知识综合方法论：克服数学建模问题的不正确性
Methodology of Syntheses of Knowledge: Overcoming Incorrectness of the Problems of Mathematical Modeling
AI总结：
针对数学物理问题建模中的不适定性，提出一种基于$l_2$空间构造性利用的新方法，将第一类Fredholm积分方程转化为第二类，并推广至线性和非线性边值问题。
摘要：
分析了J. Hadamard关于数学物理问题正确表述的思想以及相关的Banach逆算子定理。现代数值模拟技术被证明与J. Hadamard、S. Banach及其他杰出科学家的概念存在剧烈矛盾，因为优先实现了低效算法，基于不适定问题足以描述真实现象的信念。开发了一种解决传统上与第一类Fredholm积分方程相关的问题的新方法。其关键方面是构造性地利用函数空间$l_2$的可能性来确保正确性条件。通过一种特殊组合考虑了信息平滑的已知现象，该组合显式包含所求函数，并在空间$L_2$中是无穷小的。通过相对简单的变换，所概述的问题类别被简化为求解第二类Fredholm积分方程，其性质最有利于数值实现。我们展示了向第一类Fredholm积分方程的归约，并相应地将所建议的方法推广到广泛的线性和非线性边值问题及初边值问题。我们提出论据，认为基于对相应正问题中已知和未知函数的原始重命名的因果关系的确定本质上是不合法的。
英文摘要：
J. Hadamard's ideas of correct formulation of problems of mathematical physics as well as related Banach's theorem on the inverse operator are analyzed. Modern techniques of numerical simulations are shown to be in drastic contradiction to the 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 inefficient algorithms, based on a belief that ill-posed problems are adequate to real phenomena. A new method of the solution of problems, traditionally associated with Fredholm integral equations of the first kind, is developed. Its key aspect is a constructive use of possibilities of the functional space $l_2$ to ensure the conditions of correctness. A well-known phenomenon of smoothing of information is taken into account by means of a special composition that explicitly involves the sought function and is infinitesimal in the space $L_2$. By relatively simple transformations, the outlined class of problems is reduced to the solution of Fredholm integral equations of the second kind with properties most favorable for the numerical realization. We demonstrate a reduction to Fredholm integral equations of the first kind and, correspondingly, a possibility to extend the suggested approach to wide classes of linear and nonlinear boundary-value and initial-boundary-value problems. We put forward arguments that the determination of causal relationships, based on the formulation restricted to a primitive renaming of known and unknown functions of the corresponding direct problem, is essentially illegitimate.
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