2D Steady-State Heat Transfer (Part 2): Solution by Separation of Variables | انتقال حرارة

In this video, we continue the analytical solution of a two-dimensional steady-state heat-conduction problem using the Separation of Variables method. We apply the homogeneous boundary conditions to the separated equations, determine the admissible eigenvalues and eigenfunctions, and construct the solution as an infinite series using the principle of superposition. The video concludes by introducing the orthogonality of the sine functions. This property is required to apply the remaining nonhomogeneous boundary condition and determine the unknown Fourier-series coefficients. The derivation will continue in the next part, where we calculate these coefficients and obtain the complete analytical temperature distribution. Topics 2D Heat Transfer • Steady-State Heat Conduction • Separation of Variables • Laplace Equation • Boundary Conditions • Eigenvalues • Eigenfunctions • Superposition Principle • Fourier Series • Orthogonality • Partial Differential Equations • Analytical Solution #HeatTransfer #SeparationOfVariables #FourierSeries #Engineering #PDE #ThermalEngineering #انتقال_الحرارة