热传导系数跳跃的三维非Fourier温度场分布的奇摄动双参数解

Singularly perturbed solutions of a class of non-Fourier temperature field distribution

应用非Fourier热传导定律构建了温度场模型,即一类在无界域上的三维奇摄动双曲抛物方程的初边值问题.随着温度急剧变化,热传导系数发生跳跃,相应可以用非线性的具有间断系数的奇摄动双参数双曲方程表示.通过奇摄动双参数展开方法,得到了该问题的渐近解.首先应用奇摄动方法得到该问题的展开式,通过对解做出估计以及古典解的存在唯一性定理给出了内解和外解的存在性、唯一性.其次,由奇摄动理论,得到该类奇摄动双曲方程进行了初始层矫正,得到了解关于时间的导数的估计.并且通过用Fourier变换确定了热传导系数跳跃的位置表达式,从而得到了解的形式渐近展开式.最后通过余项估计,得到了渐近解的一致有效性,从而得到了热传导系数间断的温度场的分布.

Thermoelastic coupling model excited by laser is of great significance in engineering. To study the thermoelastic coupling model, the distribution of temperature field must be determined firstly. Because the laser excitation time is short(usually femtosecond), the traditional Fourier heat conduction law is no longer suitable.Therefore, it is necessary to establish the distribution of temperature field by using the non-Fourier heat conduction law. Previous studies on the temperature field model mostly use numerical analysis and computer simulation to discuss its numerical solution, but few can directly solve the analytical solution of the model. Up to now, there are few reports about using singularly perturbed analysis method to solve the asymptotic solution of temperature field model and determine the jumping position of heat conductivity coefficient. In this paper, a temperature field model is constructed by using the non-Fourier heat conduction law, i.e. a class of singularly perturbed hyperbolic equations with small parameters in an unbounded domain. The nonlinear singularly perturbed two-parameter hyperbolic equations with discontinuous coefficients are obtained when the heat conduction coefficients jump due to sharp temperature changes. By using the singularly perturbed biparametric expansion method, the asymptotic solution of the problem is obtained. First, the expansion of the problem is obtained by using singularly perturbed method. The existence and uniqueness of the internal and external solutions are obtained by estimating the maximum modulus of the internal and external solutions and the maximum modulus estimates of the time derivatives, and the formal asymptotic expansion of the solutions is obtained. Secondly, the singularly perturbed hyperbolic equation is corrected by the singular perturbation theory, and the derivative of the solution is estimated. The position expression of the jump of the thermal conductivity coefficient is determined by the Fourier transform, and the seam method is used to connect the seams of the two sides of the jump position of the thermal conductivity coefficient, thus the form asymptotic expansion of the solution is obtained. Finally, the uniform validity of the asymptotic solution is obtained by estimating the residual term, and the distribution of the temperature field with discontinuous heat conduction coefficient is obtained. In this paper, we have synthetically applied the knowledge of ordinary differential equations, partial differential equations, mathematical and physical equations, nonlinear acoustics,mathematical analysis, singular perturbation theory and so on, which enriched the study of non-Fourier temperature field model.