Handbook of mathematics for engineers and scienteists part 124 pptx

For convenience of analysis, a number λ is traditionally singled out in equation 16.4.1.1, which is called the parameter of integral equation.. Note that equations of the form 16.4.1.1 w

16.4 Linear Integral Equations of the Second Kind with

Constant Limits of Integration

16.4.1 Fredholm Integral Equations of the Second Kind Resolvent

16.4.1-1 Some definitions The eigenfunctions of a Fredholm integral equation Linear integral equations of the second kind with constant limits of integration have the general form

y (x) – λ

 b

a K (x, t)y(t) dt = f (x), (16.4.1.1)

where y(x) is the unknown function (a≤x≤b ), K(x, t) is the kernel of the integral equation, and f (x) is a given function, which is called the right-hand side of equation (16.4.1.1) For convenience of analysis, a number λ is traditionally singled out in equation (16.4.1.1), which is called the parameter of integral equation The classes of functions and kernels

under consideration were defined above in Paragraphs 16.3.1-1 and 16.3.1-2 Note that equations of the form (16.4.1.1) with constant limits of integration and with Fredholm

kernels or kernels with weak singularity are called Fredholm equations of the second kind and equations with weak singularity of the second kind, respectively.

Equation (16.4.1.1) is said to be homogeneous if f (x)≡ 0and nonhomogeneous

other-wise

A number λ is called a characteristic value of the integral equation (16.4.1.1) if there

exist nontrivial solutions of the corresponding homogeneous equation The nontrivial

solutions themselves are called the eigenfunctions of the integral equation corresponding

to the characteristic value λ If λ is a characteristic value, the number 1/λ is called an

eigenvalue of the integral equation (16.4.1.1) A value of the parameter λ is said to be regular if for this value the above homogeneous equation has only the trivial solution.

Sometimes the characteristic values and the eigenfunctions of a Fredholm integral equation

are called the characteristic values and the eigenfunctions of the kernel K(x, t).

The kernel K(x, t) of the integral equation (16.4.1.1) is called a degenerate kernel if it has the form K(x, t) = g1(x)h1(t) + · · · + g n (x)h n (t), a difference kernel if it depends on the difference of the arguments (K(x, t) = K(x – t)), and a symmetric kernel if it satisfies the condition K(x, t) = K(t, x).

The transposed integral equation is obtained from (16.4.1.1) by replacing the kernel

K (x, t) by K(t, x).

The integral equation of the second kind with difference kernel on the entire axis

(a = – ∞, b = ∞) and semiaxis (a =0, b = ∞) are referred to as an equation of convolution type of the second kind and a Wiener–Hopf integral equation of the second kind, respectively.

16.4.1-2 Structure of the solution The resolvent

The solution of equation (16.4.1.1) can be presented in the form

y (x) = f (x) + λ

 b

a R (x, t; λ)f (t) dt,

where the resolvent R(x, t; λ) is independent of f (x) and is determined by the kernel of the

integral equation

16.4.2 Fredholm Equations of the Second Kind with Degenerate

Kernel

16.4.2-1 Simplest degenerate kernel

Consider Fredholm integral equations of the second kind with the simplest degenerate kernel:

y (x) – λ

 b

a g (x)h(t)y(t) dt = f (x), a≤x≤b (16.4.2.1)

We seek a solution of equation (16.4.2.1) in the form

y (x) = f (x) + λAg(x). (16.4.2.2)

On substituting the expressions (16.4.2.2) into equation (16.4.2.1), after simple algebraic manipulations we obtain

A



1– λ

 b

a h (t)g(t) dt



=

 b

a f (t)h(t) dt. (16.4.2.3) Both integrals occurring in equation (16.4.2.3) are supposed to exist On the basis of (16.4.2.1) and (16.4.2.3) and taking into account the fact that the unique characteristic

value λ1of equation (16.4.2.1) is given by the expression

λ1=

 b

a h (t)g(t) dt

–1 , (16.4.2.4)

we obtain the following results:

1◦ If λ ≠λ1, then for an arbitrary right-hand side there exists a unique solution of

equa-tion (16.4.2.1), which can be written in the form

y (x) = f (x) + λλ1f1

λ1– λ g (x), f1=

 b

a f (t)h(t) dt. (16.4.2.5)

2◦ If λ = λ1and f1=0, then any solution of equation (16.4.2.1) can be represented in the form

y = f (x) + Cy1(x), y1(x) = g(x), (16.4.2.6)

where C is an arbitrary constant and y1(x) is an eigenfunction that corresponds to the characteristic value λ1

3◦ If λ = λ1and f1 ≠ 0, then there are no solutions

16.4.2-2 Degenerate kernel in the general case

In the general case, a Fredholm integral equation of the second kind with degenerate kernel has the form

y (x) – λ

 b

a

" n



k=1

g k (x)h k (t)

#

y (t) dt = f (x), n=2,3, (16.4.2.7)

Let us rewrite equation (16.4.2.7) in the form

y (x) = f (x) + λ

n



k=1

g k (x)

 b

a h k (t)y(t) dt, n=2,3, (16.4.2.8)

We assume that equation (16.4.2.8) has a solution and introduce the notation

A k=

 b

a h k (t)y(t) dt. (16.4.2.9)

In this case we have

y (x) = f (x) + λ

n



k=1

A k g k (x), (16.4.2.10)

and hence the solution of the integral equation with degenerate kernel is reduced to the

definition of the constants A k

Let us multiply equation (16.4.2.10) by h m (x) and integrate with respect to x from a to

b We obtain the following system of linear algebraic equations for the coefficients A k:

A m – λ

n



k=1

s mk A k = f m, m=1, , n, (16.4.2.11)

where

s mk=

 b

a h m (x)g k (x) dx, f m=

 b

a f (x)h m (x) dx; m , k =1, , n. (16.4.2.12) Once we construct a solution of system (16.4.2.11), we obtain a solution of the integral

equation with degenerate kernel (16.4.2.7) as well The values of the parameter λ at which

the determinant of system (16.4.2.11) vanishes are characteristic values of the integral

equation (16.4.2.7), and it is clear that there are just n such values counted according to

their multiplicities

Example Let us solve the integral equation

y(x) – λ

 π

–π (x cos t + t2sin x + cos x sin t)y(t) dt = x, –π≤x≤π. (16 4 2 13 ) Let us denote

A1=

 π

–π y(t) cos t dt, A2=

 π

–π

t2y(t) dt, A3=

 π

–π y(t) sin t dt, (16 4 2 14 )

where A1, A2, and A3 are unknown constants Then equation (16.4.2.13) can be rewritten in the form

y(x) = A1λx + A2λ sin x + A3λ cos x + x. (16 4 2 15 )

On substituting the expression (16.4.2.15) into relations (16.4.2.14), we obtain

A1=

 π

–π (A1λt + A2λ sin t + A3λ cos t + t) cos t dt,

A2=

 π

–π (A1λt + A2λ sin t + A3λ cos t + t)t2dt,

A3=

 π

π (A1λt + A2λ sin t + A3λ cos t + t) sin t dt.

On calculating the integrals occurring in these equations, we obtain the following system of algebraic equations

for the unknowns A1, A2, and A3 :

A1– λπA3 = 0 ,

A2+ 4λπA3= 0 , – 2λπA1– λπA2+ A3 = 2π.

(16 4 2 16 ) System (16.4.2.16) has the unique solution

A1= 2λπ2

1 + 2λ2π2, A2= – 8λπ2

1 + 2λ2π2, A3= 2π

1 + 2λ2π2.

On substituting the above values of A1, A2, and A3 into (16.4.2.15), we obtain the solution of the original integral equation:

y(x) = 2λπ

1 + 2λ2π2(λπx –4λπ sin x + cos x) + x.

16.4.3 Solution as a Power Series in the Parameter Method of

Successive Approximations

16.4.3-1 Iterated kernels

Consider the Fredholm integral equation of the second kind:

y (x) – λ

 b

a K (x, t)y(t) dt = f (x), a≤x≤b (16.4.3.1)

We seek the solution in the form of a series in powers of the parameter λ:

y (x) = f (x) +

∞



n=1

λ n ψ

Substitute series (16.4.3.2) into equation (16.4.3.1) On matching the coefficients of like

powers of λ, we obtain a recurrent system of equations for the functions ψ n (x) The solution

of this system yields

ψ1(x) =

 b

a K (x, t)f (t) dt,

ψ2(x) =

 b

a K (x, t)ψ1(t) dt =

 b

a K2(x, t)f (t) dt,

ψ3(x) =

 b

a K (x, t)ψ2(t) dt =

 b

a K3(x, t)f (t) dt, etc.

Here

K n (x, t) =

 b

a K (x, z)K n–1(z, t) dz, (16.4.3.3)

where n = 2,3, , and we have K1(x, t)≡ K (x, t) The functions K n (x, t) defined by formulas (16.4.3.3) are called iterated kernels These kernels satisfy the relation

K n (x, t) =

 b

a K m (x, s)K n–m (s, t) ds, (16.4.3.4)

where m is an arbitrary positive integer less than n.

The iterated kernels K n (x, t) can be directly expressed via K(x, t) by the formula

K n (x, t) =

 b

a

 b

a · · ·

 b

a

n–1

K (x, s1)K(s1, s2) K(s n–1, t) ds1ds2 ds n–1.

All iterated kernels K n (x, t), beginning with K2(x, t), are continuous functions on the square S ={a≤x≤b , a≤t≤b}if the original kernel K(x, t) is square integrable on S.

If K(x, t) is symmetric, then all iterated kernels K n (x, t) are also symmetric.

16.4.3-2 Method of successive approximations

The results of Subsection 16.4.3-1 can also be obtained by means of the method of successive approximations To this end, one should use the recurrent formula

y n (x) = f (x) + λ

 b

a K (x, t)y n–1(t) dt, n=1,2, , with the zeroth approximation y0(x) = f (x).

16.4.3-3 Construction of the resolvent

The resolvent of the integral equation (16.4.3.1) is defined via the iterated kernels by the formula

R (x, t; λ) =

∞



n=1

λ n–1K

n (x, t), (16.4.3.5)

where the series on the right-hand side is called the Neumann series of the kernel K(x, t).

It converges to a unique square integrable solution of equation (16.4.3.1) provided that

|λ|< 1

! b

a

 b

2(x, t) dx dt. (16.4.3.6)

If, in addition, we have

 b

2(x, t) dt≤A, a≤x≤b,

where A is a constant, then the Neumann series converges absolutely and uniformly on [a, b].

A solution of a Fredholm equation of the second kind of the form (16.4.3.1) is expressed

by the formula

y (x) = f (x) + λ

 b

a R (x, t; λ)f (t) dt, a≤x≤b (16.4.3.7) Inequality (16.4.3.6) is essential for the convergence of the series (16.4.3.5) However,

a solution of equation (16.4.3.1) can exist for values|λ|>1/Bas well

Example Let us solve the integral equation

y(x) – λ

 1

0 xty(t) dt = f (x), 0 ≤x≤ 1 ,

by the method of successive approximations Here we have K(x, t) = xt, a =0, and b =1 We successively define

K1(x, t) = xt, K2(x, t) =

 1

0 (xz)(zt) dz = xt

3 , K3(x, t) =

1 3

 1

0 (xz)(zt) dz = xt

3 2, , K n (x, t) = xt

3n–1 According to formula (16.4.3.5) for the resolvent, we obtain

R(x, t; λ) =

∞



n=1

λ n–1K n (x, t) = xt

∞



n=1



λ

3

n–1

= 3xt

3– λ,

where |λ| < 3 , and it follows from formula (16.4.3.7) that the solution of the integral equation can be rewritten

in the form

y(x) = f (x) + λ

 1 0

3xt

3– λ f(t) dt, 0 ≤x≤ 1 , λ≠ 3

16.4.4 Fredholm Theorems and the Fredholm Alternative

16.4.4-1 Fredholm theorems

THEOREM1 If λ is a regular value, then both the Fredholm integral equation of the

second kind and the transposed equation are solvable for any right-hand side, and both the equations have unique solutions The corresponding homogeneous equations have only the trivial solutions

THEOREM2 For the nonhomogeneous integral equation to be solvable, it is necessary

and sufficient that the right-hand side f (x) satisfies the conditions

 b

a f (x)ψ k (x) dx =0, k=1, , n, where ψ k (x) is a complete set of linearly independent solutions of the corresponding transposed homogeneous equation

THEOREM3 If λ is a characteristic value, then both the homogeneous integral equation

and the transposed homogeneous equation have nontrivial solutions The number of linearly independent solutions of the homogeneous integral equation is finite and is equal to the number of linearly independent solutions of the transposed homogeneous equation THEOREM4 A Fredholm equation of the second kind has at most countably many characteristic values, whose only possible accumulation point is the point at infinity

16.4.4-2 Fredholm alternative

The Fredholm theorems imply the so-called Fredholm alternative, which is most frequently used in the investigation of integral equations

THEFREDHOLM ALTERNATIVE Either the nonhomogeneous equation is solvable for any right-hand side or the corresponding homogeneous equation has nontrivial solutions The first part of the alternative holds if the given value of the parameter is regular and the second if it is characteristic

Remark The Fredholm theory is also valid for integral equations of the second kind with weak singularity.

16.4.5 Fredholm Integral Equations of the Second Kind with

Symmetric Kernel

16.4.5-1 Characteristic values and eigenfunctions

Integral equations whose kernels are symmetric, that is, satisfy the condition K(x, t) =

K (t, x), are called symmetric integral equations.

Each symmetric kernel that is not identically zero has at least one characteristic value

For any n, the set of characteristic values of the nth iterated kernel coincides with the set of nth powers of the characteristic values of the first kernel.

The eigenfunctions of a symmetric kernel corresponding to distinct characteristic values are orthogonal, i.e., if

ϕ1(x) = λ1

 b

a K (x, t)ϕ1(t) dt, ϕ2(x) = λ2

 b

a K (x, t)ϕ2(t) dt, λ1≠λ2,

then

(ϕ1, ϕ2) =0, (ϕ, ψ)≡

 b

a ϕ (x)ψ(x) dx.

The characteristic values of a symmetric kernel are real

The eigenfunctions can be normalized; namely, we can divide each characteristic func-tion by its norm If several linearly independent eigenfuncfunc-tions correspond to the same

characteristic value, say, ϕ1(x), , ϕ n (x), then each linear combination of these

func-tions is an eigenfunction as well, and these linear combinafunc-tions can be chosen so that the corresponding eigenfunctions are orthonormal

Indeed, the function

ψ1(x) = ϕ1(x)

ϕ1, ϕ1 =

(ϕ1, ϕ1),

has the norm equal to one, i.e.,ψ1 =1 Let us form a linear combination αψ1+ ϕ2and

choose α so that

(αψ1+ ϕ2, ψ1) =0, i.e.,

α= –(ϕ2, ψ1)

(ψ1, ψ1) = –(ϕ2, ψ1).

The function

ψ2(x) = αψ αψ11+ ϕ + ϕ22

is orthogonal to ψ1(x) and has the unit norm Next, we choose a linear combination

αψ1+ βψ2+ ϕ3, where the constants α and β can be found from the orthogonality relations

(αψ1+ βϕ2+ ϕ3, ψ1) =0, (αψ1+ βψ2+ ϕ3, ψ2) =0

For the coefficients α and β thus defined, the function

ψ3= αψ αψ11+ βψ + βϕ22+ ϕ + ϕ23

is orthogonal to ψ1and ψ2and has the unit norm, and so on

is orthogonal to ψ1and ψ2and has the unit norm, and so on

αψ1+ βψ2+ ϕ3, where the constants α and β can be... (αψ1+ βψ2+ ϕ3, ψ2) =0

For the coefficients α and β thus defined, the function

ψ3= αψ