摘要： 针对现代科学与工程仿真遇到愈来愈多难以用经典微积分建模方法描述的复杂问题，在理论研究和工程实践中提出各种含有多个经验参数的唯象偏微分方程模型，或直接采用统计模型来描述.这些模型的物理意义不是很清楚且参数多，其中部分人为参数缺乏物理意义.因此，利用描述问题的基本解或统计分布构造隐式微积分控制方程.这里“隐式”是指可以不需要或难以推导出该控制方程的显式微积分表达式.该方法仅需微积分控制方程的基本解和相应的边界条件就可以进行数值仿真计算.称该方法为隐式微积分方程建模.考虑多相软物质热传导的幂律行为，采用分数阶里斯（Riesz）势核函数为基本解构造稳态问题的隐式分数阶微积分方程模型并进行数值验证.此外，以列维（Lévy）稳态统计分布的概率密度函数为基本解，构造反常扩散现象的隐式分数阶微积分方程模型.该研究的主要数值计算技术基于径向基函数的配点方法.
关键词： 隐式微积分方程建模； 唯象模型； 统计模型； 基本解； 经验参数
中图分类号： O39；O241.8文献标志码： AAbstract： As to a growing number of complex scientific and engineering problems which are not easy to be described by classical calculus modeling methodology， a variety of phenomenological partial differential equation models including multiple empirical parameters have been proposed in theoretical research and engineering practice. In some cases， the statistical models are even used to substitute for the calculus models. These models are not clearly interpreted in physics and require more parameters in which the artificial parameters have no physical significance. Therefore， the fundamental solution or statistical distribution which can describe the problem is employed to construct the implicit calculus governing equation. It is noted that “implicit” in the study suggests that the explicit calculus expression of this governing equation is not required or difficult to derive. The fundamental solution of calculus governing equation and corresponding boundary conditions are sufficient to perform numerical simulation. This strategy is called the implicit calculus equation modeling. Considering the power law behaviors of heat conduction in multiple phase soft materials， the kernel function of fractional Riesz potential is used as the fundamental solution to build the implicit fractional calculus equation model for steadystate problems. The numerical experiments verify the model. In addition， the statistical density function of Lévy stable statistical distribution is used as the fundamental solution to build the implicit calculus equation of fractional order to describe anomalous diffusion. The major numerical technique in the research is the radial basis function based collocation methods.
Key words： implicit calculus equation modeling； phenomenological model； statistical model； fundamental solution； empirical parameter
0引言
微积分是现代数学和古典数学的分水岭，数学的发展和应用自此发生根本性变化.[1]经典的微积分方程建模方法在力学、声学、电磁学、热传输和扩散理论中，甚至在现代量子力学和相对论中取得巨大成功.然而，社会学家、经济学家、物理学家和力学家也发现愈来愈多难以用经典微积分方程建模的所谓“反常”现象[23]，如在扩散和耗散中广泛观察到的幂律现象[34]以及非高斯非马尔科夫过程[56]等.