Terminal value problem for the system of fractional differential equations with additional restrictions
Abstract
This paper deals with the study of terminal value problem for the system of fractional differential equations with Caputo derivative. Additional conditions are imposed on the solutions of this problem in the form of a linear vector functional. Using the theory of pseudo-inverse matrices, we obtain the necessary and sufficient conditions for the solvability and the general form of the solution of this boundary-value problem. In the one-dimensional case, the obtained results are generalized to the case of a multi-point boundary-value problem. The issue of obtaining similar results for the terminal value problem for the system of fractional differential equations with tempered and Ψ–tempered fractional derivatives of Caputo type is considered.
Keyword : terminal value problem, fractional differential equation, Caputo derivative, pseudoinverse Moore-Penrose matrix, Fredholm integral equation
How to Cite
Boichuk, O., & Feruk, V. (2025). Terminal value problem for the system of fractional differential equations with additional restrictions. Mathematical Modelling and Analysis, 30(1), 120–141. https://doi.org/10.3846/mma.2025.20814
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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N.J. Ford, M.L. Morgado and M. Rebelo. A nonpolynomial collocation method for fractional terminal value problems. J. Comput. Appl. Math., 275:392–402, 2015. https://doi.org/10.1016/j.cam.2014.06.013
C.Y. Gu, G.C. Wu and B. Shiri. An inverse problem approach to determine possible memory length of fractional differential equations. Fract. Calc. Appl. Anal., 24(6):1919–1936, 2021. https://doi.org/10.1515/fca-2021-0083
Z. Gu and Y. Kong. Spectral collocation method for Caputo fractional terminal value problems. Numer. Algorithms, 88:93–111, 2021. https://doi.org/10.1007/s11075-020-01031-3
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
N.O. Kozlova and V.A. Feruk. Noetherian boundary-value problems for integral equations. J. Math. Sci., 222:266–275, 2017. https://doi.org/10.1007/s10958-017-3298-3
C. Li, W. Deng and L. Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete Contin. Dyn. Syst. Ser. B, 24(4):1989–2015, 2019. https://doi.org/10.3934/dcdsb.2019026
M. Medveď and E. Brestovanská. Differential equations with tempered ψ-Caputo fractional derivative. Math. Model. Anal., 26(4):631–650, 2021. https://doi.org/10.3846/mma.2021.13252
S.G. Mikhlin. Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology. 2nd ed., Pergamon, New York, 1964.
M.L. Morgado and M. Rebelo. Well-posedness and numerical approximation of tempered fractional terminal value problems. Fract. Calc. Appl. Anal., 20(5):1239–1262, 2017. https://doi.org/10.1515/fca-2017-0065
I. Podlubny. Fractional Differential Equations. Academic Press, New York, 1999.
F. Sabzikar, M.M. Meerschaert and J. Chen. Tempered fractional calculus. J. Comput. Physics, 293:14–28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024
H.A.H. Salem and M. Cichon´. Analysis of tempered fractional calculus in Ho¨lder and Orlicz spaces. Symmetry, 14(8):1581, 2022. https://doi.org/10.3390/sym14081581
H.A.H. Salem and M. Väth. An abstract Gronwall lemma and applications to global existence results for functional differential and integral equations of fractional order. J. Integral Equations Appl., 16(4):411–439, 2004. https://doi.org/10.1216/jiea/1181075299
B. Shiri, G.C. Wu and B. Baleanu. Terminal value problems for the nonlinear systems of fractional differential equations. Appl. Numer. Math., 170:162–178, 2021. https://doi.org/10.1016/j.apnum.2021.06.015
H.B. Soots, K. Lätt and A. Pedas. Collocation based approximations for a class of fractional boundary value problems. Math. Model. Anal., 28(2):218–236, 2023. https://doi.org/10.3846/mma.2023.16359
J.R.L. Webb. Compactness of nonlinear integral operators with discontinuous and with singular kernels. J. Math. Anal. Appl., 509(2):126000, 2022. https://doi.org/10.1016/j.jmaa.2022.126000
G.C. Wu, B. Shiri, Q. Fan and H.R. Feng. Terminal value problems of nonhomogeneous fractional linear systems with general memory kernels. J. Nonlinear Math. Phys., 30:303–314, 2023. https://doi.org/10.1007/s44198-022-00085-2
G.C. Wu, J.L. Wei and M. Luo. Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis. J. Difference Equ. Appl., 29(9–12):1140– 1155, 2023. https://doi.org/10.1080/10236198.2023.2198043
G. Yang, B. Shiri, H. Kong and G.C. Wu. Intermediate value problems for fractional differential equations. Comput. Appl. Math., 40(1):195, 2021. https://doi.org/10.1007/s40314-021-01590-8
M.A. Zaky. Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J. Comput. Appl. Math., 357:103–122, 2019. https://doi.org/10.1016/j.cam.2019.01.046
R. Almeida, A.B. Malinowska and M.T.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Methods Appl. Sci., 41(1):336–352, 2018. https://doi.org/10.1002/mma.4617
T.M. Atanacković, S. Pilipović, B. Stanković and D. Zorica. Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. John Willey & Sons Inc., Hoboken, 2014. https://doi.org/10.1002/9781118577530
M. Benchohra, S. Bouriah and J.J. Nieto. Terminal value problem for differential equations with Hilfer–Katugampola fractional derivative. Symmetry, 11(5):672, 2019. https://doi.org/10.3390/sym11050672
A.A. Boichuk and A.M. Samoilenko. Generalized Inverse Operators and Fredholm Boundary-Value Problems. 2nd ed., Walter de Gruyter, Berlin, Boston, 2016. https://doi.org/10.1515/9783110944679
O.A. Boichuk and V.A. Feruk. Linear boundary-value problems for weakly singular integral equations. J. Math. Sci., 247:248–257, 2020. https://doi.org/10.1007/s10958-020-04800-6
O.A. Boichuk and V.A. Feruk. Boundary-value problems for weakly singular integral equations. Discrete Contin. Dyn. Syst. Ser. B, 27(3):1379–1395, 2022. https://doi.org/10.3934/dcdsb.2021094
O.A. Boichuk and V.A. Feruk. Fredholm boundary-value problem for the system of fractional differential equations. Nonlinear Dyn., 111:7459–7468, 2023. https://doi.org/10.1007/s11071-022-08218-4
M. Caputo. Linear models of dissipation whose Q is almost frequency independent - II. Geophys. J. Astronom. Soc., 13(5):529–539, 1967. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
M. Cichoń and H.A.H. Salem. On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems. J. Pseudo-Differ. Oper. Appl., 11:1869–1895, 2020. https://doi.org/10.1007/s11868-020-00345-z
M. Cichoń, H.A.H. Salem and W. Shammakh. On the generalization of temperedHilfer fractional calculus in the space of Pettis-integrable functions. Mathematics, 11(13):2875, 2023. https://doi.org/10.3390/math11132875
N.D. Cong and H.T. Tuan. Generation of nonlocal fractional dynamical systems by fractional differential equations. J. Integral Equations Appl., 29(4):585–608, 2017. https://doi.org/10.1216/JIE-2017-29-4-585
K. Diethelm. The Analysis of Fractional Differential Equations. Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2
K. Diethelm. Increasing the efficiency of shooting methods for terminal value problems of fractional order. J. Comput. Phys., 293:135–141, 2015. https://doi.org/10.1016/j.jcp.2014.10.054
K. Diethelm and N.J. Ford. A note on the well-posedness of terminal value problems for fractional differential equations. J. Integral Equations Appl., 30(3):371– 376, 2018. https://doi.org/10.1216/JIE-2018-30-3-371
K. Diethelm and F. Uhlig. A new approach to shooting methods for terminal value problems of fractional differential equations. J. Sci. Comput., 97:38, 2023. https://doi.org/10.1007/s10915-023-02361-9
M. Dobkevich and F. Sadyrbaev. On different type solutions of boundary value problems. Math. Model. Anal., 21(5):659–667, 2016. https://doi.org/10.3846/13926292.2016.1204367
J.S. Duan, L. Lu, L. Chen and Y.L. An. Fractional model and solution for the Black–Scholes equation. Math. Methods Appl. Sci., 41(2):697–704, 2018. https://doi.org/10.1002/mma.4638
H.M. Fahad, A. Fernandez, M.U. Rehman and M. Siddiqi. Tempered and Hadamard-type fractional calculus with respect to functions. Mediterr. J. Math., 18:143, 2021. https://doi.org/10.1007/s00009-021-01783-9
N.J. Ford, M.L. Morgado and M. Rebelo. High order numerical methods for fractional terminal value problems. Comput. Methods Appl. Math., 14(1):55–70, 2014. https://doi.org/10.1515/cmam-2013-0022
N.J. Ford, M.L. Morgado and M. Rebelo. A nonpolynomial collocation method for fractional terminal value problems. J. Comput. Appl. Math., 275:392–402, 2015. https://doi.org/10.1016/j.cam.2014.06.013
C.Y. Gu, G.C. Wu and B. Shiri. An inverse problem approach to determine possible memory length of fractional differential equations. Fract. Calc. Appl. Anal., 24(6):1919–1936, 2021. https://doi.org/10.1515/fca-2021-0083
Z. Gu and Y. Kong. Spectral collocation method for Caputo fractional terminal value problems. Numer. Algorithms, 88:93–111, 2021. https://doi.org/10.1007/s11075-020-01031-3
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
N.O. Kozlova and V.A. Feruk. Noetherian boundary-value problems for integral equations. J. Math. Sci., 222:266–275, 2017. https://doi.org/10.1007/s10958-017-3298-3
C. Li, W. Deng and L. Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete Contin. Dyn. Syst. Ser. B, 24(4):1989–2015, 2019. https://doi.org/10.3934/dcdsb.2019026
M. Medveď and E. Brestovanská. Differential equations with tempered ψ-Caputo fractional derivative. Math. Model. Anal., 26(4):631–650, 2021. https://doi.org/10.3846/mma.2021.13252
S.G. Mikhlin. Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology. 2nd ed., Pergamon, New York, 1964.
M.L. Morgado and M. Rebelo. Well-posedness and numerical approximation of tempered fractional terminal value problems. Fract. Calc. Appl. Anal., 20(5):1239–1262, 2017. https://doi.org/10.1515/fca-2017-0065
I. Podlubny. Fractional Differential Equations. Academic Press, New York, 1999.
F. Sabzikar, M.M. Meerschaert and J. Chen. Tempered fractional calculus. J. Comput. Physics, 293:14–28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024
H.A.H. Salem and M. Cichon´. Analysis of tempered fractional calculus in Ho¨lder and Orlicz spaces. Symmetry, 14(8):1581, 2022. https://doi.org/10.3390/sym14081581
H.A.H. Salem and M. Väth. An abstract Gronwall lemma and applications to global existence results for functional differential and integral equations of fractional order. J. Integral Equations Appl., 16(4):411–439, 2004. https://doi.org/10.1216/jiea/1181075299
B. Shiri, G.C. Wu and B. Baleanu. Terminal value problems for the nonlinear systems of fractional differential equations. Appl. Numer. Math., 170:162–178, 2021. https://doi.org/10.1016/j.apnum.2021.06.015
H.B. Soots, K. Lätt and A. Pedas. Collocation based approximations for a class of fractional boundary value problems. Math. Model. Anal., 28(2):218–236, 2023. https://doi.org/10.3846/mma.2023.16359
J.R.L. Webb. Compactness of nonlinear integral operators with discontinuous and with singular kernels. J. Math. Anal. Appl., 509(2):126000, 2022. https://doi.org/10.1016/j.jmaa.2022.126000
G.C. Wu, B. Shiri, Q. Fan and H.R. Feng. Terminal value problems of nonhomogeneous fractional linear systems with general memory kernels. J. Nonlinear Math. Phys., 30:303–314, 2023. https://doi.org/10.1007/s44198-022-00085-2
G.C. Wu, J.L. Wei and M. Luo. Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis. J. Difference Equ. Appl., 29(9–12):1140– 1155, 2023. https://doi.org/10.1080/10236198.2023.2198043
G. Yang, B. Shiri, H. Kong and G.C. Wu. Intermediate value problems for fractional differential equations. Comput. Appl. Math., 40(1):195, 2021. https://doi.org/10.1007/s40314-021-01590-8
M.A. Zaky. Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J. Comput. Appl. Math., 357:103–122, 2019. https://doi.org/10.1016/j.cam.2019.01.046