approximation solution to the heat conduction equation in a rectangular channel influenced by airflow using the chebyshev pseudo-spectral method

In this paper, the approximation of the solution to the one-dimensional convection-diffusion equation is studied as a model for heat transfer in a rectangular channel influenced by airflow. first, the convection-diffusion equation is derived using basic principles of conservation, including convection and diffusion. then, the existence and uniqueness of the solution to this equation are briefly discussed, taking into account the properties of the convective and diffusive coefficients, boundary conditions, and initial conditions. numerical solution methods for the problem have been explored by researchers, leading to various approaches. as examples, cite{kho,baz1,ism,smit1,mor1} used the chebyshev pseudo-spectral method for spatial discretization and the fourth-order runge-kutta (rk4) method for temporal discretization to solve this equation. the numerical characteristics of this method, including accuracy, stability, and convergence rate, are analyzed using eigenvalue analysis of the system and stability regions. the results obtained include temperature distribution, absolute error, and a three-dimensional analysis of the temperature distribution in space and time. additionally, the impact of the time step on the stability of the numerical method has been investigated, and it is shown that the proposed method can achieve desirable accuracy and stability with proper parameter adjustments. this study confirms the effectiveness of the chebyshev pseudo-spectral method in solving dynamic problems such as heat transfer and related applications providing a foundation for using this method in more complex problems.