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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Loewy decomposition ： ウィキペディア英語版 Loewy decomposition In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced by Alfred Loewy.〔A. Loewy, Ueber vollstaendig reduzible lineare homogene Differentialgleichungen, Mathematische Annalen, 56, pages 89–117 (1906)〕 Solving differential equations is one of the most important subfields in mathematics. Of particular interest are solutions in closed form. Breaking ODEs into largest irreducible components, reduces the process of solving the original equation to solving irreducible equations of lowest possible order. This procedure is algorithmic, so that the best possible answer for solving a reducible equation is guaranteed. A detailed discussion may be found in.〔, F.Schwarz, Loewy Decomposition of Linear Differential Equations, Springer, 2012〕 Loewy's results have been extended to linear partial differential equations (PDEs) in two independent variables. In this way, algorithmic methods for solving large classes of linear pde's have become available. == Decomposing linear ordinary differential equations == Let $D\backslash equiv\backslash frac$ denote the derivative w.r.t. the variable $x$. A differential operator of order $n$ is a polynomial of the form : $L\backslash equiv\; D^n+a\_1D^+\backslash cdots\; +a\_D+a\_n$ where the coefficients $a\_i$, $i=1,\backslash ldots,n$ are from some function field, the ''base field'' of $L$. Usually it is the field of rational functions in the variable $x$, i.e. $a\_i\backslash in(x)$. If $y$ is an indeterminate with $\backslash frac\backslash neq\; 0$, $Ly$ becomes a differential polynomial, and $Ly=0$ is the differential equation corresponding to $L$. An operator $L$ of order $n$ is called ''reducible'' if it may be represented as the product of two operators $L\_1$ and $L\_2$, both of order lower than $n$. Then one writes $L=L\_1L\_2$, i.e. juxtaposition means the operator product, it is defined by the rule $Da\_i=a\_iD+a\_i\text{'}$; $L\_1$ is called a left factor of $L$, $L\_2$ a right factor. By default, the coefficient domain of the factors is assumed to be the base field of $L$, possibly extended by some algebraic numbers, i.e. $$ be a derivative and $a\_i\backslash in$. A differential operator : $L\backslash equiv\; D^n+a\_1D^+\backslash cdots\; +a\_D+a\_n$ of order $n$ may be written uniquely as the product of completely reducible factors $L^\_k$ of maximal order $d\_k$ over $(x)$ in the form : $L=L\_m^L\_^\backslash ldots\; L\_1^$ with $d\_1+\backslash ldots+d\_m=n$. The factors $L^\_k$ are unique. Any factor $L^\_k$, $k=1,\backslash ldots,m$ may be written as : $L^\_k=Lclm(l^\_,l^\_,\backslash ldots,l^\_)$ with $e\_1+e\_2+\backslash ldots+e\_k=d\_k$; $l^\_$ for $i=1,\backslash ldots,k$, denotes an irreducible operator of order $e\_i$ over $(x)$. The decomposition determined in this theorem is called the ''Loewy decomposition'' of $L$. It provides a detailed description of the function space containing the solution of a reducible linear differential equation $Ly=0$. For operators of fixed order the possible Loewy decompositions, differing by the number and the order of factors, may be listed explicitly; some of the factors may contain parameters. Each alternative is called a ''type of Loewy decomposition''. The complete answer for $n=2$ is detailed in the following corollary to the above theorem.〔F. Schwarz, Loewy Decomposition of linear Differential Equations, Bulletin of Mathematical Sciences, 3, page 19–71 (2013); http://link.springer.com/article/10.1007/s13373-012-0026-7〕 Corollary 1 Let $L$ be a second-order operator. Its possible Loewy decompositions are denoted by $^2\_0,\backslash ldots^2\_3$, they may be described as follows; $l^$ and $l^\_j$ are irreducible operators of order $i$; $C$ is a constant. : $^2\_1:\; L=l^\_2l^\_1;$     : $^2\_2:\; L=Lclm(l^\_2,l^\_1);$    : $^2\_3:\; L=Lclm(l^(C)).$ The decomposition type of an operator is the decomposition $^2\_i$ with the highest value of $i$. An irreducible second-order operator is defined to have decomposition type $^2\_0$. The decompositions $^2\_0$, $^2\_2$ and $^2\_3$ are completely reducible. If a decomposition of type $^2\_i$, $i=1,2$ or $3$ has been obtained for a second-order equation $Ly=0$, a fundamental system may be given explicitly. Corollary 2 Let $L$ be a second-order differential operator, $D\backslash equiv\backslash frac$, $y$ a differential indeterminate, and $a\_i\backslash in(x)$. Define $\backslash varepsilon\_i(x)\backslash equiv\backslash exp$ for $i=1,2$ and $\backslash varepsilon(x,C)\backslash equiv\backslash exp$, $C$ is a parameter; the barred quantities $\backslash bar$ and $\backslash bar\backslash neq\backslash bar^2\_1$: $Ly=(D+a\_2)(D+a\_1)y=0$;    $y\_1=\backslash varepsilon\_1(x),$ : $y\_2=\backslash varepsilon\_1(x)\backslash int\backslash frac\backslash ,dx.$ : $^2\_2$: $Ly=Lclm(D+a\_2,D+a\_1)y=0;$ : $y\_i=\backslash varepsilon\_i(x);$ $a\_1$ is not equivalent to $a\_2$. : $^2\_3$: $Ly=Lclm(D+a(C))y=0;$ : $y\_1=\backslash varepsilon(x,\backslash bar)$} : $y\_2=\backslash varepsilon(x,\backslash bar(x)$ are called ''equivalent'' if there exists another rational function $r\backslash in(x)$ such that :$p-q=\backslash frac$. There remains the question how to obtain a factorization for a given equation or operator. It turns out that for linear ode's finding the factors comes down to determining rational solutions of Riccati equations or linear ode's; both may be determined algorithmically. The two examples below show how the above corollary is applied. Example 1 Equation 2.201 from Kamke's collection.〔E. Kamke, Differentialgleichungen I. Gewoehnliche Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, 1964〕 has the $^2\_2$ decomposition $y\text{'}\text{'}+(2+\backslash frac)y\text{'}-\backslash frac-\backslash frac\}\},$ D+2+\frac-\frac}}\Big)y=0. The coefficients $a\_1=2+\backslash frac-\backslash frac\}$ and $a\_2=\backslash frac-\backslash frac\}$ are rational solutions of the Riccati equation $a\text{'}-a^2+\backslash big(2+\backslash frac\backslash big)+\backslash frac=0$, they yield the fundamental system : $y\_1=\backslash frac-\backslash frac+\backslash frac,$ : $y\_2=\backslash frac+\backslash frace^.$ Example 2 An equation with a type $^2\_3$ decomposition is : $y\text{'}\text{'}-\backslash fracy=Lclm\backslash big(D+\backslash frac-\backslash frac\backslash big)y=0.$ The coefficient of the first-order factor is the rational solution of $a\text{'}-a^2+\backslash frac=0$. Upon integration the fundamental system $y\_1=x^3$ and $y\_2=\backslash frac$ for $C=0$ and $C\backslash rightarrow\backslash infty$ respectively is obtained. These results show that factorization provides an algorithmic scheme for solving reducible linear ode's. Whenever an equation of order 2 factorizes according to one of the types defined above the elements of a fundamental system are explicitly known, i.e. factorization is equivalent to solving it. A similar scheme may be set up for linear ode's of any order, although the number of alternatives grows considerably with the order; for order $n=3$ the answer is given in full detail in.〔 If an equation is irreducible it may occur that its Galois group is nontrivial, then algebraic solutions may exist.〔M. van der Put, M.Singer, Galois theory of linear differential equations}, Grundlehren der Math. Wiss. 328, Springer, 2003〕 If the Galois group is trivial it may be possible to express the solutions in terms of special function like e.g. Bessel or Legendre functions, see 〔M.Bronstein, S.Lafaille, Solutions of linear ordinary differential equations in terms of special functions, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation; T.Mora, ed., ACM, New York, 2002, pp. 23–28〕 or.〔F. Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations, CRC Press, 2007, page 39〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Loewy decomposition」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.