Spectral Approximations Optimized by Flower Pollination Algorithm for Solving Differential Equations

This study introduces Chebyshev Metaheuristic Solver Approach (CMSA), a new computational approach, to get approximate solutions with high-accuracy to a vast range of linear and non-linear differential equations (DEs).The main idea is changing the differential problem into a continuous optimization task.First the approximate solution was written as a truncated series of Chebyshev polynomials, where they are chosen due to their numerical stability and optimal approximation properties.The undetermined coefficients of this series turn into the decision variables in an optimization task.The objective function is derived from the residual of the differential equation, integrated with penalty terms to achieve initial or boundary conditions enforcement.Then the Flower Pollination Algorithm (FPA), a nature-inspired metaheuristic algorithm, is used to find the optimal polynomial coefficients via the minimization of this objective function.This hybrid approach symbiotically integrates the spectral method's exponential convergence properties with the metaheuristic's powerful global search capabilities.The demonstration of the efficiency and robustness of the approach is done through rigorous computational tests on benchmark problems, involving integro-differential and non-linear boundary value problems.A comparison of the computed results with known exact solutions, validates this optimization-driven spectral technique, showing excellent accordance.The approach is simple to implement and displays outstanding potential for tackling complex DE systems where traditional methods maybe stick.