The unique solvability issues of the Cauchy problem with a fractional derivative is considered in the case when the coefficient at the desired function is a continuous function. The solution of the problem is found in an explicit form. The uniqueness theorem is proved. The existence theorem for a solution to the problem is proved by reducing it to a Volterra equation of the second kind with a singularity in the kernel, and the necessary and sufficient conditions for the existence of a solution to the problem are obtained.
This research was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09259780, 2021-2023.)
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[29] K. Mpungu, A.M. Nass, Symmetry Analysis of Time Fractional Convection-reactiondiffusion Equation with a Delay, Results in Nonlinear Analysis 2 (2019) No. 3, 113-124.
[30] A.M. Nakhushev. Fractional Calculus and its Applications (In Russian). Fizmatlit, Moscow, 2003.
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[32] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, SanăDiego-Boston-NewăYork-London-TokyoToronto, 1999.
[33] Walter Rudin, Principles of mathematical analysis, Library of Congress Cataloging in Publication Data, USA 1976.
[34] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993.
[35] Asmae Tajani, El Alaoui Fatima-Zahrae, Boutoulout Ali, Regional Controllability for Caputo Type Semi-Linear Time-Fractional Systems, Advances in the Theory of Nonlinear Analysis and its Applications, 1(1) (2022), 1–13.
[36] V.V. Vasil’ev, L.A. Simak, Drobnoe ischislenie i approksimacionnye metody v modelirovanii dinamicheskih system, NAN Ukrainy, Kiev, 2008.
[37] A. Wazwaz, First Course In Integral Equations, A (Second Edition), World Scientific Publishing Company, USA, 2015.
Year 2023,
Volume: 7 Issue: 1, 232 - 242, 31.03.2023
[1] F.A. Aliev, N.A. Aliev, N.A. Safarova, Transformation of the Mittag-Leffler Function to an Exponential Function and Some of its Applications to Problems with a Fractional Derivative, Appl. and Comp. Math., 18 (3) , (2019) 316-325.
[2] A. Ardjouni, A. Djoudi, Existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations, Results in Nonlinear Analysis 2 (2019) No. 3, 136-142.
[3] Jin Bangti, Fractional Differential Equations, Springer, 2021.
[4] B. Bonilla, A. Kilbas, J. Trujillo: Calculo Fraccionario y Ecuaciones Diferenciales Fraccionarias. UNED Ediciones, Madrid, 2003.
[5] R. Caponetto, G. Dongola, L. Fortuna, I. Petras: Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore, 2010.
[6] K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg (1999) 217-224.
[7] M.M. Dzhrbashyan, A.B. Nersesyan, Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Armyanskoy SSR Mat. 3, (1968) 3Ű28. (In Russian)
[9] G.M. Fikhtengol’ts, The Fundamentals of Mathematical Analysis: International Series in Pure and Applied Mathematics, Vol. 2, Pergamon, 2013.
[10] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators, Mech. Syst. Signal Process. 5 (1991) 81-88.
[11] W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach of self-similar
protein dynamics, Biophys. J. 68 (1995) 46-53.
[12] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[13] M. Japundzic, D. Rajter-Ciric, Reaction-advection-diffusion equations with space fractional derivatives and variable coefficients on infinite domain, Fract. Calc. Appl. Anal. 18(4) (2015) 911-950.
[14] M.T. Jenaliyev, M.I. Ramazanov, M.T. Kosmakova, Zh.M. Tuleutaeva, On the solution to a two-dimensional heat conduction problem in a degenerate domain, Euras. Math. J. 11 (3), (2020) 89-94.
[15] E. Karapinar, D. Kumar, R. Sakthivel, N.H. Luc and N.H. Can, Identifying the space source term problem for time-space-fractional diffusion equation, Advances in Difference Equations (2020) 2020:557.
[16] A.A. Kilbas, S.A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differentsial’nye Uravneniya, 41:1 (2005) 82-86.
doi: 10.1007/s10625-005-0137-y
[17] A.A. Kilbas, H.M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, 204, Elsevier Science BV, Amsterdam, 2006.
[18] M.T. Kosmakova, M.I. Ramazanov, L.Zh. Kasymova, To Solving the Heat Equation with Fractional Load, Lobachevskii Journal of Mathematics, 42(12) (2021) 2854–2866.
[19] Yamina Laskri, Nasser-eddine Tatar, The critical exponent for an ordinary fractional differential problem, Comp.& Math. with Appl., 59 (2010) 1266Ű1.
[20] Xian-Fang Li, Approximate solution of linear ordinary differential equations with variable coefficients, Math. Comput. Simul. 75 (2007) 113-125
[22] J.A.T. Machado,V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3) (2011) 1140-1153.
[23] R. Magin: Fractional Calculus in Bioengineering. Begell House Inc., Redding, CT, 2006.
[24] F. Mainardi: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010.
[25] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York (1997) 291-348.
[26] M. Mark Meerschaert, Erkan Nane, P. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab. 37(3) (2009) 979-1007. doi: 10.1214/08-AOP426
[27] F. Metzler, W. Schick, H.G. Kilian, Nonnenmacher T.F., Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995) 7180-7186.
[28] Ali El Mfadel, Said Melliani, M’hamed Elomari, Existence results for nonlocal Cauchy problem of nonlinear ψ-Caputo type fractional differential equations via topological degree methods, Advances in the Theory of Nonlinear Analysis and its Applications, 6(2) (2022), 270–279.
[29] K. Mpungu, A.M. Nass, Symmetry Analysis of Time Fractional Convection-reactiondiffusion Equation with a Delay, Results in Nonlinear Analysis 2 (2019) No. 3, 113-124.
[30] A.M. Nakhushev. Fractional Calculus and its Applications (In Russian). Fizmatlit, Moscow, 2003.
[31] K.B. Oldham, J. Spanier: Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order. Academic Press, Inc., New York-London, 1974
[32] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, SanăDiego-Boston-NewăYork-London-TokyoToronto, 1999.
[33] Walter Rudin, Principles of mathematical analysis, Library of Congress Cataloging in Publication Data, USA 1976.
[34] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993.
[35] Asmae Tajani, El Alaoui Fatima-Zahrae, Boutoulout Ali, Regional Controllability for Caputo Type Semi-Linear Time-Fractional Systems, Advances in the Theory of Nonlinear Analysis and its Applications, 1(1) (2022), 1–13.
[36] V.V. Vasil’ev, L.A. Simak, Drobnoe ischislenie i approksimacionnye metody v modelirovanii dinamicheskih system, NAN Ukrainy, Kiev, 2008.
[37] A. Wazwaz, First Course In Integral Equations, A (Second Edition), World Scientific Publishing Company, USA, 2015.