The Cauchy problems for q-difference equations with the Caputo fractional derivatives
DOI:
https://doi.org/10.26577/JMMCS.2022.v113.i1.05Keywords:
Cauchy type q-fractional problem, existence, uniqueness, q-derivative, q-calculus, fractional calculus, fractional derivative, Caputo fractional derivativesAbstract
The fractional differential equations play important roles due to their numerous applications and also for the important role they play not only in mathematics but also in other sciences. In the present research work, we build up the explicit solutions to linear fractional q-differential equations with the q-Caputo fractional derivative of real order a > 0. To speak more precisely, we will achieve our main results we use that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. After that, by using the method of successive approximations is applied to solve the Volterra q-integral equation we construct the the explicit solutions to linear fractional q-differential equations. In the same way we have the more general homogeneous fractional q-differential equation with the Caputo fractional q-derivative of real order a > 0 and we give other The (Mittag-Leffler) q-function. Finally, some examples are presented to illustrate our main results in cases where we can even give concrete formulas for these explicit solutions.
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[34] Annaby M.H., Mansour Z.S., "q-fractional calculus and equations Springer, Heidelberg. (2012).
[35] Shaimardan S., Persson L.E., & Tokmagambetov N.S., "Existence and uniqueness of some Cauchy type problems in fractional q-difference calculus Filomat., -Volume 34, Issue 13, (2020): Pages: 4429-4444
[2] Debnath L., "Recent applications of fractional calculus to science and engineering Int. J. Math. Math. Sci., 54 (2003): 3413–3442.
[3] Magin R.L. (2006). "Fractional Calculus in Bioengineering Begell House Publishers, Danbury, CT, USA., (2006).
[4] Sabatier J., Agrawal O.P., & Machado J.A.T. "Advances in Fractional Calculus, Theoretical Developments and
Applications in Physics and Engineering Springer, Dordrecht, The Netherlands, (2007). (Eds.)
[5] Vazquez L., Velasco M.P., Usero D., & Jimenez S. "Fractional calculus as a modeling framework Monografias Matematicas Garcia de Galdean, 41 (2018): 187-197.
[6] Hilfer R. "Applications of Fractional Calculus in Physics World Scientific, Singapore
[7] Sandev T., & Tomovski Z. "Fractional Equations and Models Theory and Applications Cham, Switzerland, Singapore, (2019).
[8] Kilbas A. A., Srivastava H. M., & Trujillo J. J. "Theory and Applications of Fractional Differential Equations Elsevier, North-Holland, Mathematics studies, (2006).
[9] Hilfer R., "Applications of Fractional Calculus in Physics". World Scientific, Singapore, (2000).
[10] Shaimardan S., "Fractional order Hardy-type inequality in fractional h-discrete calculus". Math. nequal. Appl., 22(2) (2019): 691–702.
[11] Miller K.S. & Ross B. "An Introduction to the Fractional Calculus and Fractional Differential Equations Wiley, New York. (1993).
[12] Hilfer R., "Experimental evidence for fractional time evolution in glass forming materials Chem. Phys., 284 (2002): 399-408.
[13] Tomovski Z., "Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator Nonlinear Anal., 75 (2012): 3364-3384.
[14] Bakakhani A., & Gejji V.D., "Existence of positive solutions of nonlinear fractional differential equations J. Math. Anal. Appl., 278 (2003): 434-442.
[15] Bai C.Z., "Triple positive solutions for a boundary value problem of nonlinear fractional differential equation Electron. J. Qual. Theory Diff. Equ., 24, (2008): 1-10.
[16] Lakshmikantham V., "Theory of fractional functional differential equations Nonlinear Anal., 69 (2008): 3337-3343.
[17] Kosmatov N., "A singular boundary value problem for nonlinear differential equations of fractional order J. Appl. Math. Comput. 29, no. 1-2, (2009): 125-135.
[18] Persson L.-E., Ragusa M.A., Samko N., & Wall P., "Commutators of Hardy operators in vanishing Morrey spaces AIP Conference Proceedings 1493, (2012): 859-866.
[19] Shaimardan S., & Persson L. E., "Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator J. Inequal. Appl., 296, (2015): 1-17.
[20] Shaimardan S., "Hardy-type inequalities for the fractional integral operator in q-analysis Eurasian Math. J. 7, no.1, (2016): 5-16.
[21] Jackson F.H., "On q-functions and a certain difference operator Trans. Roy. Soc. Edin., 46, (1908): 253-281.
[22] Jackson F.H., "On a q-definite integrals Quart. J. Pure Appl. Math., 41, (1910): 193-203.
[23] Carmichael R.D., "The general theory of linear q-difference equations Amer. J. Math., 34, (1912): 147-168.
[24] Cheung P., Kac V. "Quantum calculus Edwards Brothers, Inc., Ann Arbor, MI, USA., (2000).
[25] Ernst T., "A comprehensive treatment of q-calculus Birkh¨auser/Springer, Basel AG, Basel.,(2012).
[26] Ernst T., "A new method of q-calculus Doctoral thesis, Uppsala university.(2002).
[27] Al-Salam W., "Some fractional q-integrals and q-derivatives Proc. Edinb. Math. Soc., 15, (1966/1967): 135-140.
[28] Agarwal R.P. "Certain fractional q-integrals and q-derivatives"Proc. Camb. Philos. Soc., 66, (1969): 365-370.
[29] Rajkovic’ P.M., Marinkovic’ S.D., & Stankovic’ M.S., "On q–fractional derivatives of Riemann–Liouville and Caputo type September; arXiv: arXiv:0909.0387, (2009).
[30] Rajkovic’ P.M., Marinkovic’ S.D. & Stankovic’ M.S. "Fractional integrals and derivatives in q–calculus Applicable Analysis and Discrete Mathematics, 1, (2007): 311-323.
[31] Zhao Y., Chen H., & Zhang Q. "Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions Adv. Differ. Equ. 48, (2013). (https://advancesindifferenceequations.springeropen.com/articles/10.1186/1687-1847-2013-48).
[32] Ferreira R.A.C., "Positive solutions for a class of boundary value problems with fractional q-differences Comput. Math. Appl. 61, (2011): 367-373.
[33] Ferreira R.A.C., "Nontrivials solutions for fractional q-difference boundary value problems Electron. J. Qual. Theory Differ. Equ., 70, (2010): 1-10.
[34] Annaby M.H., Mansour Z.S., "q-fractional calculus and equations Springer, Heidelberg. (2012).
[35] Shaimardan S., Persson L.E., & Tokmagambetov N.S., "Existence and uniqueness of some Cauchy type problems in fractional q-difference calculus Filomat., -Volume 34, Issue 13, (2020): Pages: 4429-4444
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How to Cite
Shaimardan, S., Tokmagambetov, N. S., & Temirkhanova, A. M. (2022). The Cauchy problems for q-difference equations with the Caputo fractional derivatives. Journal of Mathematics, Mechanics and Computer Science, 113(1). https://doi.org/10.26577/JMMCS.2022.v113.i1.05
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