An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and sources. The main purpose this is to eliminate limitations previous add sources system. restriction that values initial at four corners region should be zero. First, value problem 2D system transformed into dimensionless form. Second, temperature function so temperatures endpoints become Dividing two one-dimensional (1D) subsystems solving them by combining proposed shifting method eigenfunction expansion theorem, complete series form obtained through superposition subsystem solutions. Three examples are studied illustrate efficiency reliability method. For convenience, functions used considered separable space–time domain. linear, parabolic, sine chosen as space-dependent functions, sine, cosine, exponential time-dependent functions. solutions literature verify correctness derived using method, results completely consistent. parameter influence on variation also investigated. includes type harmonic type. function, smaller decay constant leads greater drop. higher frequency more frequent fluctuation change.