Investigation the boundary and initial value problems including fractional integro-differential equations with singular kernels

Document Type : Original Paper


1 Department of Industrial Engineering, Apadana Institute of Higher Education, Shiraz, Iran

2 Faculty of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

3 Faculty of Mathematics, Azarbaijan Shahid Madani University Tabriz, Iran


In this paper, the initial and boundary value problems which includes singular fractional integro-differential equations, are investigated. The fractional derivative which is considered in this article, is the Caputo fractional derivative. The integral equations which are discussed in this paper either without any singularity or contain singular kernels that can be weak or strong. In addition to, in this paper to check and study the singularity and regularity of this type of integral equations are paid. Also, the given integral equations are in the form of initial and boundary value problems, which are discussed in terms of the number and manner of boundary conditions. Finally, some examples are provided for the accuracy and efficiency of the method.


Main Subjects

[1] Jahanshahi, M., Ahmadkhanlu, A. (2014). On Well-Posed of Boundary Value Problems Including Fractional Order Differential Equation, Asian. Bull. Math., 36, 53-59.
[2] Chu, J., O’Regan, D. (2010). Singular integral equation and applications to conjugate problems, Taiwan. J. Math., 14, 329-345.
[3] Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type: Springer Science & Business Media, 2010.
[4] Dehghan, M., Solution of a partial integro-differential equation arising from viscoelasticity, Inter. J. Comput. Math., 83, 123-129.
[5] Hamlin, D., Leary, R. (1987). Methods for using an integro-differential equation as a model of tree height growth, Can. J. For. Res., 17, 353-356.
[6] Kilbas, A. A., Saigo, M., Saxena, R. K. (2004). Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral. Transform. Spec. Funct., 15, 31-49.
[7] Jahanshahi, S., Babolian, E., Torres, D. F., Vahidi, A. (2015). Solving Abel equations of kind first kind via fractional calculus, J. King. Saud. Univ. Sci., 27, 161-167.
[8] kondo, J. (1991). Integral Equations, Kodansha Tokyo, Clarendon Press Oxford.
[9] Keshavarz, E., Ordokhani, Y. (2019). A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro-differential equations with weakly singular kernels, Math. Methods. Appl. Sci., 42, 4427-4443.
[10] Kanwal, R. P. (2013). Linear integral equations, Springer Science & Business Media.
[11] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier (North-Holland) Science Publishers, Amsterdam.
[12] Leonard, A., Mullikin, T. W. (1964). An application of singular integral equation theory to a linearized problem in couette flow, Ann. Phys., 30, 235-248.
[13] Makroglou, A. (2003). Integral equations and actuarial risk management: Some models and numerics, Math. Modell. Anal. 8, 143-54.
[14] Nemati, S., Lima, P. (2018). Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions, Appl. Math. Comput., 327, 79-92.
[15] Nemati, S., Sedaghat, S., Mohammadi, I. (2016). A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels, J. Comput. Appl. Math., 308, 231-242.
[16] Susahab, D. N., Shahmorad, S., Jahanshahi, M. (2015). Efficient quadrature rules for solving nonlinear fractional integro-differential equations of the Hammerstein type, Appl. Math. Model. 39, 5452-5458.
[17] Peskin, E. N., Daniel, V.(1995). Schroeder, An Introduction to Quantum Field Theory, Perseus Books Publishing, L.L.C.
[18] Sabermahani, S., Ordokhani, Y. (2020). A new operational matrix of Müntz-Legendre polynomials and Petrov Galerkin method for solving fractional Volterra-Fredholm integrodifferential equations, Comput. Methods. . Differ. Equ. 8, 408-423.
[19] Sabermahani, S., Ordokhani, Y., Yousefi, S. A. (2018). Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations, Comput. Appl. Math. 37, 3846-3868.
[20] Volterra, V. (1959). Theory of functionals and of integral and integro-differential equations, Dover Publications.
[21] Wang, Y., Zhu, L. (2016). SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput. 275, 72-80.
[22] Zhao, X. Q. (2003). Dynamical systems in population biology: Springer.