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A Class of Linear Non-Homogenous Higher Order Matrix Fractional Differential Equations: Analytical Solutions and New Technique

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Abstract

In this paper, our formulation generalizes the fractional power series to the matrix form and a new version of the matrix fractional Taylor’s series is also considered in terms of Caputo’s fractional derivative. Moreover, several significant results have been realignment to these generalizations. Finally, to demonstrate the capability and efficiency of our theoretical results, we present the solutions of three linear non-homogenous higher order (m − 1 < α ≤ m, m ∈ N) matrix fractional differential equations by using our new approach.

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References

  1. Z. Al-Zhour, The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense. Alexandaria Eng. J. 55 (2016), 1675–1681.

    Article  Google Scholar 

  2. Z. Al-Zhour, Efficient solutions of coupled matrix and matrix differential equations. Intell. Cont. Autom. 3, No 2 (2012), 176–187.

    Article  Google Scholar 

  3. Z. Al-Zhour, The general (vector) solutions of such linear (coupled) matrix fractional differential equations by using Kronecker structures. Appl. Math. Comput. 232 (2014), 498–510.

    MathSciNet  MATH  Google Scholar 

  4. Z. Al-Zhour, New techniques for solving some matrix and matrix differential equations. Ain Shams Eng. J. 6 (2015), 347–354.

    Article  Google Scholar 

  5. S. Barnett, Introduction to Mathematical Control Theory, Oxford University Press, Oxford (1975).

    MATH  Google Scholar 

  6. A. Bhrawy, T. Taha, J. Machado, A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dynam. 81 (2015), 1023–1052.

    Article  MathSciNet  Google Scholar 

  7. S. Campbell, Singular Systems of Differential Equations II. Pitman, London (1982).

    MATH  Google Scholar 

  8. A. Carpinteri, F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien (1997), 291–348.

    Chapter  Google Scholar 

  9. R. Caponetto, G. Dongola, L. Fortuna, Fractional Order System: Modeling and Control Applications. World Scientific (2010).

    Book  Google Scholar 

  10. J. Claeyssen, M. Gutierrez, Power series solution for the mth-order-matrix ordinary differential equation. Quarterly Appl. Math. J. (1980), 447–450.

    Google Scholar 

  11. K. Diethelm, N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265, No 2 (2002), 229–248.

    Article  MathSciNet  Google Scholar 

  12. A. El-Ajou, O. Abu Arqub, Z. Al-Zhour, S. Momani, New results on fractional power series: theories and applications. Entropy 15 (2013), 5305–5323.

    Article  MathSciNet  Google Scholar 

  13. A. El-Ajou, O. Abu Arqub, M. Al-Smadi, A general form of the generalized Taylor’s formula with some applications. Appl. Math. Comput. 256 (2015), 851–859.

    MathSciNet  MATH  Google Scholar 

  14. A. El-Ajou, O. Abu Arqub, S. Momani, D. Baleanu, A. Alsaedi, Novel expansion iterative method for solving linear partial differential equations of fractional order. Appl. Math. Comput. 257 (2015), 119–133.

    MathSciNet  MATH  Google Scholar 

  15. A. El-Ajou, O. Abu Arqub, S. Momani, Approximate analytical solution of the nonlinear fractional KdV-Burgers equation a new iterative algorithm. J. Comput. Phys. 293 (2015), 81–95.

    Article  MathSciNet  Google Scholar 

  16. A. El-Ajou, M. N. Oqielat, Z. Al-Zhour, S. Momani, Analytical numerical solutions of the fractional multi-pantograph system: Two attractive methods and comparisons. Results Phys. 14 (2019), Art. 102500; DOI: 10.1016/j.rinp.2019.102500.

  17. W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics. Biophysical J. 68, No 1 (1995), 46–53.

    Article  Google Scholar 

  18. R. Horn, C. Johnson, Topics in Matrix Analysis. 1st Ed., Cambridge University Press, Cambridge (1991).

    Book  Google Scholar 

  19. H. Jafari, H. Tajadodi, D. Baleanu, A numerical approach for fractional order Riccati differential equation using B-spline operational matrix. Fract. Calc. Appl. Anal. 18, No 2 (2015), 387–399; DOI: 10.1515/fca-2015-0025; https://www.degruyter.com/view/journals/fca/18/2/fca.18.issue-2.xml.

    Article  MathSciNet  Google Scholar 

  20. T. Kailath, A.H. Sayed, Displacement structure: Theory and application. SIAM Review 37, No 3 (1995), 297–386.

    Article  MathSciNet  Google Scholar 

  21. A. Kilicman, Z. Al-Zhour, Vector least-square solutions of coupled singular matrix equations. J. Comput. Appl. Math. 206 (2007), 1051–1069.

    Article  MathSciNet  Google Scholar 

  22. A. Kilicman, Z. Al-Zhour, Note on the numerical solutions of the general matrix convolution equations by using the iterative methods and box convolution products. Abstr. Appl. Anal. 2010 (2010); DOI: 10.1155/2010/106192.

  23. A. Kilicman, W.A. Ahmad, On matrix fractional differential equations. Advance Mech. Eng. 9, No 1 (2017), 1–7.

    Google Scholar 

  24. A. Kilicman, Z. Al-Zhour, Kronecker operational matrices for fractional calculus and some application. Appl. Math. Comput. 187 (2007), 250–265.

    MathSciNet  MATH  Google Scholar 

  25. S. Kukla, I. Zamorska, Power series solution of first order matrix differential equations. J. Appl. Math. Comput. Mechanics 13, No 3 (2014), 123–128.

    Article  Google Scholar 

  26. J. Liu, X. Li, L. Wu, An operational matrix of fractional differentiation of the second kind of Chebyshev polynomial for solving multi-term variable order fractional differential equation. Math. Prob. Eng. 2016 (2016); DOI: 10.1155/2016/7126080.

  27. J.R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Economics. John Wiley and Sons Ltd. (1999).

    MATH  Google Scholar 

  28. J. Mandelkern, A matrix formulation of Frobenius power series solution using products of 4*4 matrices. Electronic J. Diff. Eq. 212 (2015), 1–16.

    MathSciNet  MATH  Google Scholar 

  29. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Willey and Sons, Inc., New York (1993).

    MATH  Google Scholar 

  30. K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Press, New York (1974).

    MATH  Google Scholar 

  31. I. Podlubny, Matrix approach to discrete fractional calculus. Frac. Calc. Appl. Anal. 3, No 4 (2000), 359–386.

    MathSciNet  MATH  Google Scholar 

  32. I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).

    MATH  Google Scholar 

  33. Y. Rossikhin, M. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50 (1997), 15–67.

    Article  Google Scholar 

  34. A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional order differential equations. Comput. Math. Appl. 59 (2010), 1326–1336.

    Article  MathSciNet  Google Scholar 

  35. F. Zhang, Matrix Theory: Basic Results and Techniques. Springer-Verlag, New York (1999).

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science Al Balqa Applied University, Salt, 19117, Jordan

    Ahmad El-Ajou & Moa’ath N. Oqielat

  2. Department of Mathematics, Faculty of Science Taibah University, Madina, Saudi Arabia

    Ahmad El-Ajou

  3. Department of Basic Sciences and Humanities, College of Engineering Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, 31441, Saudi Arabia

    Zeyad Al-Zhour

  4. Department of Mathematics, Faculty of Science University of Jordan, Amman, 11942, Jordan

    Shaher Momani

  5. Department of Mathematics and Sciences College of Humanities and Sciences, Ajman University Ajman, Ajman, United Arab Emirates

    Shaher Momani

Authors
  1. Ahmad El-Ajou
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  2. Moa’ath N. Oqielat
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  3. Zeyad Al-Zhour
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  4. Shaher Momani
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Correspondence to Ahmad El-Ajou.

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El-Ajou, A., Oqielat, M.N., Al-Zhour, Z. et al. A Class of Linear Non-Homogenous Higher Order Matrix Fractional Differential Equations: Analytical Solutions and New Technique. Fract Calc Appl Anal 23, 356–377 (2020). https://doi.org/10.1515/fca-2020-0017

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  • DOI: https://doi.org/10.1515/fca-2020-0017

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