In the study of ordinary differential equations and their associated boundary value problems, 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[edit]
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]
The operator L and its adjoint operator L* are given by:
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:
where S is the surface bounding the volume Ω and n is the unit outward normal to the surface S.
Ordinary differential equations[edit]
Any second order ordinary differential equation of the form:
This general form motivates introduction of the Sturm–Liouville operator L, defined as an operation upon a function f such that:
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]
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]
where , , and are functions of . and having continuous second derivatives on the interval .
Proof of form for ordinary differential equations[edit]
We have:
Subtracting:
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,
References[edit]
- ^ Paul DuChateau, David W. Zachmann (1986). "§8.3 Elliptic boundary value problems". Schaum's outline of theory and problems of partial differential equations. McGraw-Hill Professional. p. 103. ISBN 0-07-017897-6.
- ^ a b Derek Richards (2002). "§10.4 Sturm–Liouville systems". Advanced mathematical methods with Maple. Cambridge University Press. p. 354. ISBN 0-521-77981-2.
- ^ Norman W. Loney (2007). "Equation 6.73". Applied mathematical methods for chemical engineers (2nd ed.). CRC Press. p. 218. ISBN 978-0-8493-9778-3.
- ^ M. A. Al-Gwaiz (2008). "Exercise 2.16". Sturm–Liouville theory and its applications. Springer. p. 66. ISBN 978-1-84628-971-2.
- ^ William E. Boyce and Richard C. DiPrima (2001). "Boundary Value Problems and Sturm–Liouville Theory". Elementary Differential Equations and Boundary Value Problems (7th ed.). New York: John Wiley & Sons. p. 630. ISBN 0-471-31999-6. OCLC 64431691.
- ^ Gerald Teschl (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.