Summability of formal solutions for a family of generalized moment integro-differential equations

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.Search in 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).10.1142/10044Search in Google Scholar

[3] W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Universitext, Springer-Verlag, New York (2000).Search in 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), 411–434.10.1619/fesi.53.411Search in 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), 151–180.10.1007/s11075-015-9990-9Search in 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), 367–373.10.1016/j.cam.2017.05.022Search in 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.Search in 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.Search in 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), 309–356.10.5802/afst.1185Search in Google Scholar

[10] G.K. Immink, Accelero-summation of the formal solutions of nonlinear difference equations. Ann. Inst. Fourier (Grenoble) 61 (2011), No 1, 1–51.10.5802/aif.2596Search in 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), 1572–1599.10.1016/j.jmaa.2016.11.069Search in Google Scholar

[12] A. Kilbas, Partial fractional differential equations and some of their applications. Analysis 30, No 1 (2010), 35–66.10.1524/anly.2010.0934Search in 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).Search in 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.Search in 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.Search in Google Scholar

[16] A. Lastra, S. Malek, J. Sanz, Summability in general Carleman ultraholomorphic classes. J. Math. Anal. Appl. 430 (2015), 1175–1206.10.1016/j.jmaa.2015.05.046Search in 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), 581–614.10.1007/s00025-015-0493-8Search in 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), # 125094.10.1016/j.jmaa.2021.125094Search in 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.10.1007/s00025-020-01324-ySearch in Google Scholar

[20] S. Michalik, Summability and fractional linear partial differential equations. J. Dyn. Control Syst. 16, No 4 (2010), 557–584.10.1007/s10883-010-9107-7Search in Google Scholar

[21] S. Michalik, Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients. J. Dyn. Control Syst. 18 (2012), 103–133.10.1007/s10883-012-9136-5Search in Google Scholar

[22] S. Michalik, Analytic solutions of moment partial differential equations with constant coefficients. Funkcial. Ekvac. 56 (2013), 19–50.10.1619/fesi.56.19Search in Google Scholar

[23] S. Michalik, Summability of formal solutions of linear partial differential equations with divergent initial data. J. Math. Anal. Appl. 406 (2013), 243–260.10.1016/j.jmaa.2013.04.062Search in Google Scholar

[24] S. Michalik, Analytic summable solutions of inhomogeneous moment partial differential equations. Funkcial. Ekvac. 60 (2017), 325–351.10.1619/fesi.60.325Search in 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), 391–408.10.1515/9783110611427-016Search in Google Scholar

[26] S. Michalik, B. Tkacz, The Stokes phenomenon for some moment partial differential equations, J. Dyn. Control Syst. 25 (2019), 573–598.10.1007/s10883-018-9424-9Search in 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.Search in 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), 693–711.10.1007/s10883-015-9301-8Search in 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), 853–878.10.1007/s10883-017-9371-xSearch in 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.Search in 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), 199–262.10.1007/978-3-319-52842-7_4Search in 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), 355–370.10.1007/s10883-020-09504-3Search in Google Scholar

[33] V. Thilliez, Division by flat ultradifferentiable functions and sectorial extensions. Results Math. 44 (2003), 169–188.10.1007/BF03322923Search in Google Scholar

[34] A. Yonemura, Newton polygons and formal Gevrey classes. Publ. RIMS Kyoto Univ. 26 (1990), 197–204.10.2977/prims/1195171666Search in Google Scholar