Капуто бөлшек туындысы бар q-айырымдық теңдеулер үшін Коши есептері
DOI:
https://doi.org/10.26577/JMMCS.2022.v113.i1.05Кілттік сөздер:
Коши типтес q-бөлшек есеп, бар болу, жалғыз болу, q-туынды, q-есептеу, бөлшек есептеу, бөлшек туынды, Капутоның бөлшек туындысыАннотация
Бөлшек туындылары бар теңдеулер өздерінің көп салаларда қолданылуына байланысты маңызды рөл атқарады, сонымен қатар олар тек математикада ғана емес, сонымен қатар басқа ғылымдарда да маңызды рөл атқарады. Бұл зерттеу жұмысында біз a> 0 нақты ретті Капутоның q-бөлшек туындысы бар бөлшек-сызықтық q дифференциалдық теңдеулердің нақты шешімдерін құрамыз. Нақтырақ айтсақ, біз осы Коши типтес q-бөлшек есептің сәйкес Вольтердің q-интегралдық теңдеуіне эквивалентті болатындығын қолдана отырып, негізгі нәтижелерге қол жеткіземіз. Осыдан кейін Вольтердің q-интегралдық теңдеуінің шешіміне тізбектеп жуықтау әдісін қолдана отырып, бөлшек-сызықтық q-дифференциалдық теңдеулердің нақты шешімдерін құрамыз. Сол сияқты бізде а> 0 нақты ретті Капутоның бөлшек q-туындысы бар жалпы біртекті бөлшек q-дифференциалдық теңдеу бар және біз басқа q функциясын береміз (Миттаг-Леффлер). Соңында, біз осы нақты шешімдерге нақты формулалар бере алатын жағдайларда негізгі нәтижелерімізді көрсететін бірнеше мысалдар келтірілген.
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[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).
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[28] Agarwal R.P. "Certain fractional q-integrals and q-derivatives"Proc. Camb. Philos. Soc., 66, (1969): 365-370.
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[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).
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[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). Капуто бөлшек туындысы бар q-айырымдық теңдеулер үшін Коши есептері. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 113(1). https://doi.org/10.26577/JMMCS.2022.v113.i1.05
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