×
Karimov, Erkinjon; Ruzhansky, Michael; Tokmagambetov, Niyaz

Cauchy type problems for fractional differential equations. (English) Zbl 1490.34067

Integral Transforms Spec. Funct. 33, No. 1, 47-64 (2022).
Summary: While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann-Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
35R11 Fractional partial differential equations

Keywords:

wave equation; Cauchy type problem; inner problem; inner-boundary problem; well-posedness
PDF BibTeX XML Cite
\textit{E. Karimov} et al., Integral Transforms Spec. Funct. 33, No. 1, 47--64 (2022; Zbl 1490.34067)
Full Text: DOI arXiv
  • logo of hbz
  • logo of WorldCat

References:

[1] Pitcher, E.; Sewel, WE., Existence theorems for solutions of differential equations of non-integral order, Bull Amer Math Soc, 44, 2, 100-108 (1938) · Zbl 0018.30701
[2] Kilbas, AA; Trujillo, JJ., Differential equation of fractional order: methods, results and problems, J Appl Anal, 78, 1-2, 153-192 (2001) · Zbl 1031.34002
[3] Tomovski, Z., Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonlinear Anal, 75, 3364-3384 (2012) · Zbl 1241.33020
[4] Hilfer, R.; Luchko, Y.; Tomovki, Z., Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract Calc Appl Anal, 12, 3, 299-318 (2009) · Zbl 1182.26011
[5] Al-Musalhi, F.; Al-Salti, N.; Karimov, ET., Initial boundary value problems for a fractional differential equation with hyper-Bessel operator, Fract Calc Appl Anal, 21, 1, 200-219 (2018) · Zbl 1439.35515
[6] Pskhu, AV., Initial-value problem for a linear ordinary differential equation of noninteger order, Sb Math, 202, 4, 571-582 (2011) · Zbl 1226.34005
[7] Nakhushev, AM., Sturm-Liouville problem for a second-order ordinary differential equation with fractional derivatives in the lowest terms, Dokl Akad Nauk SSSR, 234, 2, 308-311 (1977) · Zbl 0376.34015
[8] Gadzova, LK., Dirichlet and Neumann problems for a fractional ordinary differential equation with constant coefficients, Differ Equ, 51, 12, 1556-1562 (2015) · Zbl 1337.34012
[9] Gadzova, LK., Neumann problem for a fractional-order ordinary differential equation, Vladikavkaz Mat Zh, 18, 3, 22-30 (2016) · Zbl 1466.34008
[10] Mazhgikhova, MG., Initial and boundary value problems for ordinary differential equation of fractional order with delay, Chelyabinsk Phys Math J, 3, 1, 27-37 (2018) · Zbl 1465.34089
[11] Gadzova, LK., Boundary value problem for a linear ordinary differential equation with a fractional discretely distributed differentiation operator, Differ Equ, 54, 2, 178-184 (2018) · Zbl 1395.34003
[12] Karimov, E.; Mamchuev, M.; Ruzhansky, M., Non-local initial problem for second order time-fractional and space-singular equation, Hokkaido Math J, 49, 349-361 (2020) · Zbl 1452.35241
[13] Ruzhansky, M.; Tokmagambetov, N.; Torebek, BT., Bitsadze-Samarskii type problem for the integro-differential diffusion-wave equation on the Heisenberg group, Integral Transforms Spec Funct, 31, 1-9 (2020) · Zbl 1442.45010
[14] Ruzhansky, M.; Tokmagambetov, N.; Torebek, BT., On a non-local problem for a multi-term fractional diffusion-wave equation, Fract Calc Appl Anal, 23, 2, 1-32 (2020) · Zbl 1442.45010
[15] Kilbas, AA; Srivastava, HM; Trujillo, JJ., Theory and applications of fractional differential equations (2006), North-Holland: Elsevier, North-Holland · Zbl 1092.45003
[16] Dimovski, IH., Convolutional calculus (1982), Sofia: Bulgarian Academy of Sciences, Sofia · Zbl 0517.44012
[17] Luchko, Y.; Gorenflo, R., An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math Vietnam, 24, 207-233 (1999) · Zbl 0931.44003
[18] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999), San Diego: Academic Press, San Diego · Zbl 0924.34008
[19] Gorenflo, R.; Kilbas, AA; Mainardi, F., Mittag-Leffler functions, related topics and applications (2014), Berlin: Springer, Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.