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首页> 外文期刊>International Journal of Theoretical and Mathematical Physics >On the Superiority of the Method of Integration by Parts for Solving Exact and Semi-exact Equations (With Physical Applications)
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On the Superiority of the Method of Integration by Parts for Solving Exact and Semi-exact Equations (With Physical Applications)

机译：求解精确方程和半精确方程的部分积分方法的优越性（在物理应用中）

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Further to our previous work on exact equations (published in this journal; 2015;5(5);69-75), this work is concerned with another class of differential equations that are not exact initially, but can be turned exact by multiplication by the so-called integrating factors . It seems appropriate to give these equations a name, and we will call them semi-exact equations . As was the case of the exact equations in the previous paper, the semi-exact ones likewise have two tests for establishing their semi-exactness and determining their integrating factors, as well as the same three methods of solution . These tests and methods will be contrasted again (this time in connection with the semi-exact equations). All the notions discussed previously concerning the exact equations hold true for the semi-exact ones as well and some other notions will be added in this second paper. One basic goal of this work is to see how to test the latter equations for semi-exactness and determine their integrating factors by applying the short and unwritten three-rectification test by sight based on the integrability of the semi-exact equations by parts. Beside being abbreviated and very fast this test moreover applies to both of the equations with one-variable, and those with two-variable integrating factors. This is unlike the other partial differentiation t est for semi-exactness in current use which test is comparatively prolonged and moreover applies only to the equations with one-variable, and not to those with two-variable integrating factors as will be shown later. This wide difference in efficiency between the two tests in establishing the semi-exactness and determining the integrating factors is furthermore complimented by a similarly wide difference between the current and the proposed methods of solution. Our ultimate goal will be to solve the semi-exact equations by the two-, or three- step integration by parts as a straightforward and by far shorter and less laborious solution which is expected to replace the prolonged solutions by separation of the variables , and by integration in total differentials , as was the case of the exact equations in the previous paper. Again a number of examples and exercises will be quoted for contrasting the currently applied prolonged methods of solution to the newly introduced much shorter and simpler ones, and explaining the new notions introduced in this work. These notions will open new vistas for further research in related fields as will be revealed in our next papers. Many physical applications of the equations with semi-exact first and second order doublets can be quoted from classical and quantum mechanics. To keep this paper within a reasonable number of pages and avoid its becoming excessively lengthy (although this would not matter much to the curious reader) we unfortunately have to postpone to the next paper three of the seven reworked examples of [part B] of this paper. These examples contain the selected physical applications of the semi-exact equations and will serve as the subject-matter of the next paper. The reader is assumed to be familiar with the previous paper and to recall the new notions introduced in it such as: singlets and non-singlets, doublets and triplets, dependent and independent arms of a doublet, exact, semi-exact, and non-exact equations, …etc.
机译：继我们先前关于 精确方程的工作（在该杂志上发布; 2015; 5（5）; 69-75）之后，该工作与另一类最初不是精确的微分方程有关，但可以通过乘以所谓的 积分因子可以精确地得出。给这些方程起一个名字似乎是适当的，我们将它们称为 半精确方程。与上一篇论文中的精确方程一样，半精确方程也有两个测试来确定其半精确度并确定其积分因子，而相同的 三种方法 解决。这些测试和方法将再次进行对比（这次结合半精确方程式）。先前讨论的所有与精确方程有关的概念都适用于半精确方程，在第二篇文章中还将添加一些其他概念。这项工作的一个基本目标是，了解如何通过应用简短且未写的 三次校正 test <根据半精确方程的可分性，通过 观察b。除了缩写和非常快速之外，该测试还适用于两个具有一变量的方程和具有二变量积分因子的方程。这与当前使用的半精确性的其他 偏微分t est 不同，该测试相对延长，而且仅适用于一变量方程，而不是那些具有二元积分因子的人，这将在后面显示。在建立半精确度和确定积分因子的两个测试之间，效率的巨大差异还得到了当前解决方案和拟议解决方法之间类似的巨大差异的补充。我们的最终目标将是通过零件的两步或三步积分来解决半精确方程，这是一种直接的，更短且更省力的解决方案，有望取代长期的解决方案。变量的 分离，以及通过总微分的 积分，就像前一篇论文中的精确方程式一样。再次引用大量示例和练习，以将当前应用的扩展解决方案与新近引入的更短和更简单的方法进行对比，并解释这项工作中引入的新概念。这些概念将为相关领域的进一步研究开辟新的前景，这将在我们的下一篇论文中揭示。具有半精确的一阶和二阶对偶的方程的许多物理应用可以从经典和量子力学中引用。为了使本文保持在合理的页数内，并避免篇幅过长（尽管这对好奇的读者来说并没有太大关系），我们很遗憾地不得不将本部分[B部分]的七个示例中的三个推迟到下一篇论文中纸。这些示例包含半精确方程的选定物理应用，并将作为下一篇论文的主题。假定读者熟悉先前的论文，并回想了其中引入的新概念，例如：单重态和非单重态，双峰态和三重态，双重态的独立臂和独立臂，精确，半精确和非双臂。精确方程式，等等。

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