Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. Approximate approaches have been utilized when obtaining analytical (exact) solutions requires substantial computational effort and often is not an attainable task. Hence, the importance of approximation methods, particularly, metaheuristic algorithms are understood. In this paper, a novel approach is suggested for solving engineering ordinary differential equations (ODEs). With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (error function) of the ODEs. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance and inverted generational distance metrics are used for evaluation and assessment of the approximate solutions versus the exact (numerical) solutions. Longitudinal fins having rectangular, trapezoidal, and concave parabolic profiles are considered as studied ODEs. The optimization task is carried out using three different optimizers, including the genetic algorithm, the particle swarm optimization, and the harmony search. The approximate solutions obtained are compared with the differential transformation method (DTM) and exact (numerical) solutions. The optimization results obtained show that the suggested approach can be successfully applied for approximate solving of engineering ODEs. Providing acceptable accuracy of the proposed technique is considered as its important advantage against other approximate methods and may be an alternative approach for approximate solving of ODEs. 
Metaheuristic algorithms for approximate solution to ordinary differential equations of longitudinal fins having various profiles