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Fractional Quantization of Singular Lagrangian systems with Second-Order Derivatives Using WKB Approximation

In this paper, the theory of the fractional singular Lagrangian systems is investigated with second order derivatives. The fractional quantization for these systems is examined using the WKB approximation. The Hamilton Jacobi treatment can be applied for these systems. The equations of motion are obtained. The Hamilton Jacobi partial differential equations are solved to determine the action function. By finding the action function, the wave function for these systems is constructed. We achieved that the quantum results are agreement with the classical results. Besides, to demonstrate the theory; two mathematical examples are examined.