A novel Lerch-hybrid approach for distributed-order fractional optimal control problems

This study introduces an innovative numerical approach employing the hybrid Lerch functions to solve distributed-order fractional optimal control problems. The proposed approach combines the block-pulse functions with the Lerch polynomials to create an efficient approximation scheme. The Laplace transform method is employed to formulate the Riemann–Liouville fractional integral operator corresponding to the hybrid functions. The unknown components are represented through the hybrid Lerch function expansion, and the application of Gauss–Legendre quadrature reformulates the original model into algebraic expressions. These expressions are subsequently resolved employing the Newton–Raphson iterative procedure. A comprehensive error analysis is conducted in Sobolev norms. The effectiveness of the proposed strategy is assessed through four representative computational experiments, with results showing superior precision and computational efficiency compared to existing methods. Furthermore, the method exhibits significantly lower computational time than conventional techniques, making it particularly suitable for complex distributed-order fractional systems.