Задачи Коши для q-разностных уравнений с дробными производными Капуто
DOI:
https://doi.org/10.26577/JMMCS.2022.v113.i1.05Ключевые слова:
q-дробная задача типа Коши, существование, единственность, q-производная, q-исчисление,, дробное исчисление, дробная производная, дробные производные КапутоАннотация
Уравнения с дробными производными играют важную роль из-за их многочисленных применений, а также из-за той важной роли, которую они играют не только в математике, но и в других науках. В данной исследовательской работе мы строим явные решения дробно-линейных q-дифференциальных уравнений с q-дробной производной Капуто действительного порядка a> 0. Точнее говоря, мы достигнем наших основных результатов, используя то, что эта q-дробная задача типа Коши эквивалентна соответствующему q-интегральному уравнению Вольтерра. После этого, применяя метод последовательных приближений к решению q-интегрального уравнения Вольтерра, строим явные решения дробно-линейных q-дифференциальных уравнений. Таким же образом у нас есть более общее однородное дробное q-дифференциальное уравнение с дробной q-производной Капуто действительного порядка a> 0, и мы даем другую q-функцию (Миттаг-Леффлера). Наконец, представлены некоторые примеры, иллюстрирующие наши основные результаты в тех случаях, когда мы даже можем дать конкретные формулы для этих явных решений.
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[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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Shaimardan, S., Tokmagambetov, N. S., & Temirkhanova, A. M. (2022). Задачи Коши для q-разностных уравнений с дробными производными Капуто. Вестник КазНУ. Серия математика, механика, информатика, 113(1). https://doi.org/10.26577/JMMCS.2022.v113.i1.05
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