# Initial value problems for some classes of linear evolution equations with several fractional derivatives

• Fedorov Vladimir E., kar@csu.ru Chelyabinsk State University, Mathematical Analysis Department, 129 Brothers Kashirin Street, Chelyabinsk 454021, Russia; South Ural State University, Laboratory of Functional Materials, 76 Lenin Avenue, Chelyabinsk 454080, Russia
• Boyko Kseniya V., kvboyko@mail.ru Chelyabinsk State University, Mathematical Analysis Department, 129 Brothers Kashirin Street, Chelyabinsk 454021, Russia
• Phuong Ta D., tdphuong@math.ac.vn Institute of Mathematics of the Vietnamese Academy of Science and Technology, Department of Numerical Analysis and Scientific Computing Hanoi, Vietnam
Keywords: fractional order differential equation, Gerasimov-Caputo fractional derivative, degenerate evolution equation, Cauchy problem, initial-boundary value problem

### Abstract

The problems of unique solvability of initial problems for linear inhomogeneous equations of a general form with several Gerasimov-Caputo fractional derivatives in Banach spaces are investigated. The Cauchy problem is considered for an equation solved with respect to the highest fractional derivative containing bounded operators at the lowest derivatives. The solution is presented with the use of Dunford-Taylor type integrals. The obtained result allowed us to study an initial problem for a linear inhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that, with respect to this operator, the operator at the second largest derivative is 0-bounded. Abstract results are applied to the study of a class of initial-boundary value problems for equations with several Gerasimov-Caputo time derivatives and with polynomials with respect to a self-adjoint elliptic differential operator in space variables.

### References

[1]
Uchaykin V. V., Method of Fractional Derivatives [in Russian], Artishok, Ul’yanovsk (2008).

[2]
Tarasov V. E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York (2011).

[3]
Samko S. G., Kilbas A. A., and Marichev O. I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Sci. Publ., Yveron (1993).

[4]
Pruss J., Evolutionary Integral Equations and Applications, Birkhäuser-Verl., Basel (1993).

[5]
Podlubny I., Fractional Differential Equations, Acad. Press, Boston (1999).

[6]
Kiryakova V., Generalized Fractional Calculus and Applications, Longman Scientific & Technical, Harlow (1994); copublished in John Wiley & Sons Inc., New York.

[7]
Bajlekova E. G., Fractional Evolution Equations in Banach Spaces, PhD Thes., Eindhoven Univ. Technol., Univ. Press Facilities, Eindhoven (2001).

[8]
Pskhu A. V., Partial Differential Equations of Fractional Order [in Russian], Nauka, Moscow (2005).

[9]
Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier Sci. Publ., Amsterdam; Boston; Heidelberg (2006).

[10]
KostićM., Abstract Volterra Integro-Differential Equations, CRC Press, Boca Raton (2015).

[11]
Jiang H., Liu F., Turner I., Burrage K., “Analitical solutions for the multi-term time-space Caputo–Riesz fractional advection-diffussion equations on a finite domain,” J. Math. Anal. Appl., 389, No. 2, 1117–1127 (2012).

[12]
Liu F., Meerschaert M. M., McGough R. J., Zhuang P., and Liu Q., “Numerical methods for solving the multi-term time-fractional wave-diffussion equation,” Fract. Calc. Appl. Anal., 16, No. 1, 9–25 (2013).

[13]
Alvarez-Pardo E. and Lizama C., “Mild solutions for multi-term time-fractional differential equations with nonlocal initial conditions,” Electron. J. Differ. Equ., 2014, No. 39, 1–10 (2014).

[14]
Singh V. and Pandey D. N., “Existence results for multi-term time-fractional impulsive differential equations with fractional order boundary conditions,” Malaya J. Math., 5, No. 4, 625–635 (2017).

[15]
Singh V. and Pandey D. N., “Mild solutions for multi-term time-fractional impulsive differential systems,” Nonlinear Dyn. Syst. Theory, 18, No. 3, 307–318 (2018).

[16]
Glushak A. V., “A Cauchy-type problem for an abstract differential equation with fractional derivatives,” Math. Notes, 77, No. 1, 26–38 (2005).

[17]
Lizama C. and Prado H., “Fractional relaxation equations on Banach spaces,” Appl. Math. Lett., 23, 137–142 (2010).

[18]
Karczewska A. and Lizama C., “Solutions to stochastic fractional oscillation equations,” Appl. Math. Lett., 2, 1361–1366 (2010).

[19]
Li C.-G., KostićM., and Li M., “Abstract multi-term fractional differential equations,” Kragujevac J. Math., 38, No. 1, 51–71 (2014).

[20]
Fedorov V. E. and KostićM., “On a class of abstract degenerate multi-term fractional differential equations in locally convex spaces,” Euras. Math. J., 9, No. 3, 33–57 (2018).

[21]
Plehanova M. V., “Start control problems for fractional order evolution equations [in Russian],” Chelyab. Fiz.-Mat. Zhurn., 1, No. 3, 15–36 (2016).

[22]
Fedorov V. E., Plekhanova M. V., and Nazhimov R. R., “Degenerate linear evolution equations with the Riemann–Liouville fractional derivative,” Sib. Math. J., 59, No. 1, 136–146 (2018).

[23]
Baybulatova G. D., “Start control problem for a class of degenerate equations with lower order fractional derivatives [in Russian],” Chelyab. Fiz.-Mat. Zhurn., 5, No. 3 271–284 (2020).

[24]
Fedorov V. E. and Avilovich A. S., “A Cauchy type problem for a degenerate equation with the Riemann–Liouville derivative in the sectorial case,” Sib. Math. J., 60, No. 2, 359–372 (2019).

[25]
Fedorov V. E., Gordievskih D. M., Baleanu D., and Tash K., “Criterion of the approximate controllability of a class of degenerate distributed systems with the Riemann–Liouville derivative [in Russian],” Mat. Zametki SVFU, 26, No. 2, 41–59 (2019).

[26]
Fedorov V. E. and Nagumanova A. V., “Linear inverse problems for degenerate evolution equations with the Gerasimov–Caputo derivatives in sectorial case [in Russian],” Mat. Zametki SVFU, 27, No. 2, 54–76 (2020).

[27]
Fedorov V. E., Phuong T. D., Kien B. T., Boyko K. V., and Izhberdeeva E. M., “A class of distributed order semilinear equations in Banach spaces [in Russian],” Chelyab. Fiz.-Mat. Zhurn., 5, No. 3, 342–351 (2020).

[28]
Sviridyuk G. A. and Fedorov V. E., Linear Sobolev Type Equations and Degenerate Semi-groups of Operators, VSP, Utrecht; Boston (2003).

[29]
Triebel H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publ. Co., Amsterdam; New York; Oxford (1978).

[30]
Fedorov V. E., “Strongly holomorphic groups of linear equations of sobolev type in locally convex spaces,” Differ. Equations, 40, No. 5, 753–765 (2004).
How to Cite
Fedorov, V., Boyko, K. and Phuong, T. (2021) “Initial value problems for some classes of linear evolution equations with several fractional derivatives”, Mathematical notes of NEFU, 28(3), pp. 85-104. doi: https://doi.org/10.25587/SVFU.2021.75.46.006.
Issue
Section
Mathematics