View source for Lagrange's identity (boundary value problem)

{{Short description|On boundary terms from integration by parts of a self-adjoint linear differential operator}}
{{other uses|Lagrange's identity (disambiguation)}}

In the study of [[ordinary differential equation]]s and their associated [[boundary value problem]]s in mathematics, '''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]].

==Statement==
In general terms, Lagrange's identity for any pair of functions ''u'' and ''v'' 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 |chapter-url=https://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 display="block">vL[u]-uL^*[v]=\nabla \cdot \boldsymbol M, </math>
where:
<math display="block">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 display="block">\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 display="block">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 display="block">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 display="block">\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 display="block">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 |chapter-url=https://books.google.com/books?id=AKsLy0rVGuwC&pg=PA354 |isbn=0-521-77981-2 |chapter=§10.4 Sturm–Liouville systems |year=2002 |publisher=Cambridge University Press }}</ref>
<math display="block">\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'' such that:
<math display="block">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'' for which the various derivatives exist, '''Lagrange's identity''' for ordinary differential equations holds:<ref name= Richards/>
<math display="block"> 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 |chapter-url=https://books.google.com/books?id=QD6hOiRrAjIC&pg=PA218 |page=218 |chapter=Equation 6.73 |edition=2nd |isbn=978-0-8493-9778-3 |date=2007 |publisher=CRC Press}}</ref><ref name=Al_Gwaiz>{{cite book |title=Sturm–Liouville theory and its applications |page=66 |author=M. A. Al-Gwaiz |chapter-url=https://books.google.com/books?id=myy5WxCtBygC&pg=PA66 |isbn=978-1-84628-971-2 |date=2008 |chapter=Exercise 2.16 |publisher=Springer}}</ref><ref name="Boyce 2001">{{cite book | author=William E. Boyce and Richard C. DiPrima|title=Elementary Differential Equations and Boundary Value Problems | edition=7th | date=2001 | publisher=John Wiley & Sons|location=New York|isbn=0-471-31999-6|oclc=64431691|chapter=Boundary Value Problems and Sturm–Liouville Theory | page=[https://archive.org/details/elementarydiffer00boyc_0/page/630 630]|chapter-url-access=registration | chapter-url=https://archive.org/details/elementarydiffer00boyc_0/page/630}}</ref><ref>{{cite book|author=Gerald Teschl|author-link=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]] | date = 2012 | isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
</ref>
<math display="block">\int_0^1 dx \ ( u L v - v L u) = \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 display="block">uLv = u \left[\frac {d}{dx} \left( p(x) \frac {dv}{dx} \right) + q(x) v \right], </math>
and
<math display="block">vLu = v \left[\frac {d}{dx} \left( p(x) \frac {du}{dx} \right) + q(x) u \right].  </math>

Subtracting:
<math display="block">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 display="block">\begin{align}
uLv-vLu &= \frac {d}{dx} \left( p(x)u \frac {dv}{dx} \right)-\frac {d}{dx} \left( v p(x) \frac {du}{dx} \right), \\
&=\frac {d}{dx}\left[p(x)\left(u \frac {dv}{dx}- v  \frac {du}{dx} \right)\right],
\end{align}</math>
which is Lagrange's identity. Integrating from zero to one:
<math display="block">\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|2}}

{{Joseph-Louis Lagrange}}

[[Category:Ordinary differential equations]]
[[Category:Mathematical identities]]