This general form motivates introduction of the Sturm–Liouville operatorL, 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 equationsedit
We have:
and
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,
which is Lagrange's identity. Integrating from zero to one:
as was to be shown.
Referencesedit
^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.
^ abDerek 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.