Shooting Method to Solve Nonlinear Boundary Value Problem in Heat Transfer

Abstract

This study investigates the application of the shooting method to solve nonlinear boundary value problems (BVP) in heat transfer focusing on a one-dimensional fin with temperature dependent conductivity and conductive-convective-radiative boundary conditions. The objective of this study are to solve the nonlinear BVP using the shooting method, validate the results by comparing with existing literature, and analyses the method’s accuracy and computational efficiency. The governing equation is converted into a dimensionless form and transformed into a system of first-order ordinary differential equations. The shooting method, implemented in MATLAB with a fourth-order Runge-Kutta (RK4) method, then iteratively adjusts the unknown initial condition at the fin tip until the boundary condition at the base is satisfied. Numerical solutions are obtained for various values of the convection parameter, , radiation parameter, , and thermal conductivity parameter, . The results show that higher radiation led to more heat loss and lower fin temperatures, while increased thermal conductivity helped retain heat. The temperature distributions closely match those reported in previous studies confirming the accuracy of the shooting method. Furthermore, the method demonstrated computational efficiency, converging within 1-6 iterations and requiring less than 0.15 seconds of Central Processing Unit (CPU) time for all cases. In conclusion, the shooting method is an accurate, and efficient numerical tool for solving nonlinear BVPs in heat transfer involving conductive, convective, and radiative effects.