Lagrange's identity (boundary value problem) explained

In the study of ordinary differential equations and their associated boundary value problems 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 - 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 function space C² (that is, twice differentiable) in n dimensions is:^[1] $$vL[u]-uL^*[v]=\backslash nabla\; \backslash cdot\; \backslash boldsymbol\; M,$$where:$$M\_i\; =\; \backslash sum\_^n\; a\_\backslash left(v\; \backslash frac\; -u\; \backslash frac\; \backslash right)\; +\; uv\; \backslash left(b\_i\; -\; \backslash sum\_^\; \backslash frac\; \backslash right),$$and$$\backslash nabla\; \backslash cdot\; \backslash boldsymbol\; M\; =\; \backslash sum\_^n\; \backslash frac\; M\_i,$$

The operator L and its adjoint operator L^* are given by:$$L[u]\; =\; \backslash sum\_^n\; a\_\; \backslash frac\; +\; \backslash sum\_^n\; b\_i\; \backslash frac\; +c\; u$$and$$L^*[v]\; =\; \backslash sum\_^n\; \backslash frac\; -\; \backslash sum\_^n\; \backslash frac\; +\; cv.$$

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:$$\backslash int\_\backslash Omega\; v\; L[u]\; \backslash ,\; d\backslash Omega\; =\; \backslash int\_\; u\; L^*[v]\backslash \; d\backslash Omega\; +\backslash int\_S\; \backslash boldsymbol\; \backslash ,\; dS,$$

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:$$a(x)\backslash frac\; +\; b(x)\backslash frac\; +c(x)y\; +\backslash lambda\; w(x)\; y\; =0,$$can be put in the form:^[2] $$\backslash frac\; \backslash left(p(x)\; \backslash frac\; \backslash right)\; +\backslash left(q(x)+\; \backslash lambda\; w(x)\; \backslash right)\; y(x)\; =\; 0.$$

This general form motivates introduction of the Sturm–Liouville operator L, defined as an operation upon a function f such that:$$L\; f\; =\; \backslash frac\; \backslash left(p(x)\; \backslash frac\; \backslash right)\; +\; q(x)\; f.$$

It can be shown that for any u and v for which the various derivatives exist, Lagrange's identity for ordinary differential equations holds:^[2] $$uLv\; -\; vLu\; =\; -\; \backslash frac\; \backslash left[p(x)\; \backslash left(v\backslash frac\{du\}\{dx\}\; -u\; \backslash frac\{dv\}\{dx\}\; \backslash right)\; \backslash right].$$

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):^{[3] [4] [5] [6]} $$\backslash int\_0^1\; dx\; \backslash \; (u\; L\; v\; -\; v\; L\; u)\; =\; \backslash left[p(x)\backslash left(u\; \backslash frac\; \{dv\}\{dx\}-\; v\; \backslash frac\; \{du\}\{dx\}\; \backslash right)\backslash right]\_0^1,$$

where

p = P(x) , q = Q(x) , u = U(x) and v = V(x) are functions of x . u and v having continuous second derivatives on the

Proof of form for ordinary differential equations

We have:$$uLv\; =\; u\; \backslash left[\backslash frac\; \{d\}\{dx\}\; \backslash left(p(x)\; \backslash frac\; \{dv\}\{dx\}\; \backslash right)\; +\; q(x)\; v\; \backslash right],$$and$$vLu\; =\; v\; \backslash left[\backslash frac\; \{d\}\{dx\}\; \backslash left(p(x)\; \backslash frac\; \{du\}\{dx\}\; \backslash right)\; +\; q(x)\; u\; \backslash right].$$

Subtracting:$$uLv-vLu\; =\; u\; \backslash frac\; \backslash left(p(x)\; \backslash frac\; \backslash right)-v\; \backslash frac\; \backslash left(p(x)\; \backslash frac\; \backslash right).$$

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,which is Lagrange's identity. Integrating from zero to one:$$\backslash int\_0^1\; dx\; \backslash \; (uLv-vLu)\; =\; \backslash left[p(x)\backslash left(u\; \backslash frac\; \{dv\}\{dx\}-\; v\; \backslash frac\; \{du\}\{dx\}\; \backslash right)\backslash right]\_0^1,$$as was to be shown.

Notes and References

1. Book: Schaum's outline of theory and problems of partial differential equations . Paul DuChateau, David W. Zachmann . 103 . https://books.google.com/books?id=h9kRkvGQCgkC&pg=PA103 . §8.3 Elliptic boundary value problems . 0-07-017897-6 . McGraw-Hill Professional . 1986.
2. Book: Advanced mathematical methods with Maple . Derek Richards . 354 . https://books.google.com/books?id=AKsLy0rVGuwC&pg=PA354 . 0-521-77981-2 . §10.4 Sturm–Liouville systems . 2002 . Cambridge University Press .
3. Book: Applied mathematical methods for chemical engineers . Norman W. Loney . https://books.google.com/books?id=QD6hOiRrAjIC&pg=PA218 . 218 . Equation 6.73 . 2nd . 978-0-8493-9778-3 . 2007 . CRC Press.
4. Book: Sturm–Liouville theory and its applications . 66 . M. A. Al-Gwaiz . https://books.google.com/books?id=myy5WxCtBygC&pg=PA66 . 978-1-84628-971-2 . 2008 . Exercise 2.16 . Springer.
5. Book: William E. Boyce and Richard C. DiPrima. Elementary Differential Equations and Boundary Value Problems . 7th . 2001 . John Wiley & Sons. New York. 0-471-31999-6. 64431691. Boundary Value Problems and Sturm–Liouville Theory . 630. registration . https://archive.org/details/elementarydiffer00boyc_0/page/630.
6. Book: Gerald Teschl. Gerald Teschl. Ordinary Differential Equations and Dynamical Systems. American Mathematical Society. . 2012 . 978-0-8218-8328-0.