Solution of linear fractional Fredholm integro- differential equation by using second kind Chebyshev wavelet Amit Setia, Yucheng Liu Department of Mechanical Engineering University of Louisiana Lafayette, Lafayette, LA-70503, USA axs5147@louisiana.edu, yxl5763@louisiana.edu A. S. Vatsala Department of Mathematics University of Louisiana Lafayette, Lafayette, LA-70504, USA vatsala@louisiana.edu Abstract—In the present paper, a numerical method is proposed to solve the fractional Fredholm integro-differential equation. The proposed method is based on the Chebyshev wavelet approximation. Using the approximation of an unknown function, its fractional derivative and its Integral operator in terms of Chebyshev wavelet, the fractional Fredholm integro-differential equation is ultimately reduced to a system of linear equations which can be solved easily. The test examples are given for illustration. The obtained results are compared for various number of basis functions in the Chebyshev wavelet. The proposed method is easy to understand, easy to implement and gives a very good accuracy. The errors are further measured with the help of different norms to show the good accuracy obtained. Keywords-Integral equation; fractional Fredholm integral equation, Chebyshev wavelet; Chebyshev polynomial, fractional calculus I. INTRODUCTION The fractional calculus, fractional differential equations and fractional integral equations have attracted a lot of researchers and scientists in the last twenty years. And, with the passage of time, more and more researchers are getting interest in this as this branch is finding application in many branches of Mathematics, Science and Engineering [1,2]. A lot of theoretical and numerical work is done time to time by different researches but plenty of it is still left. Among the recent ones, Diethelm and Ford [3] have discussed existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order. They have further investigated the dependence of the solution on the fractional order and initial conditions of the fractional differential equations. Jafari and Gejji [4] have employed Adomian decomposition method to obtain solutions of a system of nonlinear fractional differential equations. Arikoglu and Ozkol [5] have implemented a well-known transformation method to solve Bagley–Torvik, Ricatti and composite fractional oscillation equations. They have also proved many theorems which are not proved before. Salem [6] has established the existence of continuous solutions to some non-linear fractional integral and differential equations. Arikoglu and Ozkol [7] have implemented fractional differential transform method. The method is a semi analytical numerical technique and is extended to solve fractional integro-differential equations of Volterra type. Li [8] have implemented Chebyshev wavelet to solve a nonlinear fractional differential equations. He has also illustrated some examples for the validity and applicability of the technique. Nazari and Shahmorad [9] have developed the fractional differential transform method to solve fractional integro-differential equations with nonlocal boundary conditions. Rawashdeh [10] has applied the Legendre wavelets method to approximate the solution of fractional integro-differential equations. Rehman and Khan [11] have established sufficient conditions for the existence of solutions to a general class of multi-point boundary value problems for a coupled system of fractional differential equations. Li and Sun [12] have proposed a numerical solution of fractional differential equations using the generalized block pulse operational matrix. Using the Leray–Schauder alternative fixed point theorem, Santos, Arjunan and Cuevas [13] have studied the mild-solutions for a class of abstract fractional neutral integro-differential equations with state-dependent delay. A numerical scheme is proposed in [14] by Rehman and Khan. This scheme is based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations. Wang and Fan [15] have implemented Chebyshev wavelet for solving linear and nonlinear fractional differential equations. Chen, Yi and Yu [16] have presented an exact upper bound through the error analysis to solve the numerical solution of fractional differential equation with variable coefficient. Recently, Rehman and Khan have further employed Haar wavelets [17] to obtain solutions of boundary value problems for linear fractional partial differential equations. Further, Gülsu, Öztürk and Anapal [18] have proposed a new method. The method is based on the Taylor Matrix Method to give approximate solution of the linear fractional Fredholm integro- differential equations. In this paper, the fractional Fredholm integro- differential equation 2014 11th International Conference on Information Technology: New Generations 978-1-4799-3187-3/14 $31.00 © 2014 IEEE DOI 10.1109/ITNG.2014.69 465