Explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions

Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval

“…Fractional differential equations can be extensively applied to various scientific fields, such as physics, control theory, mechanics, and economic management and engineering, for the details, see [1,2] and [3]. Meanwhile, the study of boundary value problems has also received a great attention in the last decade, and a variety of results concerning the existence of solutions, based on kinds of analytic techniques, can be found in the literature [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. However, the theory of boundary value problems in the infinite interval is still in the initial phase, and many aspects of this theory need to be explored.…”

An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

“…To take the numerical method into account, the state function is approximated as x m (t) = t 2 C T m Ψ m (t) − 1, and by considering the Bernstein fractional operational matrix (12), the control input is estimated from the system dynamics as follows: In Fig. To take the numerical method into account, the state function is approximated as x m (t) = t 2 C T m Ψ m (t) − 1, and by considering the Bernstein fractional operational matrix (12), the control input is estimated from the system dynamics as follows: In Fig.…”

“…The optimal cost function is = 0. To take the numerical method into account, the state function is approximated as x m (t) = t 2 C T m Ψ m (t) − 1, and by considering the Bernstein fractional operational matrix (12), the control input is estimated from the system dynamics as follows: For m = 4, the Bernstein operational differential matrix In Fig. 4, the exact and approximate state function as well as control input are plotted.…”

“…Many real dynamical systems are better characterized using a fractional order dynamic model based on fractional calculus or, differentiation or integration of fractional order [1,2]. Finding analytical and numerical solutions for fractional differential equations has been considered as a strong topic of study; these methods include the Legendre spectral tau method [9], the waveform relaxation method [10], the collocation method [11], the explicit iterations method [12] and others [13,14]. Finding analytical and numerical solutions for fractional differential equations has been considered as a strong topic of study; these methods include the Legendre spectral tau method [9], the waveform relaxation method [10], the collocation method [11], the explicit iterations method [12] and others [13,14].…”

Existence and iteration of positive solution for fractional integral boundary value problems with p-Laplacian operator

“…By using the properties of Green function and the Guo-Krasnosel'skii fixed-point theorem on cones, several existence results of at least one or two positive solutions in terms of different eigenvalue interval are obtained. By means of the monotone iterative method, Wang [15] investigated the fractional integral boundary problem…”