Undetermined Coefficients Method for Sequential Fractional Differential Equations

Year 2023, , 44 - 50, 31.05.2023

Sertaç Erman

https://doi.org/10.34088/kojose.1145611

Abstract

The undetermined coefficients method is presented for nonhomogeneous sequential fractional differential equations involving Caputo fractional derivative of order n\alpha where n-1 n\alpha\le\ n and n\in\mathbb{N}. By employing proposed method, a particular solution of the considered equation is obtained. Some details about estimating the particular solution required to apply this method are explained. This method is shown to be particularly effective for nonhomogeneous fractional differential equations when the fractional differential equations involve some specific right-hand side functions.

Keywords

Fractional Differential Equation, Undetermined Coefficients, Caputo Derivative, Particular Solution

References

• [1] Miller K. S., Ross B., 1993. An introduction to the fractional calculus and fractional differential equations. Wiley.
• [2] Podlubny I., 1999. Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press.
• [3] Wei Z., Dong W., 2011. Periodic boundary value problems for Riemann-Liouville sequential fractional differential equations. Electronic Journal of Qualitative Theory of Differential Equations, 87, pp.1-13. https://doi.org/10.14232/ejqtde.2011.1.87
• [4] Wei Z., Li Q., Che J., 2010. Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative. Journal of Mathematical Analysis and Applications, 367(1), pp.260-272. https://doi.org/10.1016/j.jmaa.2010.01.023
• [5] Bai C., 2011. Impulsive periodic boundary value problems for fractional differential equation involving Riemann–Liouville sequential fractional derivative. Journal of Mathematical Analysis and Applications, 384(2), pp.211-231. https://doi.org/10.1016/j.jmaa.2011.05.082
• [6] Băleanu D., Mustafa O. G., Agarwal R. P., 2011. On Lp-solutions for a class of sequential fractional differential equations. Applied Mathematics and Computation, 218(5), pp.2074-2081. https://doi.org/10.1016/j.amc.2011.07.024
• [7] Klimek M., 2011. Sequential fractional differential equations with Hadamard derivative. Communications in Nonlinear Science and Numerical Simulation, 16(12), pp. 4689-4697. https://doi.org/10.1016/j.cnsns.2011.01.018
• [8] Awadalla M., Abuasbeh K., 2022. On system of nonlinear sequential hybrid fractional differential equations. Mathematical Problems in Engineering, 2022, pp.1-8. https://doi.org/10.1155/2022/8556578
• [9] Ahmad B., Nieto J. J., 2012. Sequential fractional differential equations with three-point boundary conditions. Computers & Mathematics with Applications, 64(10), pp.3046-3052. https://doi.org/10.1016/j.camwa.2012.02.036
• [10] Ahmad B., Ntouyas S. K., Alsaedi A., 2019. Sequential fractional differential equations and inclusions with semi-periodic and nonlocal integro-multipoint boundary conditions. Journal of King Saud University - Science, 31(2), pp.184-193. https://doi.org/10.1016/j.jksus.2017.09.020
• [11] Zhang H., Li Y., Yang J., 2020. New sequential fractional differential equations with mixed-type boundary conditions. Journal of Function Spaces, 2020, pp.1-9. https://doi.org/10.1155/2020/6821637
• [12] Mohamed H., 2022. Sequential fractional pantograph differential equations with nonlocal boundary conditions: Uniqueness and Ulam-Hyers-Rassias stability. Results in Nonlinear Analysis. https://doi.org/10.53006/rna.928654
• [13] Sambandham B., Vatsala A., 2015. Basic results for sequential caputo fractional differential equations. Mathematics, 3(1), pp.76-91. https://doi.org/10.3390/math3010076
• [14] Hao Z., Chen B., 2022. The unique solution for sequential fractional differential equations with integral multi-point and anti-periodic type boundary conditions. Symmetry, 14(4), 761. https://doi.org/10.3390/sym14040761
• [15] Awadalla M., 2022. Some existence results for a system of nonlinear sequential fractional differential equations with coupled nonseparated boundary conditions. Complexity, 2022, pp.1-17. https://doi.org/10.1155/2022/8992894
• [16] Benmehidi H., Dahmani Z., 2022. On a sequential fractional differential problem with Riemann-Liouville integral conditions. Journal of Interdisciplinary Mathematics, 25(4), pp.893-915. https://doi.org/10.1080/09720502.2020.1861789
• [17] Almalahi M. A., Panchal S. K., Abdo M. S., Jarad F., 2022. On atangana–baleanu-type nonlocal boundary fractional differential equations. Journal of Function Spaces, 2022, pp.1-17. https://doi.org/10.1155/2022/1812445
• [18] Baitiche Z., Guerbati K., Benchohra M., Zhou Y., 2019. Boundary value problems for hybrid caputo fractional differential equations. Mathematics, 7(3), 282. https://doi.org/10.3390/math7030282
• [19] Aydogan S. M., Sakar F. M., Fatehi M., Rezapour S., Masiha H. P., 2021. Two hybrid and non-hybrid k-dimensional inclusion systems via sequential fractional derivatives. Advances in Difference Equations, 2021(1), 449. https://doi.org/10.1186/s13662-021-03606-3
• [20] Mohammadi H., Rezapour S., Etemad S., Baleanu D., 2020. Two sequential fractional hybrid differential inclusions. Advances in Difference Equations, 2020(1), 385. https://doi.org/10.1186/s13662-020-02850-3
• [21] Khalil R., Al Horani M., Anderson D., 2016. Undetermined coefficients for local fractional differential equations. Journal of Mathematics and Computer Science, 16(02), pp.140-146. https://doi.org/10.22436/jmcs.016.02.02
• [22] Diethelm K., 2010. The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer-Verlag.
• [23] Gorenflo R., Kilbas A. A., Mainardi F., Rogosin S. V., 2014. Mittag-leffler functions, related topics and applications (1st ed. 2014). Springer Berlin Heidelberg : Imprint: Springer.
• [24] Prabhakar T. R., 1971. A singular integral equation with a generalized mittag leffler function in the kernel. Yokohama Mathematical Journal, 19, pp.7-15.
• [25] Garra R., Garrappa R., 2018. The Prabhakar or three parameter Mittag–Leffler function: Theory and application. Communications in Nonlinear Science and Numerical Simulation, 56, pp.314-329. https://doi.org/10.1016/j.cnsns.2017.08.018
• [26] Erman S., 2020. Solutions of linear fractional differential equations of order 𝒏− 𝟏 < 𝒏𝒒 < 𝒏. Journal of Scientific Reports-A, 0(45), pp.81-89.