Masujima M. Applied Mathematical Methods in Theoretical Physics

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8.1 Nonlinear Integral Equation of the Volterra Type 297

Fig. 8.2 The contour of integration C wrapping around the

branch cut of Figure 8.1 for λ>0.

Here we have changed the variable,

s = 4λt,0≤ t ≤ 1.

The integral in Eq. (8.1.6) can be explicitly evaluated:

φ(x) =

∞



n=0

(4λx)



dtt



1 − t

∞



n=0

(4λx)





n +









(n + 2)

, (8.1.7)

wherewehaveusedtheformula



dtt

n−1

(1 − t)

m−1

(n)(m)

(n ++m)

Now, the conﬂuent hypergeometric function is given by

F(a;c;z) = 1 +

a + 1

c + 1

+···=

∞



n=0

(c)

(a)

(a + n)

(c + n)

. (8.1.8)

From Eqs. (8.1.7) and (8.1.8), we ﬁnd that

φ(x) = F



;2;4λx



(8.1.9)

satisﬁes the nonlinear integral equation

φ(x) = 1 + λ



φ(x − y)φ(y)dy. (8.1.10)

Although the above is proved only for λ>0, we may verify that it is also true

for λ<0 by repeating the same argument. Alternatively we may prove this in the

298 8 Nonlinear Integral Equations

following way. Let us substitute Eq. (8.1.9) into Eq. (8.1.10). Since F(

;2;4λx)is

an entire function of λ, each side of the resulting equation is the entire function of

λ. Since this equation is satisﬁed for λ>0, it must be satisﬁed for all λ by analytic

continuation. Thus the integral equation (8.1.10) has the unique solution given by

Eq. (8.1.9), for all values of λ.

At the end of this section, we classify the nonlinear integral equations of Volterra

type in the following manner:

(1) Kernel part is nonlinear,

φ(x) −



H (x, y, φ(y))dy = f (x). (VN.1)

(2) Particular part is nonlinear,

G(φ(x)) −



K(x, y)φ(y)dy = f (x). (VN.2)

(3) Both parts are nonlinear,

G(φ(x)) −



H (x, y, φ(y))dy = f (x). (VN.3)

(4) Nonlinear Volterra integral equation of the ﬁrst kind,



H (x, y, φ(y))dy = f (x). (VN.4)

(5) Homogeneous nonlinear Volterra integral equation of the ﬁrst kind where

the kernel part is nonlinear,

φ(x) =



H (x, y, φ(y))dy. (VN.5)

(6) Homogeneous nonlinear Volterra integral equation of the ﬁrst kind where

the particular part is nonlinear,

G(φ(x)) =



K(x, y)φ(y)dy. (VN.6)

(7) Homogeneous nonlinear Volterra integral equation of the ﬁrst kind where

both parts are nonlinear,

G(φ(x)) =



H (x, y, φ(y))dy. (VN.7)

8.2 Nonlinear Integral Equation of the Fredholm Type 299

8.2

Nonlinear Integral Equation of the Fredholm Type

In this section, we shall give a brief discussion of the nonlinear integral equation of

Fredholm type. Recall that the linear algebraic equation

φ = f + λKφ (8.2.1)

has the solution

φ = (1 − λK)

−1

f . (8.2.2)

In particular, the solution of Eq. (8.2.1) exists and is unique as long as the

corresponding homogeneous equation

φ = λKφ (8.2.3)

has no nontrivial solutions.

Nonlinear equations behave quite differently. Consider, for example, the nonlin-

ear algebraic equation obtained from Eq. (8.2.1) by replacing K with φ,

φ = f + λφ

. (8.2.4a)

The solutions of Eq. (8.2.4a) are

φ =

1 ±



1 − 4λf

2λ

. (8.2.4b)

We ﬁrst observe that the solution is not unique. Indeed, if we require the solutions

to be real, then Eq. (8.2.4a) has two solutions if

1 − 4λf > 0, (8.2.5)

and no solution if

1 − 4λf < 0. (8.2.6)

Thus the number of solutions changes from 2 to 0 as the value of λ passes 14f .

The point λ = 14f is called a bifurcation point of Eq. (8.2.4a). Note that at the

bifurcation point, Eq. (8.2.4a) has only one solution.

We also observe from Eq. (8.2.4b) that another special point for Eq. (8.2.4a) is

λ = 0. At this point, one of the two solutions is inﬁnite. Since the number of

solutions remains to be two as the value of λ passes λ = 0, the point λ = 0isnota

bifurcation point. We shall call it a singular point.

300 8 Nonlinear Integral Equations

Consider now the equation

φ = λφ

, (8.2.7)

obtained from Eq. (8.2.4a) by setting f = 0. This equation always has the nontrivial

solution

φ = 1λ (8.2.8)

provided that

λ = 0. (8.2.9)

There is no connection between the existence of the solutions for Eq. (8.2.4a) and

the absence of the solutions for Eq. (8.2.4a) with f = 0, quite unlike the case of

linear algebraic equations.

Nonlinear integral equations share these properties. This is evident in the

following examples.

 Example 8.3. Solve

φ(x) = 1 + λ



(y)dy. (8.2.10)

Solution. The right-hand side of Eq. (8.2.10) is independent of x.Thusφ(x)is

constant. Let

φ(x) = a.

Then Eq. (8.2.10) becomes

a = 1 + λa

. (8.2.11)

Equation (8.2.11) is just Eq. (8.2.4a) with f = 1. Thus

φ(x) =

1 ±

√

1 − 4λ

2λ

. (8.2.12)

There are two real solutions for λ<14, and no real solutions for λ>14. Thus

λ = 14isabifurcationpoint.

 Example 8.4. Solve

φ(x) = 1 + λ



(y)dy. (8.2.13)

8.2 Nonlinear Integral Equation of the Fredholm Type 301

Solution. The right-hand side of Eq. (8.2.13) is independent of x.Thusφ(x)is

constant. Letting

φ(x) = a,

we get

a = 1 + λa

. (8.2.14a)

Equation (8.2.14a) is cubic and hence has three solutions. Not all of these solutions

are real. Let us rewrite Eq. (8.2.14a) as

a − 1

= a

, (8.2.14b)

and plot (a − 1)λ as well as a

in the ﬁgure. The points of intersection between

these two curves are the solutions of Eq. (8.2.14a). For λ negative, there is obviously

only one real root, while for λ positive and very small, there are three real roots

(two positive roots and one negative root). Thus λ = 0 is a bifurcation point . For

λ large and positive, there is again only one real root. The change of numbers of

roots can be shown to occur at λ = 427, which is another bifurcation point.

We may generalize the above considerations to

φ(x) = c + λ



K(φ(y))dy. (8.2.15a)

The right-hand side of Eq. (8.2.15a) is independent of x.Thusφ(x)isconstant.

Letting

φ(x) = a,

we get

(a − c)λ = K(a). (8.2.15b)

The roots of the above equation can be graphically obtained by plotting K(a)and

(a − c)λ.

Obviously, with a proper choice of K(a), the number of solutions as well as the

number of bifurcation points may take any value. For example, if

K(φ) = φ sin πφ, (8.2.16)

there are inﬁnitely many solutions as long as

< 1. As another example, for

K(φ) = sin φ(φ

+ 1), (8.2.17)

there are inﬁnitely many bifurcation points .

In summary, we have found the following conclusions for nonlinear integral

equations:

302 8 Nonlinear Integral Equations

(1) There may be more than one solution.

(2) There may be one or more bifurcation points.

(3) There is no signiﬁcant relationship between the integral equation with f = 0 and

the one obtained from it by setting f = 0.

The above considerations for simple examples may be extended to more general

cases. Consider the integral equation

φ(x) = f (x) +



K(x, y, φ(x), φ(y))dy. (8.2.18)

If K is separable, i.e.,

K(x, y, φ(x), φ(y)) = g(x, φ(x))h(y, φ(y)), (8.2.19)

then the integral equation (8.2.18) is solved by

φ(x) = f (x) + ag(x, φ(x)), (8.2.20)

with

a =



h(x , φ(x))dx. (8.2.21)

We may solve Eq. (8.2.20) for φ(x) and express φ(x) as a function of x and a.

There may be more than one solution. Substituting any one of these solutions into

Eq. (8.2.21), we may obtain an equation for a. Thus the nonlinear integral equation

(8.2.18) is equivalent to one or more nonlinear algebraic equations for a.

Similarly, if K is a sum of the separable terms, the integral equation is equivalent

to systems of N coupled nonlinear algebraic equations.

In closing this section, we classify the nonlinear integral equations of the

Fredholm type in the following manner:

(1) Kernel part is nonlinear,

φ(x) −



H (x, y, φ(y))dy = f (x). (FN.1)

(2) Particular part is nonlinear,

G(φ(x)) −



K(x, y)φ(y)dy = f (x). (FN.2)

(3) Both parts are nonlinear,

G(φ(x)) −



H (x, y, φ(y))dy = f (x). (FN.3)

8.3 Nonlinear Integral Equation of the Hammerstein Type 303

(4) Nonlinear Fredholm integral equation of the ﬁrst kind,



H (x, y, φ(y))dy = f (x). (FN.4)

(5) Homogeneous nonlinear Fredholm integral equation of the ﬁrst kind,

where the kernel part is nonlinear,

φ(x) =



H (x, y, φ(y))dy. (FN.5)

(6) Homogeneous nonlinear Fredholm integral equation of the ﬁrst kind,

where the particular part is nonlinear,

G(φ(x)) =



K(x, y)φ(y)dy. (FN.6)

(7) Homogeneous nonlinear Fredholm integral equation of the ﬁrst kind,

where both parts are nonlinear,

G(φ(x)) =



H (x, y, φ(y))dy. (FN.7)

8.3

Nonlinear Integral Equation of the Hammerstein Type

The inhomogeneous term f (x) in Eq. (8.2.18) is actually not particularly meaningful.

This is because we may deﬁne

ψ(x) ≡ φ(x) − f (x), (8.3.1)

then Eq. (8.2.18) is of the form

ψ(x) =



K(x, y, ψ(x) + f (x), ψ(y) + f (y))dy. (8.3.2)

The nonlinear integral equation of the Hammerstein type is a special case of

Eq. (8.3.2),

ψ(x) =



K(x, y)f (y, ψ(y))dy. (8.3.3)

For the remainder of this section, we discuss this latter equation, (8.3.3). We shall

show that if f satisﬁes uniformly a Lipschitz condition of the form



f (y, u

) − f (y, u

)



< C(y)

− u

, (8.3.4)

304 8 Nonlinear Integral Equations

and if



A(y)C

(y)dy = M

< 1, (8.3.5)

then the solution of Eq. (8.3.3) is unique and can be obtained by iteration. The

function A(x) in Eq. (8.3.5) is given by

A(x) =



(x, y)dy. (8.3.6)

We begin by setting

(x) = 0 (8.3.7)

and

(x) =



K(x, y)f (y, ψ

n−1

(y))dy. (8.3.8)

If f (y,0) = 0, then Eq. (8.3.3) is solved by ψ(x) = 0. If

f (y,0)= 0, (8.3.9)

we have

(x) ≤ A(x)



(y,0)dy = A(x)



, (8.3.10)

where



≡



(y,0)dy. (8.3.11)

Also, as a consequence of the Lipschitz condition (8.3.4),



(x) − ψ

n−1

(x)







K(x, y)



C(y)



n−1

(y) − ψ

n−2

(y)



dy.

Thus

[ψ

(x) − ψ

n−1

(x)]

≤ A(x)



(y)[ψ

n−1

(y) − ψ

n−2

(y)]

dy. (8.3.12)

From Eqs. (8.3.4) and (8.3.5), we get

[ψ

(x) − ψ

(x)]

≤ A(x)



8.4 Problems for Chapter 8 305

By induction,

[ψ

(x) − ψ

n−1

(x)]

≤ A(x)



)

n−1

. (8.3.13)

Thus the series

(x) + [ψ

(x) − ψ

(x)] + [ψ

(x) − ψ

(x)] +···

is convergent, due to Eq. (8.3.5).

The proof of uniqueness will be left to the reader.

8.4

Problems for Chapter 8

8.1. (due to H. C.). Prove the uniqueness of the solution to the nonlinear

integral equation of the Hammerstein type

ψ(x) =



K(x, y)f (y, ψ(y))dy.

8.2. (due to H. C.). Consider

x (t) + x(t) =

(t), t > 0,

with the initial conditions

x (0) = 0and



t=0

= 1.

(a) Transform this nonlinear ordinary differential equation to an integral

equation.

(b) Obtain an approximate solution accurate to a few percent.

8.3. (due to H. C.). Consider



∂

∂x

∂

∂y



φ(x, y) =

(x, y), x

+ y

< 1,

with

φ(x, y) = 1onx

+ y

= 1.

306 8 Nonlinear Integral Equations

(a) Construct a Green’s function satisfying



∂

∂x

∂

∂y



G(x, y;x



, y



) = δ(x − x



)δ(y − y



with the boundary condition

G(x, y;x



, y



) = 0onx

+ y

= 1.

Prove that

G(x, y;x



, y



) = G(x



, y



;x, y).

Hint: To construct Green’s function G(x, y;x



, y



), which vanishes on

theunitcircle,usethemethodofimages.

(b) Transform the above nonlinear partial differential equation to an

integral equation, and obtain an approximate solution accurate to a few

percent.

8.4. (due to H. C.).

(a) Discuss a phase transition of ρ(θ) which is given by the nonlinear

integral equation,

ρ(θ ) = Z

−1

exp[β



2π

cos(θ − φ)ρ(φ)dφ],

β = Jk

T,0≤ θ ≤ 2π ,

where Z is the normalization constant such that ρ(θ)isnormalizedto

unity,



2π

ρ(θ )dθ = 1.

(b) Determine the nature of the phase transition.

Hint: You may need the following special functions:

(z) =

∞



m=0

(m!)





(z) =

∞



m=0

m!(m + 1)!





2m+1