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    An efficient algorithm for solving fractional differential equations with boundary conditions
    (Sciendo, 2016) Alkan, Sertan; Yildirim, Kenan; Secer, Aydin
    In this paper, a sinc-collocation method is described to determine the approximate solution of fractional order boundary value problem (FBVP). The results obtained are presented as two new theorems. The fractional derivatives are defined in the Caputo sense, which is often used in fractional calculus. In order to demonstrate the efficiency and capacity of the present method, it is applied to some FBVP with variable coefficients. Obtained results are compared to exact solutions as well as Cubic Spline solutions. The comparisons can be used to conclude that sinc-collocation method is powerful and promising method for determining the approximate solutions of FBVPs in different types of scenarios.
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    A time-delay equation: well-posedness to optimal control
    (De Gruyter Open Ltd, 2016) Yildirim, Kenan; Alkan, Sertan
    In this paper, well-posedness, controllability and optimal control for a time-delay beam equation are studied. The equation of motion is modeled as a time-delayed distributed parameter system(DPS) and includes Heaviside functions and their spatial derivatives due to the finite size of piezoelectric patch actuators used to suppress the excessive vibrations based on displacement and moment conditions. The optimal control problem is defined with the performance index including a weighted quadratic functional of the displacement and velocity which is to be minimized at a given terminal time and a penalty term defined as the control voltage used in the control duration. Optimal control law is obtained by using Maximum principle and hence, the optimal control problem is transformed the into a boundary-, initial and terminal value problem. The explicit solution of the control problem is obtained by eigenfunction expansions of the state and adjoint variables. Numerical results are presented to show the effectiveness and applicability of the piezoelectric control.