Page 166 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 9: Adjoint and Self-Adjoint Equations

This definition encompasses a wide class of second order differential operators. Notes
For example, if

d 2 d
x
x
x
L 1 a ( ) a ( ) a ( ) ...(2)
2 2 1 0
dx dx
is non-singular on [a, b], we can write it in self-adjoint form by defining
t
x
t
x a ( ) a ( ) x a ( )
p ( ) exp 1 dt , ( ) 0 exp 1 dt ...(3)
x
q
x
t
a 2 ( ) a 2 ( ) a 2 ( )
t
x
Note that p(x) 0 for x [a, b]. By studying inhomogeneous boundary value problems of the
form Ly = f, or
d dy
x
x
p ( ) q ( )y f ( ) ...(4)
x
dx dx
we are therefore considering all second order, non-singular, linear differential operators. For
example, consider Hermite’s equations.
2
d y dy
2x y 0, ...(5)
dx 2 dx
for < x < . This is not in self-adjoint form, but, if we follow the above procedure, the self-
adjoint form of the equation is
d x 2 dy x 2
e e y 0
dx dx
2
x 2
This can be simplified, and kept in self-adjoint form, by writing u e y to obtain
2
d u 2
(x 1)u u ...(6)
dx 2
9.2 Boundary Conditions

To complete the definition of a boundary value problem associated with (4), we need to know
the boundary conditions. In general these will be of the form
, ,
y(a) + y(b) + y (a) + y (b) = 0,
1 2 3 4
, ,
y(a) + y(b) + y (a) + y (b) = 0. ...(7)
1 2 3 4
,
Since each of these is dependent on the values of y and y at each end of [a, b], we refer to these
as mixed or coupled boundary conditions. It is unnecessarily complicated to work with the
boundary conditions in this form, and we can start to simplify matters by deriving Lagrange’s
identity.
Lagrange’s Identity: If L is the linear differential operator given by (1) on [a, b] and if y , y C 2
1 2
[a, b], then
, , ,
y (Ly ) y (Ly ) = [p(y y y y )] . ...(8)
1 2 2 1 1 2 1 2
Proof: From the definition of L,

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