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{{for|other meanings of the term|Lagrange's identity (disambiguation)}}
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In the study of [[ordinary differential equation]]s and their associated [[boundary value problem]]s, '''Lagrange's identity''', named after [[Joseph Louis Lagrange]], gives the boundary terms arising from [[integration by parts]] of a self-adjoint linear [[differential operator]].  Lagrange's identity is fundamental in [[Sturm&ndash;Liouville theory]].  In more than one independent variable, Lagrange's identity is generalized by [[Green's second identity]].
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==Statement==
In general terms, Lagrange's identity for any pair of functions ''u'' and ''v'' &ensp;in [[Smooth_functions#The_space_of_Ck_functions|function space ''C''<sup>2</sup>]] (that is, twice differentiable) in ''n'' dimensions is:<ref name=Schaum>
 
{{cite book |title= Schaum's outline of theory and problems of partial differential equations |author=Paul DuChateau, David W. Zachmann |page=103 |url=http://books.google.com/books?id=h9kRkvGQCgkC&pg=PA103 |chapter=§8.3 Elliptic boundary value problems |isbn=0-07-017897-6 |publisher=McGraw-Hill Professional |year=1986}}
</ref>
 
:<math>vL[u]-uL^*[v]=\nabla \cdot \boldsymbol M, \ </math>
 
where:
 
:<math>M_i = \sum_{j=1}^n a_{ij}\left(
v \frac{\partial u}{\partial x_j} -u \frac{\partial v}{\partial x_j}
\right ) + uv \left(
b_i - \sum_{j=1}^{n} \frac{\partial a_{ij}}{\partial x_j} \right ), </math>
 
and
 
:<math>\nabla \cdot \boldsymbol M  = \sum_{i=1}^n \frac{\partial}{\partial x_i} M_i, </math>
 
The operator ''L'' and its [[adjoint operator]] ''L''<sup>*</sup> are given by:
 
:<math>L[u] = \sum_{i,\ j =1}^n a_{i,j}  \frac {\partial ^2 u }{\partial x_i \partial x_j} + \sum_{i=1}^n b_i \frac {\partial u}{\partial x_i} +c u </math>
 
and  
 
:<math>L^*[v] = \sum_{i,\ j =1}^n  \frac {\partial ^2 (a_{i,j} v) }{\partial x_i \partial x_j} - \sum_{i=1}^n \frac {\partial (b_i  v)}{\partial x_i} + cv. \, </math>
 
If Lagrange's identity is integrated over a bounded region, then the [[divergence theorem]] can be used to form [[Green's second identity]] in the form:
 
:<math>\int_\Omega v L[u]\ d\Omega = \int_{\Omega} u L^*[v]\  d\Omega +\int_S \boldsymbol{M \cdot n } \, dS, </math>
 
where ''S'' is the surface bounding the volume ''Ω'' and '''''n''''' is the unit outward normal to the surface ''S''.
 
===Ordinary differential equations===
Any second order [[ordinary differential equation]] of the form:
 
:<math>a(x)\frac{d^2y}{dx^2} + b(x)\frac {dy}{dx} +c(x)y +\lambda w(x) y =0, </math>
 
can be put in the form:<ref name= Richards>
 
{{cite book |title=Advanced mathematical methods with Maple |author=Derek Richards |page=354 |url=http://books.google.com/books?id=AKsLy0rVGuwC&pg=PA354 |isbn=0-521-77981-2 |chapter=§10.4 Sturm&ndash;Liouville systems |year=2002 |publisher=Cambridge University Press }}
 
</ref>
 
:<math>\frac {d}{dx} \left( p(x) \frac {dy}{dx} \right )  +\left( q(x)+ \lambda w(x) \right) y(x) = 0. </math>
 
This general form motivates introduction of the [[Sturm–Liouville theory|Sturm–Liouville operator]] ''L'', defined as an operation upon a function ''f&ensp;'' such that:
 
:<math>L f = \frac {d}{dx} \left( p(x) \frac {df}{dx} \right) + q(x) f. </math>
 
It can be shown that for any ''u'' and ''v''&ensp;for which the various derivatives exist, '''Lagrange's identity''' for ordinary differential equations holds:<ref name= Richards/>
 
:<math> uLv - vLu = - \frac{d}{dx} \left[ p(x) \left(v\frac{du}{dx} -u \frac{dv}{dx} \right ) \right]. </math>
 
For ordinary differential equations defined in the interval [0, 1], Lagrange's identity can be integrated to obtain an integral form (also known as Green's formula):<ref name=Loney>
 
{{cite book |title=Applied mathematical methods for chemical engineers |author=Norman W. Loney |url=http://books.google.com/books?id=QD6hOiRrAjIC&pg=PA218 |page=218 |chapter=Equation 6.73 |edition=2nd |isbn=0-8493-9778-2 |year=2007 |publisher=CRC Press}}
 
</ref><ref name=Al_Gwaiz>
 
{{cite book |title=Sturm&ndash;Liouville theory and its applications |page=66 |author=M. A. Al-Gwaiz |url=http://books.google.com/books?id=myy5WxCtBygC&pg=PA66 |isbn=1-84628-971-8 |year=2008 |chapter=Exercise 2.16 |publisher=Springer}}
 
</ref><ref name="Boyce 2001">{{cite book|last=Boyce|first=William E.|authorlink= |coauthors=Richard C. DiPrima|editor= |others= |title=Elementary Differential Equations and Boundary Value Problems|origdate= |origyear= |origmonth= |url= |format=PDF|accessdate= |edition=7th ed.|series= |volume= |date= |year=2001|month= |publisher=John Wiley & Sons|location=New York|isbn=0-471-31999-6|oclc=64431691|doi= |id= |chapter=Boundary Value Problems and Sturm–Liouville Theory |chapterurl= |quote= |page=630 }}
</ref><ref>{{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
</ref>
:<math>\int_0^1 \ dx \ ( uLv-vLu) = \left[p(x)\left(u \frac {dv}{dx}- v  \frac {du}{dx} \right)\right]_0^1, </math>
 
where <math>\ p=P(x)</math>, <math>\ q=Q(x)</math>, <math>\ u=U(x)</math> and <math>\ v=V(x)</math> are functions of <math>\ x</math>. <math>\ u</math> and <math>\ v</math> having continuous second derivatives on the {{nowrap|interval <math>\ [0,1] </math>.}}
 
=== Proof of form for ordinary differential equations ===
We have:
:<math>uLv = u \left[\frac {d}{dx} \left( p(x) \frac {dv}{dx} \right) + q(x) v \right], </math>
 
and
 
:<math>vLu = v \left[\frac {d}{dx} \left( p(x) \frac {du}{dx} \right) + q(x) u \right].  </math>
 
Subtracting:
 
:<math>uLv-vLu = u \frac {d}{dx} \left( p(x) \frac {dv}{dx} \right)-v \frac {d}{dx} \left( p(x) \frac {du}{dx} \right). </math>
 
The leading multiplied ''u'' and ''v'' can be moved ''inside'' the differentiation, because the extra differentiated terms in ''u'' and ''v'' are the same in the two subtracted terms and simply cancel each other. Thus,
 
:<math>uLv-vLu = \frac {d}{dx} \left( p(x)u \frac {dv}{dx} \right)-\frac {d}{dx}  \left( v p(x) \frac {du}{dx} \right), </math>
::<math>=\frac {d}{dx}\left[p(x)\left(u \frac {dv}{dx}- v  \frac {du}{dx} \right)\right], </math>
 
which is Lagrange's identity. Integrating from zero to one:
 
:<math>\int_0^1 \ dx \ ( uLv-vLu) = \left[p(x)\left(u \frac {dv}{dx}- v  \frac {du}{dx} \right)\right]_0^1, </math>
as was to be shown.
 
==References==
{{reflist}}
 
[[Category:Ordinary differential equations]]
[[Category:Mathematical identities]]

Latest revision as of 08:53, 7 January 2015

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