知识综合方法论：克服数学建模问题的不正确性（修订版，2005年3月）
Methodology of Syntheses of Knowledge: Overcoming Incorrectness of the Problems of Mathematical Modeling (revised version, March 2005)
AI总结：
针对数学物理问题中不适定性与真实物理现象的矛盾，提出将第一类Fredholm积分方程转化为第二类方程的新方法，并论证了广泛问题可归约为二维第一类Fredholm积分方程，支持Hadamard关于物理问题适定性的观点。
摘要：
分析了J. Hadamard关于数学物理问题正确提法的思想，并涉及与之直接相关的Banach逆算子定理的不同解释。研究表明，当代数学建模工具与J. Hadamard、S. Banach及其他杰出科学家的概念存在严重矛盾，因为优先考虑算法的实现实际上意味着不适定问题足以描述真实现象。开发了一种新方法，用于解决传统上与第一类Fredholm积分方程相关的问题，该方法允许将其简化为具有最有利于数值实现性质的第二类Fredholm积分方程。证明了广泛的问题可以归约为二维第一类Fredholm积分方程，包括具有变系数、非规范定义域等的线性边值问题和初边值问题。所提出的算法可直接应用于这些问题，并可用于检验其可解性。在讨论数学物理问题的提法时，特别关注方法论方面。论证了当问题的求解归结为对相应正问题中已知和未知函数的简单重命名时，关于因果关系的结论本质上是不合理的。本工作的目的是建设性地实现J. Hadamard的观点，即具有物理意义的问题总是适定的。
英文摘要：
J. Hadamard's ideas about the correct formulation of the problems of mathematical physics have been analyzed. In this connection various interpretations of the directly related Banach theorem about the inverse operator has been touched. The contemporary apparatus of mathematical modeling is shown to be in a drastic contradiction with concepts of J. Hadamard, S. Banach and a number of other outstanding scientists in the sense that the priority is given to the realization of algorithms, which actually imply that ill-posed problems are adequate to real phenomena. A new method is developed for solving problems traditionally associated with the Fredholm integral equation of the first kind that admits of their reduction to Fredholm integral equation of the second kind with properties most favorable for the numerical realization. It is demonstrated that a wide circle of problems can be reduced to two-dimensional Fredholm integral equations of the first kind; these are linear boundary-value and initial-boundary-value problems with variable coefficients, non-canonical domain of definition and others. The elaborated algorithm is shown to be directly applicable to them and may be used for testing their solvability. In discussing the formulation of problems of mathematical physics, considerable attention is paid to methodological aspects. Conclusions about cause-and-effect relations are argued to be essentially illegitimate when the solution of a problem reduces to a primitive renaming of known and unknown functions of a corresponding direct problem. The aim of this work is a constructive realization of J. Hadamard's opinion that physically meaningful problems are always well-posed.
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