Abstract
In the present paper, a direct numerical technique for solving fractional optimal control problems based on an orthonormal wavelet, is introduced. First we approximate the involved functions by Sine-Cosine wavelet basis; then, an operational matrix is used to transform the given problem into a linear system of algebraic equations, which is easier. In fact operational matrix of Riemann-Liouville fractional integration and derivative of Sine-Cosine wavelet are employed to achieve a linear algebraic equation. The mentioned matrices are derived via hat functions. The solution of transformed system, gives us the solution of original problem. Two numerical examples are also given. Finally, the paper is ended with conclusion
Highlights
In recent years, fractional calculus is one of the interesting issues that attract many scientists
Abel applied the fractional calculus in the solution of an integral equation which arises in the formulation of the problem of finding the shape of a frictionless wire lying in a vertical plane such that the time of a bead placed on the wire slides to the lowest point of the wire in the same time regardless of where the bead is placed [6]
Two new operational matrices are introduced and a direct method based on Sine-Cosine wavelet with their fractional integration and differentiation operational matrix is proposed to solve a fractional optimal control problem (FOCP) and a variational problem
Summary
Fractional calculus is one of the interesting issues that attract many scientists. The main idea of applying an orthogonal basis is reduction of considered problem into a system of algebraic equations, by truncating series of orthogonal basis functions for the solution of the problem and applying operational matrix of integration and differentiation to eliminate the integral and derivative operations whenever needed, greatly simplifying the problem. These matrices can be uniquely determined based on the particular orthogonal functions. Two new operational matrices are introduced and a direct method based on Sine-Cosine wavelet with their fractional integration and differentiation operational matrix is proposed to solve a FOCP and a variational problem.
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