^{1}

^{*}

^{2}

^{3}

The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3
*D* Fredholm-Stieltjes integral equations,
*i.e. *3
*D* Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, relevant to the problem of computing the eigenvalues of a body charged by a finite number of masses concentrated on points, curves or surfaces lying in.

The theory of Fredholm integral equations is strictly connected with the birth of functional analysis. A background of this theory can be found in classical books (see e.g. [

In [

In the one-dimensional case, the mechanical interpretation of these equations is connected with the problem of the free vibrations of a string charged by a finite number of cursors, and is related to an extension of the classical orthogonality property of eigensolutions, called the “Sobolev-type” orthogonality (see e.g. [

In preceding articles [

In this article, after briefly recalling the main results about the theory of eigenvalues for charged Fredholm integral equations, we mention how to obtain (in the particular case of a symmetric and strictly positive operator), the lower and upper approximations of these eigenvalues by means of the Rayleigh-Ritz method [

Consider the interval A and the Hilbert space L d M 2 ≡ L d M 2 ( A ) equipped with the scalar product

[ U , V ] d M : = ( U , V ) ρ ( P ) d P + ( U , V ) Δ r + ( U , V ) Δ γ n + ( U , V ) Δ Σ m (1)

where [ U , V ] d M ≡ ( U , V ) L d M 2 and the subscripts “ ρ ( P ) d P ”, “ Δ r ”, “ Δ γ n ”, “ Δ Σ m ” refer to the following definitions:

( U , V ) ρ ( P ) d P : = ∫ A U ( P ) V ( P ) ρ ( P ) d P (2)

( U , V ) Δ r : = ∑ h = 1 r m h U ( A h ) V ( A h ) (3)

( U , V ) Δ γ n : = ∑ k = 1 n ∫ γ k σ k ( P ( s ) ) U ( P ( s ) ) V ( P ( s ) ) d s (4)

and

( U , V ) Δ Σ m : = ∑ l = 1 m ∫ Σ l τ l ( P ( u , v ) ) U ( P ( u , v ) ) V ( P ( u , v ) ) d σ , (5)

(obviously d σ : = E G − F 2 d u d v ), so that the Stieltjes measure is the sum of the ordinary Lebesgue measure plus a finite sum of charges m h concentrated on points A h ⊂ A ¯ , ( h = 1 , 2 , ⋯ , r ) , plus a finite sum of continuous charges belonging to the curves Σ l ⊂ A ¯ , with densities σ k ( ⋅ ) , ( k = 1 , 2 , ⋯ , n ) , plus a finite sum of continuous charges belonging to the surfaces Σ l ⊂ A ¯ , with densities τ k ( ⋅ ) , ( l = 1 , 2 , ⋯ , m ) .

It is worth noting that L d M 2 ( A ) is constituted by functions of the ordinary space L 2 ( A ) , a (complete) Hilbert space, which does not have singularities (i.e. discontinuities) at points A h , ( h = 1 , 2 , ⋯ , r ) , at curves γ k , ( k = 1 , 2 , ⋯ , n ) , and surfaces Σ l , ( l = 1 , 2 , ⋯ , m ) , according to the usual condition for existence of the relevant Stieltjes integrals (see e.g. [

Let us consider, for example, the computation of integrals with respect to the above Dirac-type measures:

∫ A K ( P , Q ) φ ( Q ) m h δ ( A h ) = m h K ( P , A h ) φ (Ah)

∫ A K ( P , Q ) φ ( Q ) ∫ γ k σ k ( Q ( s ) ) δ ( Q ( s ) ) d s = ∫ γ k K ( P , Q ( s ) ) φ ( Q ( s ) ) τ l ( Q ( s ) ) d s

∫ A K ( P , Q ) φ ( Q ) ∫ Σ l τ l ( Q ( u , v ) ) δ ( Q ( u , v ) ) d σ = ∫ Σ l K ( P , Q ( u , v ) ) φ ( Q ( u , v ) ) τ l ( Q ( u , v ) ) d σ

These formulas will be useful in the following.

The Eigenvalue Problem for 3D “Charged” OperatorsConsider in L d M 2 ( A ) the eigenvalue problem

K φ = μ φ (6)

K φ ( ⋅ ) : = ∫ A K ( ⋅ , Q ) φ ( Q ) d M Q (7)

where K ( P , Q ) is a symmetric kernel, and

d M Q = ρ ( Q ) d Q + ∑ h = 1 r m h δ ( A h ) + ∑ k = 1 n ∫ γ k σ k ( Q ( s ) ) δ ( Q ( s ) ) d s + ∑ l = 1 m ∫ Σ l τ l ( Q ( u , v ) ) δ ( Q ( u , v ) ) d σ (8)

( δ denoting the usual Dirac-Delta function).

According to the above positions, we have

( K φ ) ( P ) = ∫ A K ( P , Q ) φ ( Q ) ρ ( Q ) d Q + ∑ h = 1 r m h K ( P , A h ) φ ( A h ) + ∑ k = 1 n ∫ A ∫ γ k K ( P , Q ) φ ( Q ( s ) ) σ k ( Q ( s ) ) δ ( Q ( s ) ) d s + ∑ l = 1 m ∫ A ∫ Σ l K ( P , Q ) φ ( Q ( s ) ) τ l ( Q ( u , v ) ) δ ( Q ( u , v ) ) d σ = ( K ( P , Q ) , φ ( Q ) ) ρ ( Q ) d Q + ( K ( P , Q ) , φ ( Q ) ) Δ r ( Q ) + ( K ( P , Q ) , φ ( Q ) ) Δ γ n ( Q ) + ( K ( P , Q ) , φ ( Q ) ) Δ Σ m ( Q ) (9)

so that

[ K U , V ] d M = ( ( K U ) ( P ) , V ( P ) ) ρ ( P ) d P + ( ( K U ) ( P ) , V ( P ) ) Δ r ( P ) + ( ( K U ) ( P ) , V ( P ) ) Δ γ n (P)

i.e., by using scalar products, with respect to the relevant measures:

[ K U , V ] d M = ( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) ρ ( P ) d P + ( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) ρ ( P ) d P + ( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) ρ ( P ) d P + ( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) ρ ( P ) d P + ( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) Δ r ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) Δ r ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) Δ r ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) Δ r (P)

+ ( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) Δ γ n ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) Δ γ n ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) Δ γ n ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) Δ γ n ( P ) + ( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) Δ Σ m ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) Δ Σ m ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) Δ Σ m ( P ) + ( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) Δ Σ m (P)

Furthermore

( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) ρ ( P ) d P = ∫ ∫ A × A K ( P , Q ) U ( Q ) V ( P ) ρ ( P ) ρ ( Q ) d P d Q

( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) ρ ( P ) d P = ∑ h = 1 r m h U ( A h ) ∫ A K ( P , A h ) V ( P ) ρ ( P ) d P

( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) Δ r ( P ) = ∑ h = 1 r m h V ( A h ) ∫ A K ( P , A h ) U ( P ) ρ ( P ) d P

where we used the symmetry of the kernel,

( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) ρ ( P ) d P = ∑ k = 1 n ∫ γ k ∫ A K ( P , Q ( s ) ) U ( Q ( s ) ) τ l ( Q ( s ) ) V ( P ) ρ ( P ) d s d P

( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) Δ γ n ( P ) = ∑ k = 1 n ∫ γ k ∫ A K ( P , Q ( s ) ) V ( Q ( s ) ) σ k ( Q ( s ) ) U ( P ) ρ ( P ) d s d P

( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) Δ Σ m ( P ) = ∑ j = 1 m ∑ l = 1 m ∫ Σ j ∫ Σ l K ( P ( t , w ) , Q ( u , v ) ) U ( Q ( u , v ) ) × V ( P ( t , w ) ) τ l ( Q ( u , v ) ) τ l ( P ( t , w ) ) d σ 1 d σ 2

where we used the symmetry of the kernel,

( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) ρ ( P ) d P = ∑ l = 1 m ∫ Σ l ∫ A K ( P , Q ( u , v ) ) U ( Q ( u , v ) ) τ l ( Q ( u , v ) ) V ( P ) ρ ( P ) d σ d P

( ( K ( P , Q ) , U ( Q ) ) ρ ( Q ) d Q , V ( P ) ) Δ Σ m ( P ) = ∑ l = 1 m ∫ Σ l ∫ A K ( P , Q ( u , v ) ) V ( Q ( u , v ) ) τ l ( Q ( u , v ) ) U ( P ) ρ ( P ) d σ d P

where we used the symmetry of the kernel,

( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) Δ r ( P ) = ∑ h = 1 r ∑ j = 1 r m h m j K ( A h , A j ) U ( A h ) V (Aj)

( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) Δ r ( P ) = ∑ h = 1 r m h V ( A h ) ∑ k = 1 n ∫ γ k K ( A h , Q ( s ) ) U ( Q ( s ) ) σ k ( Q ( s ) ) d s

( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) Δ γ n ( P ) = ∑ h = 1 r m h U ( A h ) ∑ k = 1 n ∫ γ k K ( A h , Q ( s ) ) V ( Q ( s ) ) σ k ( Q ( s ) ) d s

where we used the symmetry of the kernel,

( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) Δ r ( P ) = ∑ h = 1 r m h V ( A h ) ∑ l = 1 m ∫ Σ l K ( A h , Q ( u , v ) ) U ( Q ( u , v ) ) τ l ( Q ( u , v ) ) d σ

( ( K ( P , Q ) , U ( Q ) ) Δ r ( Q ) , V ( P ) ) Δ Σ m ( P ) = ∑ h = 1 r m h U ( A h ) ∑ l = 1 m ∫ Σ l K ( A h , Q ( u , v ) ) V ( Q ( u , v ) ) τ l ( Q ( u , v ) ) d σ

where we used the symmetry of the kernel,

( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) Δ γ n ( P ) = ∑ k = 1 n V ( P ( s ) ) σ k ( P ( s ) ) ∑ l = 1 m ∫ Σ l K ( P ( s ) , Q ( u , v ) ) U ( Q ( u , v ) ) τ l ( Q ( u , v ) ) d σ d s

( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) Δ Σ m ( P ) = ∑ k = 1 n U ( P ( s ) ) σ k ( P ( s ) ) ∑ l = 1 m ∫ Σ l K ( P ( s ) , Q ( u , v ) ) V ( Q ( u , v ) ) τ l ( Q ( u , v ) ) d σ d s

where we used the symmetry of the kernel, and lastly

( ( K ( P , Q ) , U ( Q ) ) Δ γ n ( Q ) , V ( P ) ) Δ γ n ( P ) = ∑ k = 1 n ∑ l = 1 n ∫ γ k ∫ γ l K ( P ( σ ) , Q ( s ) ) U ( Q ( s ) ) V ( P ( σ ) ) σ k ( Q ( s ) ) σ l ( P ( σ ) ) d s d σ

( ( K ( P , Q ) , U ( Q ) ) Δ Σ m ( Q ) , V ( P ) ) Δ Σ m ( P ) = ∑ j = 1 m ∑ l = 1 m ∫ Σ j ∫ Σ l K ( P ( t , w ) , Q ( u , v ) ) U ( Q ( u , v ) ) × V ( P ( t , w ) ) τ l ( Q ( u , v ) ) τ l ( P ( t , w ) ) d σ 1 d σ 2

The computation of the eigenvalues of second kind Fredholm integral equations is usually performed by using the Rayleig-Ritz method [

We will not describe herewith the three above mentioned methods, because they are essentially independent of the dimension of the considered vibrating item (string, membrane or body). We refer for shortness to the above mentioned articles [

Let A ≡ [ 0, a ] × [ 0, b ] × [ 0, c ] , O ≡ ( 0 , 0 , 0 ) , U ≡ ( a , b , c ) ,

P ≡ ( x P , y P , z P ) , Q ≡ ( x Q , y Q , z Q ) , R ≡ ( x P , y P , z Q ) , S ≡ ( x P , y Q , z Q )

T ≡ ( x Q , y P , z P ) , L ≡ ( x Q , y Q , z P ) , M ≡ ( x P , y Q , z P ) , N ≡ ( x Q , y P , z Q )

0 ≤ x P ≤ a , 0 ≤ y P ≤ b , 0 ≤ z P ≤ c

0 ≤ x Q ≤ a , 0 ≤ y Q ≤ b , 0 ≤ z Q ≤ c

K ( P , Q ) = { O P ¯ ⋅ U Q ¯ , if x P ≤ x Q , y P ≤ y Q , z P ≤ z Q , O Q ¯ ⋅ U P ¯ , if x P ≥ x Q , y P ≥ y Q , z P ≥ z Q , O R ¯ ⋅ U L ¯ , if x P ≤ x Q , y P ≤ y Q , z P ≥ z Q , O S ¯ ⋅ U T ¯ , if x P ≤ x Q , y P ≥ y Q , z P ≥ z Q , O T ¯ ⋅ U S ¯ , if x P ≥ x Q , y P ≤ y Q , z P ≤ z Q , O L ¯ ⋅ U R ¯ , if x P ≥ x Q , y P ≥ y Q , z P ≤ z Q , O M ¯ ⋅ U N ¯ , if x P ≤ x Q , y P ≥ y Q , z P ≤ z Q , O N ¯ ⋅ U M ¯ , if x P ≥ x Q , y P ≤ y Q , z P ≥ z Q , (1)

i.e.

Obviously K ( P , Q ) = K ( Q , P ) , i.e. the kernel is symmetric, and even positive definite, since K ( P , Q ) > 0 , ∀ ( P , Q ) ∈ A .

Consider in L d M 2 ( A ) the eigenvalue problem

K φ = μ φ (2)

K φ : = ∫ A K ( ⋅ , Q ) φ ( Q ) d M Q (3)

where

d M Q = ρ ( Q ) d Q + ∑ h = 1 r m h δ ( A h ) + ∑ k = 1 n ∫ γ k σ k ( Q ( s ) ) δ ( Q ( s ) ) d s + ∑ l = 1 m ∫ Σ l τ l ( Q ( u , v ) ) δ ( Q ( u , v ) ) d σ

( δ denoting the usual Dirac-Delta function).

The considered operator is compact and strictly positive, since it is connected with free vibrations of a body charged by a finite number of masses concentrated on points, curves, or surfaces contained in A .

Numerical Example 1

Let a = b = c = 1 , and d M = ρ ( x , y , z ) d x d y d z with ρ ( x , y , z ) = 1 + 2 x + 3 y + 4 z (see

Numerical Example 2

Let a = b = c = 1 , and d M = ρ ( x , y , z ) d x d y d z with ρ ( x , y , z ) = 8 π 3 e − ( x + y + z ) 2 sin ( π x y z ) (see

Numerical Example 3

Let a = b = c = 1 , and

d M = ρ ( x , y , z ) d x d y d z + δ ( x − 1 2 ) δ ( y − 1 2 ) δ ( z − 1 2 ) d x d y d z

with ρ ( x , y , z ) = e − x − y − z (see

Numerical Example 4

Let a = b = c = 1 , and d M = ρ ( x , y , z ) d x d y d z + ∑ h m h δ ( x − x h ) δ ( y − y h ) δ ( z − z h ) d x d y d z with ρ ( x , y , z ) = 2 log ( 1 + z ( 1 + y ( 1 + x ) ) ) , and m h = 1 / ( x h + y h + z h ) ,

x h = y h = z h = 1 H ( h − 1 2 ) for h = 1 , 2 , 3 = H (see

tion, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in

Numerical Example 5

Let a = b = c = 1 , and d M = ρ ( x , y , z ) d x d y d z + ∑ h m h δ ( x − x h ) δ ( y − y h ) δ ( z − z h ) d x d y d z with

ρ ( x , y , z ) = 1 + cos [ π 2 ( 1 − 2 x − 3 y − 4 z ) ] , and m h = 3 x h + 2 y h + z h ,

x 1 = x 2 = x 5 = x 6 = y 1 = y 3 = y 5 = y 7 = z 1 = z 2 = z 3 = z 4 = 1 4 ,

x 3 = x 4 = x 7 = x 8 = y 2 = y 4 = y 6 = y 8 = z 5 = z 6 = z 7 = z 8 = 3 4 for h = 1 , 2 , ⋯ , 8 = H

(see

Numerical Example 6

Let a = b = c = 1 , and d M = ρ ( x , y , z ) d x d y d z + σ ( x , y , z ) δ ( z − ζ s ) d x d y d z + ∑ h δ ( x − x h ) δ ( y − y h ) δ ( z − z h ) d x d y d z with ρ ( x , y , z ) = [ cosh ( 2 y − x ) + sin ( 2 x − y ) ] / ( 1 + e x + y ) , σ ( x , y , z ) = e − 2 x − 3 y + 4 z , ζ s = 1 4 , and x h = 1 − y h = 1 H ( h − 1 2 ) , z h = 4 5 for h = 1 , 2 = H (see

Under this assumption, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in

Numerical Example 7

Let a = b = c = 1 , and d M = ρ ( x , y , z ) d x d y d z + σ ( x , y , z ) δ ( x − ξ s ) d x d y d z + ∑ h m h δ ( x − x h ) δ ( y − y h ) δ ( z − z h ) d x d y d z with ρ ( x , y , z ) = sech ( 4 [ ( x − 1 / 2 ) 2 + ( y − 1 / 2 ) 2 + ( z − 1 / 2 ) 2 ] 1 / 2 ) , σ ( x , y , z ) = 5 x + sin ( 2 y ) cos ( 3 z ) , ξ s = 1 5 , and m h = y h + z h , x h = 3 4 ,

y h = z h = 1 H ( h − 1 2 ) for h = 1 , 2 , 3 = H (see

Numerical Example 8

Let a = b = c = 1 , and

d M = ρ ( x , y , z ) d x d y d z + σ ( x , y ) δ ( z − ζ s ( x , y ) ) d x d y d z + δ ( x − 3 4 ) δ ( y − 3 4 ) δ ( z − 3 4 ) d x d y d z

with ρ ( x , y , z ) = [ 3 + 2 cos ( x z ) + sin ( y z ) ] / ( 1 + 2 x + 3 y + 4 z ) , and

σ ( x , y ) = ( 1 + 2 x + 3 y ) / [ 7 + c o s ( 6 y ) + s i n ( 5 x ) ] ,

ζ s ( x , y ) = 1 2 [ 1 + cos ( x ) − sin ( 3 y ) ] (see

approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in

Numerical Example 9

Let a = b = c = 1 , and

d M = ρ ( x , y , z ) d x d y d z + σ ( x , y ) δ ( z − ζ s ( x , y ) ) d x d y d z + τ ( z ) δ ( x − ξ l ( z ) ) δ ( y − η l ( z ) ) d x d y d z + ∑ h m h δ ( x − x h ) δ ( y − y h ) δ ( z − z h ) d x d y d z

with ρ ( x , y , z ) = 4 + 3 cos ( 3 x ) + 2 sin ( 2 y ) + tan ( z ) , σ ( x , y ) = 2 log ( 1 + y + x y ) , ζ s ( x , y ) = 1 4 ( 1 − x + 2 y ) , τ ( z ) = 2 z , ξ l ( z ) = η l ( z ) = 4 z − 3 , and

m h = x h + y h + z h , x 1 = x 2 = y 1 = y 3 = 1 4 , x 3 = x 4 = y 2 = y 4 = 3 4 , z h = 2 3 for

h = 1 , 2 , 3 , 4 = H (see

This study has been partly carried out in the framework of the research and

development program running at The Antenna Company, and is funded by the Competitiveness Enhancement Program grant at Tomsk Polytechnic University.

Caratelli, D., Natalini, P. and Ricci, P.E. (2017) Computation of the Eigenvalues of 3D “Charged” Integral Equations. Journal of Applied Mathematics and Physics, 5, 2051-2071. https://doi.org/10.4236/jamp.2017.510171