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Summability of Formal Solutions for a Family of Generalized Moment Integro-Differential Equations

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Abstract

Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem.

A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.

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References

  1. R. Agarwal, S. Histova, D. O’Regan, Some stability properties related to initial time difference for Caputo fractional differential equations. Fract. Calc. Appl. Anal. 21, No 1 (2018), 72–93; 10.1515/fca-2018-0005; https://www.degruyter.com/journal/key/fca/21/1/html.

    Article  MathSciNet  Google Scholar 

  2. D. Baleanu, Dumitru, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus. Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos 5. Hackensack, NJ: World Scientific (2017).

    MATH  Google Scholar 

  3. W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Universitext, Springer-Verlag, New York (2000).

    MATH  Google Scholar 

  4. W. Balser, M. Yoshino, Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients. Funkcial. Ekvac. 53 (2010), 411–434.

    Article  MathSciNet  Google Scholar 

  5. A.H. Bhrawy, M.A. Zaky, R.A. Van Gorder, A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation., Numer. Algorithms 71, No 1 (2016), 151–180.

    Article  MathSciNet  Google Scholar 

  6. M.A. Firoozjaee, H. Jafari, A. Lia, D. Baleanu, Numerical approach of Fokker-Planck equation with Caputo-Fabrizio fractional derivative using Ritz approximation. J. Comput. Appl. Math. 339 (2018), 367–373.

    Article  MathSciNet  Google Scholar 

  7. A.O. Gel’fond, A.F. Leont’ev, On a generalization of the Fourier series (in Russian). Mat. Sbornik 29, No 71 (1951), 477–500.

    MathSciNet  Google Scholar 

  8. M.I. Gomoyunov, On representation formulas for solutions of linear differential equations with Caputo fractional derivatives. Fract. Calc. Appl. Anal. 23, No 4 (2020), 1141–1160; 10.1515/fca-2020-0058; https://www.degruyter.com/journal/key/fca/23/4/html.

    Article  MathSciNet  Google Scholar 

  9. G.K. Immink, Exact asymptotics of nonlinear difference equations with levels 1 and 1+. Ann. Fac. Sci. Toulouse XVII, No 2 (2008), 309–356.

    Article  MathSciNet  Google Scholar 

  10. G.K. Immink, Accelero-summation of the formal solutions of nonlinear difference equations. Ann. Inst. Fourier (Grenoble) 61 (2011), No 1, 1–51.

    Article  MathSciNet  Google Scholar 

  11. J. Jiménez-Garrido, J. Sanz, G. Schindl, Log-convex sequences and nonzero proximate orders. J. Math. Anal. Appl. 448. No 2 (2017), 1572–1599.

    Article  MathSciNet  Google Scholar 

  12. A. Kilbas, Partial fractional differential equations and some of their applications. Analysis 30, No 1 (2010), 35–66.

    Article  MathSciNet  Google Scholar 

  13. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Stud. 204, Elsevier, Amsterdam (2006).

    MATH  Google Scholar 

  14. V. Kiryakova, An open problem of Ljubomir Iliev related to the Mittag-Leffler function and fractional calculus operators. In: “Complex Analysis and Applications ’13. Proc. Intern. Conference, Sofia, Oct. 31-Nov. 2, 2013”, Bulg. Acad. Sci., Inst. Math. Inform. (2013), 139–153; at http://www.math.bas.bg/complan/caa13/ → Proc. CAA ’13, E-Book.

    Google Scholar 

  15. V. Kiryakova, Gel’fond-Leont’ev integration operators of fractional (multi-)order generated by some special functions. AIP Conf. Proc. 2048 (2018), # 050016; 10.1063/1.5082115.

  16. A. Lastra, S. Malek, J. Sanz, Summability in general Carleman ultraholomorphic classes. J. Math. Anal. Appl. 430 (2015), 1175–1206.

    Article  MathSciNet  Google Scholar 

  17. A. Lastra, S. Malek, J. Sanz, Strongly regular multi-level solutions of singularly perturbed linear partial differential equations. Results Math. 70 (2016), 581–614.

    Article  MathSciNet  Google Scholar 

  18. A. Lastra, S. Michalik, M. Suwińska, Estimates of formal solutions for some generalized moment partial differential equations. J. Math. Anal. Appl. 500, No 1 (2021), # 125094.

    Google Scholar 

  19. A. Lastra, S. Michalik, M. Suwińska, Summability of formal solutions for some generalized moment partial differential equations. Results Math. 76 (2021), # 22, 27 pp.

  20. S. Michalik, Summability and fractional linear partial differential equations. J. Dyn. Control Syst. 16, No 4 (2010), 557–584.

    Article  MathSciNet  Google Scholar 

  21. S. Michalik, Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients. J. Dyn. Control Syst. 18 (2012), 103–133.

    Article  MathSciNet  Google Scholar 

  22. S. Michalik, Analytic solutions of moment partial differential equations with constant coefficients. Funkcial. Ekvac. 56 (2013), 19–50.

    Article  MathSciNet  Google Scholar 

  23. S. Michalik, Summability of formal solutions of linear partial differential equations with divergent initial data. J. Math. Anal. Appl. 406 (2013), 243–260.

    Article  MathSciNet  Google Scholar 

  24. S. Michalik, Analytic summable solutions of inhomogeneous moment partial differential equations. Funkcial. Ekvac. 60 (2017), 325–351.

    Article  MathSciNet  Google Scholar 

  25. S. Michalik, M. Suwińska, Gevrey estimates for certain moment partial differential equations. In: Complex Differential and Difference Equations, De Gruyter Proc. in Math. (2020), 391–408.

    MATH  Google Scholar 

  26. S. Michalik, B. Tkacz, The Stokes phenomenon for some moment partial differential equations, J. Dyn. Control Syst. 25 (2019), 573–598.

    Article  MathSciNet  Google Scholar 

  27. N. Obrechkoff, On the summation of Taylor’s series on the contour of the domain of summability (Paper reprinted in EN from original of 1930). Fract. Calc. Appl. Anal. 19, No 5 (2016), 1316–1346; 10.1515/fca-2016-0069; https://www.degruyter.com/journal/key/fca/19/5/html.

    Article  MathSciNet  Google Scholar 

  28. P. Remy, Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients. J. Dyn. Control Syst. 22 (2016), 693–711.

    Article  MathSciNet  Google Scholar 

  29. P. Remy, Gevrey order and summability of formal series solutions of certain classes of inhomogeneous linear integro-differential equations with variable coefficients. J. Dyn. Control Syst. 23 (2017), 853–878.

    Article  MathSciNet  Google Scholar 

  30. L. Ren, J. Wang, M. Fečkan, Asymptotically periodic solutions for Caputo type fractional evolution equations. Fract. Calc. Appl. Anal. 21, No 5 (2019), 1294–1312; 10.1515/fca-2018-0068; https://www.degruyter.com/journal/key/fca/21/5/html.

    Article  MathSciNet  Google Scholar 

  31. J. Sanz, Asymptotic analysis and summability of formal power series, Analytic, algebraic and geometric aspects of differential equations. In: Trends Math., Birkhäuser/Springer, Cham (2017), 199–262.

    Google Scholar 

  32. M. Suwińska, Gevrey estimates of formal solutions for certain moment partial differential equations with variable coefficients. J. Dyn. Control Syst. 27 (2021), 355–370.

    Article  MathSciNet  Google Scholar 

  33. V. Thilliez, Division by flat ultradifferentiable functions and sectorial extensions. Results Math. 44 (2003), 169–188.

    Article  MathSciNet  Google Scholar 

  34. A. Yonemura, Newton polygons and formal Gevrey classes. Publ. RIMS Kyoto Univ. 26 (1990), 197–204.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dpto. Física y Matemáticas Alcalá de Henares, Universidad de Alcalá, Madrid, Spain

    Alberto Lastra

  2. Faculty of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszyński University, Wóycickiego 1/3, 01-938, Warszawa, Poland

    Sławomir Michalik & Maria Suwińska

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  1. Alberto Lastra
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  2. Sławomir Michalik
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  3. Maria Suwińska
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Correspondence to Alberto Lastra.

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Lastra, A., Michalik, S. & Suwińska, M. Summability of Formal Solutions for a Family of Generalized Moment Integro-Differential Equations. Fract Calc Appl Anal 24, 1445–1476 (2021). https://doi.org/10.1515/fca-2021-0061

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  • DOI: https://doi.org/10.1515/fca-2021-0061

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