2D Steady-State Heat Transfer (Part 3): Orthogonality and Final Expression | انتقال حرارة

In this video, we complete the analytical solution of a two-dimensional steady-state heat-conduction problem using the Separation of Variables method. Starting from the infinite-series solution developed in Part 2, we apply the remaining nonhomogeneous boundary condition. We then use the orthogonality of the sine functions to isolate and determine the unknown Fourier coefficients. Finally, we substitute these coefficients into the series and obtain the complete analytical expression for the dimensionless temperature distribution. The result is then converted back to dimensional temperature using: T = T₁ + (T₂ − T₁)θ This completes the analytical derivation and prepares the solution for computational evaluation, visualization, and comparison with numerical methods. Video Structure 00:00 – An Essential Introduction | مقدمة لا بد منها 01:30 – Orthogonality and the Final Analytical Expression Topics 2D Heat Transfer • Steady-State Heat Conduction • Separation of Variables • Laplace Equation • Boundary Conditions • Fourier Series • Orthogonality • Fourier Coefficients • Eigenvalues • Eigenfunctions • Dimensionless Temperature • Analytical Solution • Partial Differential Equations #HeatTransfer #SeparationOfVariables #Orthogonality #FourierSeries #AnalyticalSolution #ThermalEngineering #Engineering #انتقال_الحرارة