Distribuição de temperatura em uma haste por meio do método shooting e diferenças finitas

Abstract

This study aimed to investigate the temperature distribution in a metallic rod subjected to specific boundary conditions, using two numerical methods widely applied in the solution of ordinary differential equations: the Finite Difference Method (FDM) and the Shooting Method. The problem was modeled by a nonlinear second-order ordinary differential equation that describes the thermal behavior of a rod during the heat conduction and convection process. The choice of these methods is justified by the need to obtain approximate solutions for problems whose analytical solutions are either unfeasible or complex, in addition to allowing a comparative analysis of their efficiency, accuracy, and computational implementation. The mathematical formulation of the problem involved modeling the physical phenomenon of heat transfer, considering parameters that represent the effects of conduction, convection, and thermal radiation. For the Finite Difference Method, the domain was discretized into uniform subintervals, transforming the differential equation into a nonlinear system of algebraic equations, which was numerically solved using the Newton Method. In the Shooting Method, the boundary value problem was converted into an equivalent initial value problem and solved using the fourth-order Runge–Kutta method, combined with the Secant Method, to adjust the initial slope until the imposed boundary conditions at the rod ends were satisfied. The implementation of both methods was carried out using the Python programming language, which proved to be an effective tool for algorithm execution, result visualization, and validation. The comparative analysis between the methods showed excellent agreement between the obtained solutions, presenting a similar thermal behavior along the rod’s length. The FDM stood out for its simplicity of implementation, while the Shooting Method demonstrated high accuracy and rapid convergence. Both methods proved suitable for solving nonlinear ODEs, representing viable alternatives for problems in thermal engineering and other applied sciences. It is concluded that the study contributes to a deeper understanding of the use of numerical methods in the resolution of physical problems, as well as reinforces the potential of computational tools in teaching and research. Moreover, it highlights the relevance of mathematical modeling in describing and predicting real physical phenomena with precision and efficiency.