Rapid semi-analytical evaluation of transient heat conduction integrals for prescribed surface-temperature strips

Rapid and accurate thermal analysis is essential for parametric design sweeps and rapid thermal assessment in electronic packaging, including the prediction of finite-pulse integrated thermal deficit and thermal response time. These engineering quantities involve time-integrated functionals of transient temperature fields, which ultimately reduce to definite integrals containing the lower incomplete gamma function. For arbitrary surface-temperature profiles, however, no general closed-form evaluation exists for such integrals. This paper proposes a rapid semi-analytical method for transient heat conduction in a semi-infinite substrate subject to prescribed surface-temperature profiles. The method first derives a family of closed-form identities that express a structured class of these integrals in elementary functions. A semi-analytical scheme is then developed by replacing the nested double quadrature with two single quadratures, substantially reducing the per-point computational cost. Using Taylor and Chebyshev expansions, the scheme is further extended to arbitrary analytic or continuous surface-temperature profiles. Four boundary profiles, uniform, linearly graded, quadratic, and Gaussian, demonstrate the method for thermal deficit evaluation and response-time prediction. For the uniform case, the semi-analytical scheme agrees with an independent direct double-quadrature benchmark, with an average relative difference below 10⁻⁹ 10 - 9 , while being 36 to 40 times faster at comparable accuracy. These results confirm that the method provides a practical reduced-order tool for parametric design sweeps and rapid thermal assessment in semi-infinite substrates.