A collocation-shooting method for solving fractional boundary value problems

An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem

“…, 100}. We compare our results with the results in Reference [19]. From Table 1, we see that there is no significant effect for the initial condition.…”

“…From Table 1, we see that there is no significant effect for the initial condition. In addition, the proposed method gives more accurate results than Reference [19]. The approximate and exact solutions are given in Figure 2.…”

“…Three of our numerical examples are presented in Section 4. We compare our results with the results in [19,29]. Finally, we draw some conclusions in Section 5.…”

“…Fractional derivatives have many applications, such as diffusion problems, liquid crystals, proteins, mechanics structural control, and biosystems [1][2][3][4][5][6][7][8][9]. Several analytical and numerical methods are used to solve fractional boundary and initial value problems, such as generalized differential transform, the Adomian decomposition method, the homotopy perturbation technique, fractional multistep methods, the spline approximation method, and the collocation method [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Some existence results on nonlinear fractional differential equations

“…It should be noted that most papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations in terms of special functions. Recently, there have been some papers dealing with the existence of solutions of nonlinear initial-value problems of fractional differential equations by using techniques of nonlinear analysis such as fixed-point results, the Leray-Schauder theorem and stability [11][12][13][14][15][16][17][18][19]. In fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena and aerodynamics ( [6,[20][21][22] references therein).…”