Lagrange identity (boundary value problems)

The Lagrange identity , named after Joseph Louis Lagrange (1736–1813), is used in solving ordinary second-order differential equations , particularly in Sturm-Liouville problems .

definition

The Lagrange identity for the functions , from the differentiability class and the coefficient functions , and is given by the storm-Liouville operator ${\displaystyle \phi }$ ${\displaystyle \psi }$ ${\displaystyle \phi ,\psi \in C^{2}((a,b),\mathbb {R} )}$ ${\displaystyle w,q\in C^{0}((a,b),\mathbb {R} )}$ ${\displaystyle p\in C^{1}((a,b),\mathbb {R} )}$ ${\displaystyle p,q>0}$

${\displaystyle {\mathcal {L}}={\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)}$

for which applies:

${\displaystyle {\begin{aligned}\phi {\mathcal {L}}\psi -\psi {\mathcal {L}}\phi &={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}p(\phi \psi '-\psi \phi '){\bigg )}\\&={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}pW(\phi ,\psi ){\bigg )}\end{aligned}}}$

where means the Wronsky determinant of the functions . ${\displaystyle W(\phi ,\psi )}$ ${\displaystyle \phi ,\psi }$

Derivation

Let be a Sturm-Liouville differential operator, then: ${\displaystyle {\mathcal {L}}}$

${\displaystyle \phi {\mathcal {L}}\psi =\phi {\frac {-1}{w}}{\bigg (}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(p{\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right)-q\psi {\bigg )},}$

and

${\displaystyle \psi {\mathcal {L}}\phi =\psi {\frac {-1}{w}}{\bigg (}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(p{\frac {\mathrm {d} \phi }{\mathrm {d} x}}\right)-q\phi {\bigg )}.}$

Subtracting the two equations gives:

${\displaystyle \phi {\mathcal {L}}\psi -\psi {\mathcal {L}}\phi =\phi {\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(p{\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right)-\psi {\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(p{\frac {\mathrm {d} \phi }{\mathrm {d} x}}\right).}$

Now, using the product rule for derivatives , the term is not taken into account, the following representations can be calculated and . In this way it can be seen that the second term is the same in both derivatives and disappears when the difference is formed, i.e.: ${\displaystyle {\tfrac {-1}{w}}}$ ${\displaystyle \textstyle \phi {\frac {\mathrm {d} }{\mathrm {d} x}}\left(p{\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right)={\frac {\mathrm {d} }{\mathrm {d} x}}\left(p\phi {\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right)-\left(p{\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right){\frac {\mathrm {d} \phi }{\mathrm {d} x}}}$ ${\displaystyle \textstyle \psi {\frac {\mathrm {d} }{\mathrm {d} x}}\left(p{\frac {\mathrm {d} \phi }{\mathrm {d} x}}\right)={\frac {\mathrm {d} }{\mathrm {d} x}}\left(p\psi {\frac {\mathrm {d} \phi }{\mathrm {d} x}}\right)-\left(p{\frac {\mathrm {d} \phi }{\mathrm {d} x}}\right){\frac {\mathrm {d} \psi }{\mathrm {d} x}}}$

${\displaystyle {\begin{aligned}\phi {\mathcal {L}}\psi -\psi {\mathcal {L}}\phi &={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(p\phi {\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right)-{\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(p\psi {\frac {\mathrm {d} \phi }{\mathrm {d} x}}\right)\\\\&={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}p\left(\phi {\frac {\mathrm {d} \psi }{\mathrm {d} x}}-\psi {\frac {\mathrm {d} \phi }{\mathrm {d} x}}\right){\bigg )}\\\\&={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}pW(\phi ,\psi ){\bigg )}.\\\end{aligned}}}$

literature

• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( free online version ).

• Harro Heuser : Ordinary differential equations , Vieweg + Teubner 2009 (6th edition), ISBN 978-3-8348-0705-2