Linear Partial Differential Equations
1.12 Sturm–Liouville Systems and Some General Results
If the source is located at(ξ, η, ζ, τ) = (ξξξ, τ), the desired Green function is given by
G(x, t;ξξξ, τ) = 1 4π|x−ξξξ| δ
t−τ−|x−ξξξ| c
−δ
t−τ+|x−ξξξ| c
. (1.11.97) It should be noted that Green’s function (1.11.96) for the hyperbolic equation is a generalized function, whereas in the other examples of Green’s functions, it was always a piecewise analytic function. In general, Green’s function for an elliptic function is always analytic, whereas Green’s function for a hyperbolic equation is a generalized function.
The eigenvalue problem defined by (1.12.4) and (1.12.6ab) is called the Sturm–
Liouville (SL) system. The values ofλfor which the Sturm–Liouville problem has a nontrivial solution are called the eigenvalues, and the corresponding solutions are called the eigenfunctions.
In terms of the operator L= d
dx p(x) d dx
+q(x), (1.12.7)
we can write (1.12.4) in the form,X(x) =u(x),
Lu+λρu= 0. (1.12.8)
The Sturm–Liouville equation (1.12.8) is called regular in a closed finite interval [a, b]if the functionsp(x)andρ(x)are positive in[a, b]. Thus, for a givenλ, there ex- ist two linearly independent solutions of a regular Sturm–Liouville equation (1.12.8) in[a, b].
The Sturm–Liouville equation (1.12.8) in[a, b]together with two separated end conditions
a1u(a) +a2u(a) = 0, b1u(b) +b2u(b) = 0, (1.12.9ab) wherea1,a2,b1,b2are given real constants such thata21+a22>0andb21+b22>0 is called a regular Sturm–Liouville (RSL) system.
The set of all eigenvaluesλof a regular Sturm–Liouville problem is called the spectrum of the problem.
Example 1.12.1. Consider the regular Sturm–Liouville problem
u+λu= 0, 0≤x≤π, (1.12.10)
u(0) = 0 =u(π). (1.12.11ab)
It is easy to check that, forλ≤0, this problem has no nonzero solutions. In other words, there are no negative(λ <0)or zero(λ= 0)eigenvalues of the problem.
However, whenλ >0, then the solutions of the equation are u(x) =Acos√
λx+Bsin√ λx.
The boundary conditions (1.12.11ab) give A= 0 and Bsin
π√ λ
= 0.
Sinceλ= 0, andB = 0yields a trivial solution, we must haveB = 0, and hence, sin
π√ λ
= 0.
Thus, the eigenvalues areλn=n2,n= 1,2, . . ., and the eigenfunctions are un(x) = sinnx.
Note thatλn → ∞ asn → ∞, unlike the case of self-adjoint, compact op- erators when the eigenvalues converge to zero (see Debnath and Mikusinski1999, Theorem 4.9.9 and Theorem 5.10.4).
Example 1.12.2. Consider the Cauchy–Euler equation
x2u+xu+λu= 0, 1≤x≤e (1.12.12) with the end conditions
u(1) = 0 =u(e). (1.12.13ab)
The Cauchy–Euler equation can be put into the Sturm–Liouville form d
dx
xdu dx
+1
xλu= 0.
The general solution of this equation is u(x) =C1xi
√λ+C2x−i
√λ, whereC1andC2are arbitrary constants.
In view of the fact that
xia= exp(ialnx) = cos(alnx) +isin(alnx), the solution becomes
u(x) =Acos√ λlnx
+Bsin√ λlnx
,
whereAandBare new arbitrary constants related toC1andC2. The end condition u(1) = 0givesA= 0, and the end conditionu(e) = 0gives
sin√
λ= 0, B= 0, which leads to the eigenvalues
λn= (nπ)2, n= 1,2,3, . . . , and the corresponding eigenfunctions
un(x) = sin(nπlnx), n= 1,2,3, . . . .
A Sturm–Liouville equation (1.12.8) is called singular when it is given on a semi- infinite or infinite interval, or when the coefficientp(x)or ρ(x)vanishes, or when one of the coefficients becomes infinite at one end or both ends of a finite interval.
A singular Sturm–Liouville equation together with appropriate linear homogeneous end conditions is called a singular Sturm–Liouville system. The conditions imposed in this case are not like the separated boundary conditions in the regular Sturm–
Liouville problem.
Example 1.12.3. We consider the singular Sturm–Liouville problem involving Leg- endre’s equation
d dx
1−x2 u
+λu= 0, −1< x <1, (1.12.14) with the boundary conditions thatuanduare finite asx→ ±1.
In this case,p(x) = 1−x2andρ(x) = 1, andp(x)vanishes atx =±1. The Legendre functions of the first kindPn(x),n = 0,1,2, . . ., are the eigenfunctions which are finite asx→ ±1. The corresponding eigenvalues areλn =n(n+ 1)for n = 0,1,2, . . .. We note that the singular Sturm–Liouville problem has infinitely many eigenvalues, and the eigenfunctionsPn(x)are orthogonal to each other with respect to the weight functionρ(x) = 1.
Example 1.12.4. Another example of a singular Sturm–Liouville problem is the Bessel equation for fixedν
d dx
xdu
dx
+
λx−ν2 x
u= 0, 0< x < a, (1.12.15) with the end conditions thatu(a) = 0andu,uare finite asx→0+.
In this case,p(x) = x,q(x) = −νx2, and ρ(x) = x. Herep(0) = 0,q(x)is infinite asx→0+, andρ(0) = 0. Therefore, the problem is singular. Ifλ=k2, the eigenfunctions are the Bessel functionsJν(knx)of the first kind of orderν where n= 1,2,3, . . ., and(kna)is the nth zero ofJν. The eigenvalues areλn=k2n. The Bessel functionJν and its derivative are both finite asx→0+. Thus, the problem has infinitely many eigenvalues and the eigenfunctions are orthogonal to each other with respect to the weight functionρ(x) =x.
In the preceding examples, we see that the eigenfunctions are orthogonal with respect to the weight functionρ(x). In general, the eigenfunctions of a singular SL system are orthogonal with respect to the weight functionρ(x), which will be proved later on.
Another type of problem that often arises in practice is the periodic Sturm–
Liouville system:
d dx
p(x)du
dx
+ (q+λρ)u= 0, a≤x≤b, (1.12.16) in whichp(a) =p(b), together with the periodic end conditions
u(a) =u(b), u(a) =u(b). (1.12.17ab) Example 1.12.5. Find the eigenvalues and eigenfunctions of the periodic Sturm–
Liouville system:
u+λu= 0, −π≤x≤π, (1.12.18)
u(−π) =u(π), u(π) =u(−π). (1.12.19ab)
Note that herep(x) = 1and hencep(π) = p(−π). Forλ >0, the general solution of the equation is
u(x) =Acos√
λx+Bsin√ λx,
whereAandB are arbitrary constants. Using the boundary conditions (1.12.19ab), we obtain
2Bsin√ λπ= 0, 2A√
λsin√ λπ= 0.
Thus, for nontrivial solutions, we must have sin√
λπ= 0, A= 0, B = 0.
Consequently,
λn=n2, n= 1,2,3, . . . .
So, for every eigenvalue λn = n2, there are two linearly independent solutions cosnxandsinnx.
It can easily be checked that there are no negative eigenvalues of the system.
However,λ = 0is an eigenvalue and the associated eigenfunction is the constant functionu(x) = 1. Thus, the eigenvalues are0,{n2}, and the corresponding eigen- functions are1,{cosnx},{sinnx}, wherenis a positive integer.
For the regular Sturm–Liouville problem, we denote the domain ofLbyD(L), that is, D(L) is the space of all complex-valued functionsudefined on [a, b]for whichu∈L2([a, b])and which satisfy boundary conditions (1.12.9ab).
Theorem 1.12.1 (Lagrange’s Identity). For anyu, v∈D(L), we have uLv−vLu= d
dx p
uv−vu
. (1.12.20)
Proof. We have
uLv−vLu=u d dx
pdv
dx
+quv−v d dx
pdu
dx
−quv
= d dx
p
uv−vu .
Theorem 1.12.2 (Abel’s Formula). If uandv are two solutions of the equation (1.12.8) in[a, b], then
p(x)W(u, v;x) = const., (1.12.21) whereW is the Wronskian defined by
W(u, v;x) =
uv−uv .
Proof. Sinceu,vare solutions of (1.12.8), we have d
dx pu
+ (q+λρ)u= 0, d
dx pv
+ (q+λρ)v= 0.
Multiplying the first equation byvand the second equation byu, and then sub- tracting gives
u d dx
pv
−v d dx
pu
= 0.
Integrating this equation fromatoxyields p(x)
u(x)v(x)−u(x)v(x)
−p(a)
u(a)v(a)−u(a)v(a)
= 0.
This is Abel’s formula.
Theorem 1.12.3. The Sturm–Liouville operatorLis self-adjoint. In other words, for anyu,v∈D(L), we have
Lu, v=u, Lv, (1.12.22)
where·,·denotes the inner product inL2([a, b])defined by f, g=
b a
f(x)g(x)dx. (1.12.23) Proof. Since all constants involved in the boundary conditions of a Sturm–Liouville system are real, ifv∈D(L), thenv∈D(L).
Also sincep,qandρare real valued,Lv=Lv. Consequently, we have Lu, v − u, Lv=
b a
(vLu−uLv)dx
= p
vu−uvb
a, by Lagrange’s identity(1.12.20).
(1.12.24) We shall show that the right-hand side of the above equality vanishes for both the regular and singular SL systems. Ifp(a) = 0, the result follows immediately. If p(a) > 0, thenuandv satisfy the boundary conditions of the form (1.12.9ab) at x=a. That is,
u(a) u(a) v(a) v(a)
a1 a2
= 0.
Sincea1anda2are not both zero, we have
v(a)u(a)−u(a)v(a) = 0.
A similar argument can be used to the other end pointx=b, so that the right-hand side of (1.12.24) vanishes. This proves the theorem.
Theorem 1.12.4. All eigenvalues of a Sturm–Liouville system are real.
Proof. Let λbe an eigenvalue of an SL system and letu(x)be the corresponding eigenfunction. This means thatu= 0andLu=−λρu. Then
0 =Lu, u − u, Lu= (λ−λ) b
a
ρ(x)u(x)2dx.
Sinceρ(x)>0in[a, b]andu= 0, the integral is a positive number. Thusλ=λ.
This completes the proof.
Remark. This theorem states that all eigenvalues of a regular SL system are real, but it does not guarantee that an eigenvalue exists. It has been shown by example that an SL system has an infinite sequence of eigenvalues. All preceding examples of the SL system suggest thatλ1< λ2< λ3<· · · with
nlim→∞λn=∞.
All of these results can be stated in the form of a theorem as follows.
Theorem 1.12.5. The eigenvaluesλnof an SL system can be arranged in the form λ1< λ2< λ3<· · ·,
and
nlim→∞λn=∞, (1.12.25)
so thatnrefers to the number of zeros of the eigenfunctionsun(x)in[a, b].
The proof of this theorem is beyond the scope of this book, and we refer to Deb- nath and Mikusinski (2005).
Theorem 1.12.6. The eigenfunctions corresponding to distinct eigenvalues of a Sturm–Liouville system are orthogonal with respect to the inner product with the weight functionρ(x).
Proof. Suppose u1(x)andu2(x)are eigenfunctions corresponding to eigenvalues λ1andλ2withλ1=λ2.
Thus,
Lu1=−λ1ρu1 and Lu2=−λ2ρu2. Hence
u1Lu2−u2Lu1= (λ1−λ2)ρu1u2. (1.12.26) By Theorem1.12.1, we have
u1Lu2−u2Lu1= d dx
p
u1u2−u2u1
. (1.12.27)
Combining (1.12.26) and (1.12.27) and integrating fromatobgives (λ1−λ2)
b a
ρ(x)u1(x)u2(x)dx
= p(x)
u1(x)u2(x)−u2(x)u1(x)b a= 0, by boundary conditions (1.12.9ab).
Sinceλ1=λ2, this equality shows that b
a
ρ(x)u1(x)u2(x)dx= 0.
This proves the theorem.
We consider some general results about the eigenfunction expansions and their completeness property of an SL system.
Suppose{un(x)}∞n=1is a set of orthogonal eigenfunctions of an SL system in [a, b]. The inner product of these functions with respect to the weight functionρ(x) is defined by
un, um= b
a
ρ(x)un(x)um(x)dx, (1.12.28) so that the square of the norm is
un2=un, un= b
a
ρ(x)u2n(x)dx. (1.12.29) The set of orthogonal eigenfunctions of an SL system is said to be complete if any arbitrary functionf ∈L2([a, b])can be expanded uniquely as
f(x) = ∞ n=1
anun(x), (1.12.30)
where the series converges tof(x)inL2([a, b])and the coefficientsanare given by an= f(x), un(x)
un(x), un(x) = 1
un2f, un, (1.12.31) wheren= 1,2,3, . . ..
The expansion (1.12.30) is called a generalized Fourier series off(x)and the associated scalarsanare called the generalized Fourier coefficients off(x). The set of eigenfunctions{un}∞n=1is called orthonormal ifun= 1. Obviously, the set of orthonormal eigenfunctions is said to be complete, if, for everyf ∈ L2([a, b]), the following expansion holds:
f(x) = ∞ n=1
anun= ∞ n=1
f, unun. (1.12.32)
A series of orthonormal eigenfunctions∞
n=1anun(x)is said to be convergent to f(x)inL2([a, b])if
lim
n→∞ f(x)−sn(x) = 0, (1.12.33) wheresn(x) = n
r=1arur(x)is thenth partial sum of series (1.12.32). Equiva- lently, (1.12.33) reads as
nlim→∞
b a
f(x)−
n r=1
arur(x)
2
ρ(x)dx= 0. (1.12.34) This type of convergence is called the strong convergence and is entirely different from pointwise or uniform convergence in analysis. In general, the strong conver- gence inL2([a, b])implies neither pointwise convergence nor uniform convergence.
However, the uniform convergence implies both strong convergence and pointwise convergence.
We now determine the coefficients ar such that thenth partial sum sn(x)of the series (1.12.32) represents the best approximation tof(x)in the sense of least squares, that is, we seek to minimize the integral in (1.12.34)
I(ar) = b
a
f(x)−
n r=1
arur(x) 2
ρ(x)dx
= b
a
ρ(x)f2(x)dx−2 n r=1
ar
b a
ρ(x)f(x)ur(x)dx
+ n r=1
a2r b
a
ρ(x)u2r(x)dx. (1.12.35)
This is an extremal problem. A necessary condition forI(ar)to be minimum is that the first partial derivatives ofIwith respect to the coefficientsarvanish.
Thus, we obtain
∂I
∂ar
= n r=1
−2 b
a
ρurf(x)dx+ 2ar
b a
ρu2rdx
= 0. (1.12.36) Consequently,
ar= b
a
f(x)ur(x)ρ(x)dx=f, ur. (1.12.37) If we complete the square, the right-hand side of (1.12.35) becomes
I(ar) = b
a
ρf2dx+ n r=1
ar− b
a
ρf urdx 2
− n r=1
b a
ρf urdx 2
. (1.12.38) The right-hand side shows thatIis minimum if and only ifaris given by (1.12.37).
This choice ofargives the best approximation tof(x)in the sense of least squares.
Substituting the values ofarinto (1.12.35) gives b
a
f(x)−
n r=1
arur(x) 2
ρ(x)dx= b
a
ρ(x)f2(x)dx− n r=1
a2r. (1.12.39) If the series of orthonormal eigenfunctions converges tof(x), then (1.12.34) is satisfied. Invoking the limit asn → ∞in (1.12.39) and using (1.12.34) gives the Parseval relation
∞ r=1
a2r= b
a
ρ(x)f2(x)dx=f2, (1.12.40) or equivalently,
∞ r=1
f, ur2=f2. (1.12.41)
Since the left-hand side of (1.12.39) is nonnegative, it follows from (1.12.39) that n
r=1
a2r≤ f2. (1.12.42)
Since the right-hand side of (1.12.42) is finite, the left-hand side of (1.12.42) is bounded above for anyn. Proceeding to the limit asn→ ∞gives the inequality
∞ n=1
a2n≤ f2, (1.12.43)
or equivalently,
∞ n=1
f, un2≤ f2. (1.12.44)
This is called Bessel’s inequality.
In Section1.6, the method of separation of variables or the Fourier method or the method of eigenfunction expansions has been discussed with many examples.
This is the basic method for solving partial differential equations in bounded special domains. We now illustrate the generalized Fourier method by solving more general Sturm–Liouville problems associated with a general wave equation and a general diffusion equation.
Example 1.12.6 (Solution of the Sturm–Liouville Problem Associated with the Wave Equation). We develop the generalized Fourier method by solving a more general Sturm–Liouville equation associated with the wave equation
∂2u
∂t2 = 1 ρ(x)
∂
∂x
p∂u
∂x
+qu
+F(x, t), a≤x≤b, t >0, (1.12.45) with the boundary conditions (1.12.2ab) and the initial conditions (1.12.3ab), where F(x, t)is the forcing (source) term.
In terms of the SL operator, equation (1.12.45) can be written as
utt=Lu+F(x, t), a≤x≤b, t >0, (1.12.46) where
Lu= 1 ρ(x)
∂
∂x(pux) +qu
. (1.12.47)
Following the method of separation of variables, we assume the solution of the wave equation (1.12.45) withF = 0in the formu(x, t) =φ(x)ψ(t)= 0so that equation (1.12.45) reduces to
d2ψ
dt2 =λψ, t >0, (1.12.48)
Lφ=λφ, a≤x≤b, (1.12.49)
whereλis a separation constant.
The associated boundary conditions forφ(x)are
a1φ(a) +a2φ(a) = 0, b1φ(b) +b2φ(b) = 0. (1.12.50ab) Equation (1.12.49) with (1.12.50ab) is called the associated SL problem. In general, this problem can be solved by finding the eigenvaluesλnand the orthonormal eigen- functionsφn(x),n= 1,2,3, . . .. Using the principle of superposition, we can write the solution of the linear equation (1.12.46) in the form
u(x, t) = ∞ n=1
φn(x)ψn(t), (1.12.51)
whereψn(t)are to be determined.
We further assume that the forcing term can also be expanded in terms of the eigenfunctions as
F(x, t) = ∞ n=1
φn(x)fn(t), (1.12.52)
where the generalized Fourier coefficientsfn(t)ofF(x, t)are given by
fn(t) =F, φn= b
a
F(x, t)φn(x)dx. (1.12.53) Substituting (1.12.51) and (1.12.52) into (1.12.46) gives
∞ n=1
ψ¨n(t)φn(x) =L ∞
n=1
φn(x)ψn(t)
+ ∞ n=1
φn(x)fn(t)
= ∞ n=1
ψn(t)Lφn(x) + ∞ n=1
φn(x)fn(t)
= ∞ n=1
λnψn(t) +fn(t) φn(x).
This leads to the ordinary differential equation
ψ¨n(t) +α2nψn(t) =fn(t), (1.12.54) whereλn =−α2n.
Application of the Laplace transform method leads to the solution of (1.12.54) as
ψn(t) =ψn(0) cos(αnt) + 1 αn
ψ(0) sin(α˙ nt) + 1
αn
t 0
sinαn(t−τ)fn(τ)dτ, (1.12.55) whereψn(0)andψ˙n(0)can be determined from the initial data (1.12.3ab) so that
f(x) =u(x,0) = ∞ n=1
φn(x)ψn(0), (1.12.56) g(x) =ut(x,0) =
∞ n=1
ψ˙n(0)φn(x), (1.12.57)
which give the generalized Fourier coefficientsψn(0)andψ˙n(0)as follows:
ψn(0) =f, φn= b
a
f(ξ)φn(ξ)dξ, (1.12.58) ψ˙n(0) =g, φn=
b a
g(ξ)φn(ξ)dξ. (1.12.59) Therefore, the final solution is given by
u(x, t) = ∞ n=1
ψn(t)φn(x)
= ∞ n=1
f, φncosαnt+ 1
αng, φnsinαnt + 1
αn
t 0
sinαn(t−τ)fn(τ)dτ
φn(x). (1.12.60)
This represents an infinite series solution of the wave equation (1.12.45) with the boundary and initial data (1.12.2ab) and (1.12.3ab) under appropriate conditions on the initial dataf(x),g(x)and the forcing termF(x, t).
Replacing the inner products by the integrals (1.12.58) and (1.12.59), andfn(τ) by (1.12.53) and interchanging the summation and integration, we obtain the solution in the form
u(x, t) = b
a
∞
n=1
φn(x)φn(ξ) cosαnt
f(ξ)dξ +
b a
∞
n=1
1
αnφn(x)φn(ξ) sinαnt
g(ξ)dξ +
b a
t 0
∞
n=1
1 αn
φn(x)φn(ξ) sinαn(t−τ)
F(ξ, τ)dξ dτ. (1.12.61) Example 1.12.7 (Solution of the Sturm–Liouville Problem Associated with the Dif- fusion Equation). We consider the diffusion equation with a forcing (source) term F(x, t)in the form
∂u
∂t = ∂
∂x p(x)∂u
∂x
+q(x)u+F(x, t), a≤x≤b, t >0, (1.12.62) with boundary conditions
a1u(a, t) +a2ux(a, t) = 0, b1u(b, t) +b2ux(b, t) = 0, t >0, (1.12.63ab) and the initial condition
u(x,0) =f(x), a < x < b. (1.12.64) In terms of the SL operator L, equation (1.12.62) takes the form
ut=Lu+F. (1.12.65)
We use the method of separation of variables to seek a solution of the equation (1.12.62) with F = 0in the form u(x, t) = φ(x)ψ(t) = 0 so that the equation (1.12.62) becomes
dψ
dt =λψ, t >0, (1.12.66)
Lφ=λφ, a≤x≤b, (1.12.67)
whereλis the separation constant.
The associated boundary conditions are
a1φ(a) +a2φ(a) = 0, b1φ(b) +b2φ(b) = 0. (1.12.68ab) Equation (1.12.67) with (1.12.68ab) is called the associated SL problem, which can easily be solved by finding the eigenvaluesλn and the orthonormal eigenfunctions
φn(x),n= 1,2,3, . . .. According to the linear superposition principle, we write the solution of (1.12.67) in the form
u(x, t) = ∞ n=1
φn(x)ψn(t), (1.12.69)
whereψn(t)are to be determined.
We further assume that the forcing function can also be expanded in terms of the eigenfunctions as
F(x, t) = ∞ n=1
fn(t)φn(x), (1.12.70)
where the Fourier coefficientsfn(t)are given by fn(t) =F, φ=
b a
F(ξ, t)φn(ξ)dξ. (1.12.71) Putting (1.12.69) and (1.12.70) in equation (1.12.65) yields
∞ n=1
φn(x) ˙ψn(t) =L ∞
n=1
φn(x)ψn(t)
+ ∞ n=1
fn(t)φn(x)
= ∞ n=1
ψn(t)Lφn(x) +fn(t)φn(x)
= ∞ n=1
λnψn(t) +fn(t) φn(x).
This gives an ordinary differential equation forψn(t)as
ψ˙n(t) =λnψn(t) +fn(t). (1.12.72) Applying the Laplace transform to this equation gives the solution
ψn(t) =ψn(0) exp(λnt) + t
0
exp
λn(t−τ)fn(τ)dτ
, (1.12.73) wheren= 1,2,3, . . .andψn(0)can be determined from the initial condition
f(x) =u(x,0) = ∞ n=1
φn(x)ψn(0), (1.12.74) where the Fourier coefficientsψn(0)of the functionf(x)are given by
ψn(0) =f, φn= b
a
f(ξ)φn(ξ)dξ. (1.12.75) Substituting (1.12.73) in the solution (1.12.69) gives
u(x, t) = ∞ n=1
f, φnexp(λnt) + t
0
exp
λn(t−τ)fn(τ)dτ φn(x).
(1.12.76) We next replace the inner product in (1.12.76) by (1.12.75),fn(τ)by (1.12.71) and interchange the summation and integration to obtain the final form of the solution in the form
u(x, t) = b
a
∞
n=1
φn(ξ)φn(x) exp(λnt)
f(ξ)dξ +
t 0
b a
∞ n=1
φn(ξ)φn(x) exp
λn(t−τ)
F(ξ, τ)dξ dτ.
(1.12.77) Introducing a new functionGdefined by
G(x, ξ, t) = ∞ n=1
φn(ξ)φn(x)eλnt, (1.12.78)
we can write the solution in terms ofGin the form u(x, t) =
b a
G(x, ξ, t)f(ξ)dξ +
t 0
b a
G(x, ξ, t−τ)F(ξ, τ)dξ dτ. (1.12.79) It is noted that the first term of this solution represents the contribution from the initial condition, and the second term is due to the nonhomogeneous term of the equation (1.12.62).
A typical boundary-value problem for an ordinary differential equation can be written in the operator form as
Lu=f, a≤x≤b. (1.12.80)
Usually, we seek a solution of this equation with the given boundary conditions.
One formal approach to the problem is to find the inverse operatorL−1. Then the solution of (1.12.80) can be found asu=L−1(f). It turns out that it is possible in many important cases, and the inverse operator is an integral operator of the form
u(x) = L−1f
(x) = b
a
G(x, t)f(t)dt. (1.12.81) The functionGis called Green’s function of the operatorL. The existence of Green’s function and its construction is not a simple problem in the case of the regular Sturm–
Liouville system.
Theorem 1.12.7 (Green’s Function for an SL System). Supposeλ= 0is not an eigenvalue of the following regular SL system:
Lu≡ d dx
p(x)du
dx
+q(x)u=f(x), a≤x≤b, (1.12.82) with the homogeneous boundary conditions
a1u(a) +a2u(a) = 0, b1u(b) +b2u(b) = 0, (1.12.83ab) wherepandqare continuous real-valued functions on[a, b],pis positive in[a, b], p(x)exists and is continuous in[a, b], anda1,a2,b1,b2, are given real constants such thata21+a21>0andb21+b21>0. Thus, for anyf ∈C2([a, b]), the SL system has a unique solution
u(x) = b
a
G(x, t)f(t)dt, (1.12.84) whereGis the Green function given by
G(x, t) =
⎧⎨
⎩
u2(x)u1(t)
p(t)W(t) ifa≤t < x,
u1(x)u2(t)
p(t)W(t) ifx < t≤b,
(1.12.85) whereu1andu2are nonzero solutions of the homogeneous system(f = 0)andW is the Wronskian given by
W(t) =
u1(t) u2(t) u1(t) u2(t)
.
Proof. According to the theory of ordinary differential equations, the general solu- tion of (1.12.82) is of the form
u(x) =c1u1(x) +c2u2(x) +up(x), (1.12.86) wherec1 and c2 are arbitrary constants, u1 andu2 are two linearly independent solutions of the homogeneous equationLu= 0, andup(x)is any particular solution of (1.12.82).
Using the method of variation of parameters, we obtain the particular solution up(x) =u1(x)v1(x) +u2(x)v2(x), (1.12.87) wherev1(x)andv2(x)are given by
v1(x) =−
f(x)u2(x)
p(x)W(x)dx, v2(x) =
f(x)u1(x)
p(x)W(x)dx. (1.12.88) According to Abel’s formula (see Theorem1.12.2),p(x)W(x)is a constant. Since W(x) = 0in[a, b]and p(x) is assumed to be positive, the constant is nonzero.
Denoting the constant bycso that
c= 1 p(x)W(x), then
v1(x) =−
cf(x)u2(x)dx and v2(x) =
cf(x)u1(x)dx.
Thus, the final form ofup(x)is up(x) =−cu1(x)
x b
f(t)u2(t)dt+cu2(x) x
a
f(t)u1(t)dt
= x
a
cu2(x)u1(t)f(t)dt+ b
x
cu1(x)u2(t)f(t)dt. (1.12.89) Consequently, if we denote Green’s function as
G(x, t) =
cu2(x)u1(t) ifa≤t < x,
cu1(x)u2(t) ifx < t≤b, (1.12.90) we can write
up(x) = b
a
G(x, t)f(t)dt, (1.12.91)
provided the integral exists. This follows immediately from the continuity ofG. The continuity ofGis left as an exercise.
We denote the integral operatorTgiven by (1.12.81), that is, (T f)(x) =
b a
G(x, t)f(t)dt. (1.12.92) Theorem 1.12.8. The operatorTdefined by (1.12.92) is self-adjoint fromL2([a, b]) intoC([a, b])ifG(x, t) =G(t, x).
Proof. The functionG(x, t)defined on[a, b]×[a, b]is continuous if b
a
b a
G(x, t)2dx dt <∞. We have
T f, g= b
a
b a
G(x, t)f(t)g(x)dx dt
= b
a
b a
G(x, t)f(t)g(x)dx dt
= b
a
b a
f(t)dt b
a
G(x, t)g(x)dx
=$ f, T∗g%
,
which shows that
T∗f (x) =
b a
G(t, x)f(t)dt.
Obviously,T is self-adjoint if its kernel satisfies the equalityG(x, t) =G(t, x).
Theorem 1.12.9. Under the assumptions of Theorem1.12.7,λis an eigenvalue ofL if and only if(1/λ)is an eigenvalue ofT. Furthermore, iffis an eigenfunction ofL corresponding to the eigenvalueλ, thenf is an eigenfunction ofT corresponding to the eigenvalue(1/λ).
Proof. SupposeLf =λf for some nonzerof in the domain ofL. In view of the definition ofT, and Theorem1.12.7, we have
f =L−1(λf) =T(λf).
Or equivalently, sinceλ= 0,
T f = 1 λf.
This means(1/λ)is an eigenvalue ofTwith the corresponding eigenfunctionf. Conversely, if(f = 0)is an eigenfunction ofT corresponding to the eigenvalue λ= 0, then
T f =λf.
SinceT =L−1, one has
f =L(T f) =L(λf) =λL(f).
Thus,(1/λ)is an eigenvalue ofLand the corresponding eigenfunction isf. Theorem 1.12.10 (Bilinear Expansion of Green’s Function). IfG(x, t)is Green’s function for the regular SL system (1.12.82), (1.12.83ab) and the associated eigen- value problem
Lφ=λφ, a≤x≤b, (1.12.93)
with (1.12.83ab) has infinitely many nonzero eigenvaluesλnwith the corresponding orthonormal eigenfunctionsφn, thenG(x, t)can be expanded in terms ofλnandφn as
G(x, t) = ∞ n=1
1 λn
φn(x)φn(t). (1.12.94) Proof. We assume that the solutionu(x)of (1.12.82), (1.12.83ab) is given in terms of the eigenfunctions as
u(x) = ∞ n=1
anφn(x), (1.12.95)
where the coefficientsanare to be determined.
We next express the given forcing functionf in terms of the eigenfunctions as f(x) =
∞ n=0
fnφn(x), (1.12.96)
where the coefficientsfnare
fn =f, φn= b
a
f(t)φn(t)dt. (1.12.97) Substituting (1.12.95), (1.12.96) into (1.12.82) yields
L ∞
n=1
anφn(x)
= ∞ n=1
fnφn(x). (1.12.98)
But the left-hand side of (1.12.98) is L
∞
n=1
anφn(x)
= ∞ n=1
anL φn(x)
= ∞ n=1
anλnφn(x). (1.12.99) Equating the right-hand side of (1.12.98) and (1.12.99) yields
an= 1 λn
fn= 1
λnf, φn= 1 λn
b a
f(t)φn(t)dt. (1.12.100) Consequently, (1.12.95) leads to the result
u(x) = ∞ n=1
1 λn
b a
f(t)φn(t)dt
φn(x), which, by interchanging the summation and integration,
= b
a
∞
n=1
1 λn
φn(t)φn(x)
f(t)dt. (1.12.101)
In view of (1.12.84), Green’s functionG(x, t)is then given by G(x, t) =
∞ n=1
1 λn
φn(t)φn(x). (1.12.102) This is the desired result (1.12.94).