Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions

Mehran Fatemi, Nihan Aliev, Sedaghat Shahmorad

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DOI: 10.4236/am.2011.210179                

Abstract

In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].

Keywords

Fractional Order Integro-Differential Equation, Non-Local Boundary Conditions, Fundamental Solution

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. Nazari and S. Shahmorad, “Application of Fractional Differential Transform Method to the Fractional Order Integro-Differential Equations with Nonlocal Boundary Conditions,” Journal of Computational and Applied Ma-thematics, Vol. 234, No. 3, 2010, pp. 883-891. doi:10.1016/j.cam.2010.01.053
[2]	S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integrals and De-rivatives,” Theory and Applications, Cordon and Breach, Yverdon, 1993.
[3]	C. J. Tranter, “Integral Transforms in Mathematical Physics,” London and New York, 1949.
[4]	V. S. Vladimirov, “Equation of Mathematical Physics,” Mir Publication, Moscow, 1984.
[5]	G. E. Shilov, “Mathematical Analysis. The Second Special Course,” Nauka, Moscow, 1965.
[6]	S. M. Hosseini and N. A. Aliev, “Sufficient Conditions for the Reduction of a BVP for PDE with Non-Local and Global Boundary Conditions to Fredholm Integral Equations (on a Rectan-gular Domain),” Applied Mathematics and Computation, Vol. 147, No. 3, 2004, pp. 669-685.
[7]	F. Bahrami, N. Aliev and S. M. Hosseini, “A Method for the Reduction of Four Imensional Mixed Problems with General Boundary Conditions to a System of Second Kind Fredholm Integral Equations,” Italian Journal of Pure and Applied Mathematics, No. 17, 2005, pp. 91-104.
[8]	N. Aliev and M. Jahanshehi, “Solution of Poissoins Equation with Global, Local and Non-Local Boundary Conditions,” In-ternational Journal of Mathematical Education in Science and Technology, Vol. 33, No. 2, 2002, pp. 241-247. doi:10.1080/00207390110097551