Approximate solutions of heat transfer fins with convex and exponential profiles using fourier-based optimization method

Differential equations are at the heart of physics and much of chemistry. In this paper, differential equations of convective-radiative longitudinal fins with convex and exponential profiles have been solved approximately using a Fourier-based optimization approach. Using the concepts of mathematics, Fourier series expansion, and metaheuristics, ordinary differential equations (ODEs) can be modeled as an optimization problem. The optimization's target is to minimize the weighted residual function (cost function) of the ODEs. Boundary and initial conditions of ODEs are considered as constraints for the optimization model. Generational distance metric has been used for evaluation and assessment of the approximate solutions against the exact (numerical) solutions. The optimization task has been performed using two well-known optimizers including the harmony search and particle swarm optimization. Approximate solutions obtained by the applied method have been compared with numerical and approximate methods in literature. The optimization results obtained show that the applied approach can be successfully utilized for approximately solving of longitudinal fins with convex and exponential profiles.