This paper focuses on the application of fractional backstepping control scheme for nonlinear fractional partial differential equation (FPDE). Two types of fractional derivatives are considered in this paper, Caputo and the Grünwald-Letnikov fractional derivatives. Therefore, obtaining highly accurate approximations for this derivative is of a great importance. Here, the discretized approach for the space variable is used to transform the FPDE into a system of fractional differential equations. The convergence of the closed loop system is guaranteed in the sense of Mittag-Leffler stability. An illustrative example is given to demonstrate the effectiveness of the proposed control scheme.

The two-dimensional integro-differential partial equations is one of the so difficult problems to be solved analytically and/or approximately, and therefore, a method that is efficient for solving such type of problems seems to be necessary. Therefore, in this paper, the iteration methods, which is so called the variational iteration method have been used to provide a solution to such type of problems approximately, in which the obtained results are very accurate in comparison with the exact solution for certain well selected examples which are constructed so that the exact solution exist. Main results of this work is to derive first the variational iteration formula and then analyzing analytically the error term and prove its convergence to zero as the number of iteration increases.

Abstract

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This paper focuses on the application of backstepping control scheme for fractional order partial differential equations (FPDEs) of order with . Therefore to obtain highly accurate approximations for this derivative is of great importance. Here the discretised approach for the space variable is used to transform the FPDEs into a system of differential equations. These approximations arise mainly from the Caputo definition and the Grünwald-Letnikov definition. A Lyapunov function is defined at each stage and the negativity of an overall Lyapunov function is ensured by proper selection of the control law. Illustrative example is given to demonstrate the effectiveness of the proposed control scheme.