Multi-term fractional integro-differential equations in power growth function spaces
-
Vu Kim Tuan
,
Dinh Thanh Duc
Abstract
In this paper we characterize the Laplace transform of functions with power growth square averages and study several multi-term Caputo and Riemann-Liouville fractional integro-differential equations in this space of functions.
Acknowledgements
The second author would like to thank the Vietnam Institute for Advanced Study in Mathematics VIASM for support during his visit to VIASM, and Dr. Huynh Van Ngai (Quy Nhon University), for fruitful discussions.
References
[1] M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 9th printing, Dover, New York (1972).Search in Google Scholar
[2] B. Ahmad, M.M. Matar, and O.M. El-Salmy, Existence of solutions and Ulam stability for Caputo type sequential fractional differential equations of order α ∈ (2, 3). Intern. J. Anal. Appl. 15 (2017), 86–101.Search in Google Scholar
[3] E. Bazhlekova, Completely monotone multinomial Mittag-Leffler type functions and diffusion equations with multiple time-derivatives. Fract. Calc. Appl. Anal. 24, No 1 (2021), 88–111; 10.1515/fca-2021-0005; https://www.degruyter.com/journal/key/FCA/24/1/html.Search in Google Scholar
[4] Xiao-Li Ding and J.J. Nieto, Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms. Fract. Calc. Appl. Anal. 21, No 2 (2018), 312–335; 10.1515/fca-2018-0019; https://www.degruyter.com/journal/key/FCA/21/2/html.Search in Google Scholar
[5] R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In: A. Carpinteri and F. Mainardi (Eds), Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York (1997), 223–276.10.1007/978-3-7091-2664-6_5Search in Google Scholar
[6] R. Gorenflo, A.A. Kilbas, F. Mainardi, and S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014), 2nd Ed. (2020).10.1007/978-3-662-43930-2Search in Google Scholar
[7] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar
[8] M.-Ha Kim, G.-Chol Ri, and H.-Chol O, Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives. Fract. Calc. Appl. Anal. 17, No 1 (2014), 79–95; 10.2478/s13540-014-0156-6; https://www.degruyter.com/journal/key/FCA/17/1/html.Search in Google Scholar
[9] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011); 10.1142/11107.Search in Google Scholar
[10] N. Heymans and J.C. Bauwens, Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta 33 (1994), 210–219.10.1007/BF00437306Search in Google Scholar
[11] Y. Liu, Boundary value problems of singular multi-term fractional differential equations with impulse effects. Math. Nachr. 289 (2016), 1526–1547; 10.1002/mana.201400339.Search in Google Scholar
[12] Zh. Liu, Sh. Zeng, and Y. Bai, Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications. Fract. Calc. Appl. Anal. 19, No 1 (2016), 188–211; 10.1515/fca-2016-0011; https://www.degruyter.com/journal/key/FCA/19/1/html.Search in Google Scholar
[13] R.L. Magin, Fractional Calculus in Bioengineering. Begell House Publishers (2006).Search in Google Scholar
[14] F. Mainardi, Some basic problems in continuum and statistical mechanics. In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Berlin (1997), 291–348.10.1007/978-3-7091-2664-6_7Search in Google Scholar
[15] F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: a tutorial survey. Fract. Calc. Appl. Anal. 10, No 3 (2007), 269–308; E-print http://arxiv.org/abs/0801.4914.Search in Google Scholar
[16] T. Matsuzaki and M. Nakagawa, A chaos neuron model with fractional differential equation. J. Phys. Soc. Japan. 72 (2003), 2678–2684.10.1143/JPSJ.72.2678Search in Google Scholar
[17] R.E.A.C. Paley and N. Wiener, Fourier Transforms in the Complex Domain. Amer. Math. Soc. Coll. Publ. 19 (1934).Search in Google Scholar
[18] S. Picozzi and B.J. West, Fractional Langevin model of memory in financial markets. Phys. Rev. E 66 (2002), 46–118.10.1103/PhysRevE.66.046118Search in Google Scholar
[19] R. Schumer, D. Benson, M.M. Meerschaert, and S.W. Wheatcraft, Eulerian derivative of the fractional advection-dispersion equation. J. Contam. Hydrol. 48 (2001), 69–88.10.1016/S0169-7722(00)00170-4Search in Google Scholar
[20] P.J. Torvik and R.L. Bagley, On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51 (1984), 294–298, 10.1115/1.3167615.Search in Google Scholar
[21] Vu Kim Tuan, Laplace transform of functions with bounded averages. Internat. J. of Evolution Equations 1, No 4 (2005), 429–433.Search in Google Scholar
[22] Vu Kim Tuan, Fractional integro-differential equations in Wiener spaces. Fract. Calc. Appl. Anal. 23, No 5 (2020), 1300–1328; 10.1515/fca-2020-0065; https://www.degruyter.com/journal/key/FCA/23/5/html.Search in Google Scholar
[23] D.V. Widder, The Laplace Transform. Princeton Univ. Press, Princeton (1946).Search in Google Scholar
[24] N. Wiener, Generalized harmonic analysis. Acta Math. 55 (1930), 117–258.10.1017/CBO9780511662492.007Search in Google Scholar
© 2021 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial Survey
- In memory of the honorary founding editors behind the FCAA success story
- Research Paper
- Short time coupled fractional fourier transform and the uncertainty principle
- (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
- Sharp asymptotics in a fractional Sturm-Liouville problem
- Multi-term fractional integro-differential equations in power growth function spaces
- Galerkin method for time fractional semilinear equations
- Müntz sturm-liouville problems: Theory and numerical experiments
- Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
- Nonlinear convolution integro-differential equation with variable coefficient
- An efficient localized collocation solver for anomalous diffusion on surfaces
- Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
- Sliding methods for the higher order fractional laplacians
- Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
Articles in the same Issue
- Frontmatter
- Editorial Survey
- In memory of the honorary founding editors behind the FCAA success story
- Research Paper
- Short time coupled fractional fourier transform and the uncertainty principle
- (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
- Sharp asymptotics in a fractional Sturm-Liouville problem
- Multi-term fractional integro-differential equations in power growth function spaces
- Galerkin method for time fractional semilinear equations
- Müntz sturm-liouville problems: Theory and numerical experiments
- Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
- Nonlinear convolution integro-differential equation with variable coefficient
- An efficient localized collocation solver for anomalous diffusion on surfaces
- Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
- Sliding methods for the higher order fractional laplacians
- Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
