Article  MathSciNet  Google Scholar 

  22. Karthikeyan K, Trujillo JJ (2012) Existence and uniqueness results for fractional integrodifferential equations with boundary value conditions. Commun Nonlinear Sci Numer Simul 17(11):4037–4043

    Article  MathSciNet  Google Scholar 

  23. Dimplekumar N, Chalishajar K (2013) Karthikeyan, Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving gronwall’s inequality in banach spaces. Acta Mathematica Scientia 33(3):758–772

    Article  MathSciNet  Google Scholar 

  24. Anguraj A, Karthikeyan P, Rivero M, Trujillo JJ (2014) On new existence results for fractional integro-differential equations with impulsive and integral conditions. Appl Math Comput 66(12):2587–2594

    Article  MathSciNet  Google Scholar 

  25. Archana C, Jaydev D (2014) Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition. Commun Nonlinear Sci Numer Simul 19(4):821–829

    Article  MathSciNet  Google Scholar 

  26. Zaky Mahmoud A (2019) Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Appl Numer Math 145:429–457

    Article  MathSciNet  Google Scholar 

  27. Xing-Guo L (2005) A Two-step Adomian decomposition method. Appl Math Comput 170(1):570–583

    MathSciNet  MATH  Google Scholar 

  28. Varsha D-G, Hossein J (2005) Adomian decomposition: a tool for solving a system of fractional differential equations. J Math Anal Appl 301(2):508–518

    Article  MathSciNet  Google Scholar 

  29. Haubold HJ, Mathai AM, Saxena RK (2011) Mittag-Leffler functions and their applications. J Appl Math 2011:51

    Article  MathSciNet  Google Scholar 

  30. Kamlesh K, Pandey Rajesh K, Shiva S (2017) Comparative study of three numerical schemes for fractional integro-differential equations. J Comput Appl Math 315:287–302

    Article  MathSciNet  Google Scholar 

  31. Richard S (2007) Palais, A simple proof of the Banach contraction principle. J Fixed Point Theory Appl 2(2):221–223

    Article  MathSciNet  Google Scholar 

  32. Garcia-Falset J, Latrach K, Moreno-Gàlvez E, Taoudi M-A (2012) Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness. J Differ Equ 252(5):3436–3452

    Article  MathSciNet  Google Scholar 

  33. Green JW, Valentine FA (2018) On the Arzel\(\grave{a}\)-Ascoli Theorem. Math Mag 34(4):199–202

    Google Scholar 

  34. Murad SA, Hadid S (2012) Existence and uniqueness theorem for fractional differential equation with integral boundary condition. J Fract Calculus Appl 3(6):1–9

    MATH  Google Scholar 

  35. Abdo Mohammed S, Saeed Abdulkafi M, Panchal Satish K (2019) Caputo fractional integro-differential equation with non-local condition in banach space. Int J Appl Math 32(2):279–288

    MATH  Google Scholar 

  36. Abdul-Majid W (1999) A reliable modification of Adomian decomposition method. Appl Math Comput 102(1):77–86

    MathSciNet  MATH  Google Scholar 

  37. Babolian E, Vahidi AR, Shoja A (2014) An efficient Method for non-linear fractional differential equations: Combination of the Adomian decomposition method and spectral method. Indian J Pure Appl Math 45(6):1017–1028

    Article  MathSciNet  Google Scholar 

  38. Sohrab B, Alireza H (2020) The alternative Legendre Tau method for solving nonlinear multi-order fractional differential equations. J Appl Anal Comput 10(2):442–456

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We express our sincere thanks to editor in chief, editor and reviewers for their valuable suggestions to revise this manuscript.

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  1. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh, 211004, India

    Pratibha Verma & Manoj Kumar

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  1. Pratibha Verma
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  2. Manoj Kumar
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Correspondence to Manoj Kumar.

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Verma, P., Kumar, M. Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative. Engineering with Computers 38, 661–678 (2022). https://doi.org/10.1007/s00366-020-01061-4

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  • DOI: https://doi.org/10.1007/s00366-020-01061-4

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