基于分数阶热传导方程激光加热瞬态温度场研究

基于分数阶Taylor(泰勒)级数展开原理，建立单相延迟一阶分数阶近似方程，获得分数阶热传导方程.针对短脉冲激光加热问题建立分数阶热传导方程组，并运用Laplace(拉普拉斯)变换方法进行求解，给出非Gauss(高斯)时间分布的激光内热源温度场解析解.针对具体算例数值研究温度波传播特性.结果表明热传播速度与分数阶阶次有关，分数阶阶次增加，热传播速度减小，温度变化幅度增加.分数阶方程可以用于描述介于扩散方程和热波方程间的热传输过程，且对热传播机制与分数阶热传导方程中分数阶项的关系做了深入剖析.

Study on the Transient Temperature Field Based on the Fractional Heat Conduction Equation for Laser Heating

Based on the fractional Taylor series expansion principle, the 1st-order fractional approximate heat conduction constitutive equation was formulated through expansion of the single-phase lag model. Combined with the energy equation, the fractional heat conduction equations were built for short pulse laser heating, and the Laplace transform was applied to solve the equations and obtain the analytical solution of the volumetric heat source temperature field of the non-Gauss time type. The properties of the temperature wave influenced by the fractional order were investigated based on specific examples. The thermal wave velocity decreases and its amplitude increases with the fractional order. The fractional heat conduction equation is applicable for depicting the intermediate heat conduction process between that of the Fourier diffusion equation and that of the thermal wave equation. The correlation between the heat conduction mechanism and the fractional derivative terms in the fractional heat conduction equation was also fully discussed.