ABSTRACT

The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D 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.

Keywords:

3D Fredholm-Stieltjes Integral Equations, Eigenvalues, Rayleigh-Ritz Method, Inverse Iteration Method

1. Introduction

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. [1] [2] ). For recent developments relevant to numerical computation of solutions, see [3] .

In [4] [5] , the Author shows that the Fredholm theory still holds considering the so called charged Fredholm integral equations, i.e. Fredholm equations with respect to a Stieltjes measure obtained by adding to the ordinary Lebesgue measure a finite combination of positive masses concentrated in arbitrary points of the considered interval. A topic which at present is included, as a particular case, in the theory of strictly positive compact operators.

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. [6] [7] ).

In preceding articles [8] [9] [10] , the problem of numerical computation of the above mentioned eigenvalue problems was solved, by using the so called inverse iteration method, showing applications in one and two-dimensional cases.

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 [1] , [11] and the Fichera orthogonal invariants method [11] [12] respectively. Then we conclude by showing that, even in the three-dimensional case, the lower approximations of the eigenvalues obtained by means of the Rayleigh-Ritz method can be improved by applying the inverse iteration method [13] . Numerical computations relevant to the considered case are developed in the concluding section.

2. Lebesgue-Stieltjes Measures in a 3D Interval

Consider the interval $\mathcal{A}$ and the Hilbert space $L_{dM}^2\equiv L_{dM}^2\left(\mathcal{A}\right)$ equipped with the scalar product

${\left[U,V\right]}_{dM}:={\left(U,V\right)}_{\rho \left(P\right)dP}+{\left(U,V\right)}_{{\Delta }_r}+{\left(U,V\right)}_{\Delta {\gamma }_n}+{\left(U,V\right)}_{{\Delta }_{{\Sigma }_m}}$ (1)

where ${\left[U,V\right]}_{dM}\equiv {\left(U,V\right)}_{L_{dM}^2}$ and the subscripts “ $\rho \left(P\right)\text{d}P$ ”, “ ${\Delta }_r$ ”, “ $\Delta {\gamma }_n$ ”, “ ${\Delta }_{{\Sigma }_m}$ ” refer to the following definitions:

${\left(U\mathrm{,}V\right)}_{\rho \left(P\right)dP}\mathrm{<mo>\; :\; </mo>}={\displaystyle {\int }_{\mathcal{A}}}U\left(P\right)V\left(P\right)\rho \left(P\right)\text{d}P$ (2)

${\left(U,V\right)}_{{\Delta }_r}:={\displaystyle \sum _{h=1}^r}{m}_{h}U\left(A_h\right)V\left(A_h\right)$ (3)

${\left(U,V\right)}_{\Delta {\gamma }_n}:={\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{{\gamma }_k}}{\sigma }_k\left(P\left(s\right)\right)U\left(P\left(s\right)\right)V\left(P\left(s\right)\right)\text{d}s$ (4)

and

${\left(U,V\right)}_{{\Delta }_{{\Sigma }_m}}:={\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}{\tau }_{\mathcal{l}}\left(P\left(u,v\right)\right)U\left(P\left(u,v\right)\right)V\left(P\left(u,v\right)\right)\text{d}\sigma$ , (5)

(obviously $\text{d}\sigma :=\sqrt{EG-F^2}\text{d}u\text{d}v$ ), so that the Stieltjes measure is the sum of the ordinary Lebesgue measure plus a finite sum of charges $m_h$ concentrated on points $A_h\subset \overline{\mathcal{A}}$ , $\left(h=1,2,\cdots ,r\right)$ , plus a finite sum of continuous charges belonging to the curves ${\Sigma }_{\mathcal{l}}\subset \overline{\mathcal{A}}$ , with densities ${\sigma }_k(\cdot )$ , $\left(k=1,2,\cdots ,n\right)$ , plus a finite sum of continuous charges belonging to the surfaces ${\Sigma }_{\mathcal{l}}\subset \overline{\mathcal{A}}$ , with densities ${\tau }_k(\cdot )$ , $\left(\mathcal{l}=1,2,\cdots ,m\right)$ .

It is worth noting that $L_{dM}^2\left(\mathcal{A}\right)$ is constituted by functions of the ordinary space $L^2\left(\mathcal{A}\right)$ , a (complete) Hilbert space, which does not have singularities (i.e. discontinuities) at points $A_h$ , $\left(h=1,2,\cdots ,r\right)$ , at curves ${\gamma }_k$ , $\left(k=1,2,\cdots ,n\right)$ , and surfaces ${\Sigma }_{\mathcal{l}}$ , $\left(\mathcal{l}=1,2,\cdots ,m\right)$ , according to the usual condition for existence of the relevant Stieltjes integrals (see e.g. [14] ).

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

${\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\right)\phi \left(Q\right)m_h\delta \left(A_h\right)=m_{h}K\left(P,A_h\right)\phi (Ah)$

$\begin{array}{l}{\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\right)\phi \left(Q\right){\displaystyle {\int }_{{\gamma }_k}}{\sigma }_k\left(Q\left(s\right)\right)\delta \left(Q\left(s\right)\right)\text{d}s\\ ={\displaystyle {\int }_{{\gamma }_k}}K\left(P,Q\left(s\right)\right)\phi \left(Q\left(s\right)\right){\tau }_{\mathcal{l}}\left(Q\left(s\right)\right)\text{d}s\end{array}$

$\begin{array}{l}{\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\right)\phi \left(Q\right){\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}{\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\delta \left(Q\left(u,v\right)\right)\text{d}\sigma \\ ={\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(P,Q\left(u,v\right)\right)\phi \left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\text{d}\sigma \end{array}$

These formulas will be useful in the following.

The Eigenvalue Problem for 3D “Charged” Operators

Consider in $L_{dM}^2\left(\mathcal{A}\right)$ the eigenvalue problem

$\mathcal{K}\phi =\mu \phi$ (6)

$\mathcal{K}\phi (\cdot )\mathrm{<mo>\; :\; </mo>}={\displaystyle {\int }_{\mathcal{A}}}K\left(\cdot \mathrm{,}Q\right)\phi \left(Q\right)\text{d}{M}_Q$ (7)

where $K\left(P\mathrm{,}Q\right)$ is a symmetric kernel, and

$\begin{array}{c}\text{d}{M}_Q=\rho \left(Q\right)\text{d}Q+{\displaystyle \sum _{h=1}^r}{m}_h\delta \left(A_h\right)+{\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{{\gamma }_k}}{\sigma }_k\left(Q\left(s\right)\right)\delta \left(Q\left(s\right)\right)\text{d}s\\ +{\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}{\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\delta \left(Q\left(u,v\right)\right)\text{d}\sigma \end{array}$ (8)

( $\delta$ denoting the usual Dirac-Delta function).

According to the above positions, we have

$\begin{array}{c}\left(\mathcal{K}\phi \right)\left(P\right)={\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\right)\phi \left(Q\right)\rho \left(Q\right)\text{d}Q+{\displaystyle \sum _{h=1}^r}{m}_{h}K\left(P,A_h\right)\phi \left(A_h\right)\\ +{\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{\mathcal{A}}}{\displaystyle {\int }_{{\gamma }_k}}K\left(P,Q\right)\phi \left(Q\left(s\right)\right){\sigma }_k\left(Q\left(s\right)\right)\delta \left(Q\left(s\right)\right)\text{d}s\\ +{\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{\mathcal{A}}}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(P,Q\right)\phi \left(Q\left(s\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\delta \left(Q\left(u,v\right)\right)\text{d}\sigma \\ ={\left(K\left(P,Q\right),\phi \left(Q\right)\right)}_{\rho \left(Q\right)dQ}+{\left(K\left(P,Q\right),\phi \left(Q\right)\right)}_{{\Delta }_r\left(Q\right)}\\ +{\left(K\left(P,Q\right),\phi \left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)}+{\left(K\left(P,Q\right),\phi \left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)}\end{array}$ (9)

so that

$\begin{array}{c}{\left[\mathcal{K}U,V\right]}_{dM}={\left(\left(\mathcal{K}U\right)\left(P\right),V\left(P\right)\right)}_{\rho \left(P\right)dP}+{\left(\left(\mathcal{K}U\right)\left(P\right),V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}\\ +{\left(\left(\mathcal{K}U\right)\left(P\right),V\left(P\right)\right)}_{{\Delta }_{{\gamma }_n}(P)}\end{array}$

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

$\begin{array}{l}{\left[\mathcal{K}U,V\right]}_{dM}\\ ={\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{\rho \left(P\right)dP}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{\rho \left(P\right)dP}\\ +{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)},V\left(P\right)\right)}_{\rho \left(P\right)dP}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{\rho \left(P\right)dP}\\ +{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}\\ +{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_r(P)}\end{array}$

$\begin{array}{l}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{\Delta {\gamma }_n\left(P\right)}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{\Delta {\gamma }_n\left(P\right)}\\ +{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)},V\left(P\right)\right)}_{\Delta {\gamma }_n\left(P\right)}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{\Delta {\gamma }_n\left(P\right)}\\ +{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}\\ +{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}+{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}(P)}\end{array}$

Furthermore

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{\rho \left(P\right)dP}\\ ={\displaystyle \int }{\displaystyle {\int }_{\mathcal{A}\times \mathcal{A}}}K\left(P,Q\right)U\left(Q\right)V\left(P\right)\rho \left(P\right)\rho \left(Q\right)\text{d}P\text{d}Q\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{\rho \left(P\right)dP}\\ ={\displaystyle \sum _{h=1}^r}{m}_{h}U\left(A_h\right){\displaystyle {\int }_{\mathcal{A}}}K\left(P,A_h\right)V\left(P\right)\rho \left(P\right)\text{d}P\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}\\ ={\displaystyle \sum _{h=1}^r}{m}_{h}V\left(A_h\right){\displaystyle {\int }_{\mathcal{A}}}K\left(P,A_h\right)U\left(P\right)\rho \left(P\right)\text{d}P\end{array}$

where we used the symmetry of the kernel,

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)},V\left(P\right)\right)}_{\rho \left(P\right)dP}\\ ={\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{{\gamma }_k}}{\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\left(s\right)\right)U\left(Q\left(s\right)\right){\tau }_{\mathcal{l}}\left(Q\left(s\right)\right)V\left(P\right)\rho \left(P\right)\text{d}s\text{d}P\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{\Delta {\gamma }_n\left(P\right)}\\ ={\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{{\gamma }_k}}{\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\left(s\right)\right)V\left(Q\left(s\right)\right){\sigma }_k\left(Q\left(s\right)\right)U\left(P\right)\rho \left(P\right)\text{d}s\text{d}P\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}={\displaystyle \sum _{j=1}^m}{\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_j}}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(P\left(t,w\right),Q\left(u,v\right)\right)U\left(Q\left(u,v\right)\right)\\ \times V\left(P\left(t,w\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(P\left(t,w\right)\right)\text{d}{\sigma }_1\text{d}{\sigma }_2\end{array}$

where we used the symmetry of the kernel,

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{\rho \left(P\right)dP}\\ ={\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}{\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\left(u,v\right)\right)U\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)V\left(P\right)\rho \left(P\right)\text{d}\sigma \text{d}P\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\rho \left(Q\right)dQ},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}\\ ={\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}{\displaystyle {\int }_{\mathcal{A}}}K\left(P,Q\left(u,v\right)\right)V\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)U\left(P\right)\rho \left(P\right)\text{d}\sigma \text{d}P\end{array}$

where we used the symmetry of the kernel,

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}\\ ={\displaystyle \sum _{h=1}^r}{\displaystyle \sum _{j=1}^r}{m}_h{m}_{j}K\left(A_h,A_j\right)U\left(A_h\right)V(Aj)\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\gamma }_n}\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}\\ ={\displaystyle \sum _{h=1}^r}{m}_{h}V\left(A_h\right){\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{{\gamma }_k}}K\left(A_h,Q\left(s\right)\right)U\left(Q\left(s\right)\right){\sigma }_k\left(Q\left(s\right)\right)\text{d}s\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\gamma }_n}\left(P\right)}\\ ={\displaystyle \sum _{h=1}^r}{m}_{h}U\left(A_h\right){\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{{\gamma }_k}}K\left(A_h,Q\left(s\right)\right)V\left(Q\left(s\right)\right){\sigma }_k\left(Q\left(s\right)\right)\text{d}s\end{array}$

where we used the symmetry of the kernel,

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_r\left(P\right)}\\ ={\displaystyle \sum _{h=1}^r}{m}_{h}V\left(A_h\right){\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(A_h,Q\left(u,v\right)\right)U\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\text{d}\sigma \end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_r\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}\\ ={\displaystyle \sum _{h=1}^r}{m}_{h}U\left(A_h\right){\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(A_h,Q\left(u,v\right)\right)V\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\text{d}\sigma \end{array}$

where we used the symmetry of the kernel,

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\gamma }_n}\left(P\right)}\\ ={\displaystyle \sum _{k=1}^n}V\left(P\left(s\right)\right){\sigma }_k\left(P\left(s\right)\right){\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(P\left(s\right),Q\left(u,v\right)\right)U\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\text{d}\sigma \text{d}s\end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}\\ ={\displaystyle \sum _{k=1}^n}U\left(P\left(s\right)\right){\sigma }_k\left(P\left(s\right)\right){\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(P\left(s\right),Q\left(u,v\right)\right)V\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\text{d}\sigma \text{d}s\end{array}$

where we used the symmetry of the kernel, and lastly

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{\Delta {\gamma }_n\left(Q\right)},V\left(P\right)\right)}_{\Delta {\gamma }_n\left(P\right)}\\ ={\displaystyle \sum _{k=1}^n}{\displaystyle \sum _{\mathcal{l}=1}^n}{\displaystyle {\int }_{{\gamma }_k}}{\displaystyle {\int }_{{\gamma }_{\mathcal{l}}}}K\left(P\left(\sigma \right),Q\left(s\right)\right)U\left(Q\left(s\right)\right)V\left(P\left(\sigma \right)\right){\sigma }_k\left(Q\left(s\right)\right){\sigma }_{\mathcal{l}}\left(P\left(\sigma \right)\right)\text{d}s\text{d}\sigma \end{array}$

$\begin{array}{l}{\left({\left(K\left(P,Q\right),U\left(Q\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(Q\right)},V\left(P\right)\right)}_{{\Delta }_{{\Sigma }_m}\left(P\right)}\\ ={\displaystyle \sum _{j=1}^m}{\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_j}}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}K\left(P\left(t,w\right),Q\left(u,v\right)\right)U\left(Q\left(u,v\right)\right)\\ \times V\left(P\left(t,w\right)\right){\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right){\tau }_{\mathcal{l}}\left(P\left(t,w\right)\right)\text{d}{\sigma }_1\text{d}{\sigma }_2\end{array}$

3. Computation of the Eigenvalues for Charged Integral Equations

The computation of the eigenvalues of second kind Fredholm integral equations is usually performed by using the Rayleig-Ritz method [1] [11] for lower bounds, and the Fichera orthogonal invariants method [11] [12] for upper bounds. An alternative procedure, called the inverse iteration method, can be used to improve the lower approximations previously obtained by means of the Rayleigh-Ritz method. This approach was already considered in [13] , and will be applied even in the present case.

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 [8] [9] [11] [12] .

4. Applications

Let $\mathcal{A}\equiv \left[\mathrm{0,}a\right]\times \left[\mathrm{0,}b\right]\times \left[\mathrm{0,}c\right]$ , $O\equiv \left(0,0,0\right),U\equiv \left(a,b,c\right)$ ,

$P\equiv \left(x_P\mathrm{,}{y}_P\mathrm{,}{z}_P\right)\mathrm{,}Q\equiv \left(x_Q\mathrm{,}{y}_Q\mathrm{,}{z}_Q\right)\mathrm{,}R\equiv \left(x_P\mathrm{,}{y}_P\mathrm{,}{z}_Q\right)\mathrm{,}S\equiv \left(x_P\mathrm{,}{y}_Q\mathrm{,}{z}_Q\right)$

$T\equiv \left(x_Q\mathrm{,}{y}_P\mathrm{,}{z}_P\right)\mathrm{,}L\equiv \left(x_Q\mathrm{,}{y}_Q\mathrm{,}{z}_P\right)\mathrm{,}M\equiv \left(x_P\mathrm{,}{y}_Q\mathrm{,}{z}_P\right)\mathrm{,}N\equiv \left(x_Q\mathrm{,}{y}_P\mathrm{,}{z}_Q\right)$

$0\le x_P\le a\mathrm{,}0\le y_P\le b\mathrm{,}0\le z_P\le c$

$0\le x_Q\le a\mathrm{,}0\le y_Q\le b\mathrm{,}0\le z_Q\le c$

$K\left(P,Q\right)=\{\begin{array}{ll}\overline{OP}\cdot \overline{UQ},\hfill & \text{if}{x}_P\le x_Q,y_P\le y_Q,z_P\le z_Q,\hfill \\ \overline{OQ}\cdot \overline{UP},\hfill & \text{if}{x}_P\ge x_Q,y_P\ge y_Q,z_P\ge z_Q,\hfill \\ \overline{OR}\cdot \overline{UL},\hfill & \text{if}{x}_P\le x_Q,y_P\le y_Q,z_P\ge z_Q,\hfill \\ \overline{OS}\cdot \overline{UT},\hfill & \text{if}{x}_P\le x_Q,y_P\ge y_Q,z_P\ge z_Q,\hfill \\ \overline{OT}\cdot \overline{US},\hfill & \text{if}{x}_P\ge x_Q,y_P\le y_Q,z_P\le z_Q,\hfill \\ \overline{OL}\cdot \overline{UR},\hfill & \text{if}{x}_P\ge x_Q,y_P\ge y_Q,z_P\le z_Q,\hfill \\ \overline{OM}\cdot \overline{UN},\hfill & \text{if}{x}_P\le x_Q,y_P\ge y_Q,z_P\le z_Q,\hfill \\ \overline{ON}\cdot \overline{UM},\hfill & \text{if}{x}_P\ge x_Q,y_P\le y_Q,z_P\ge z_Q,\hfill \end{array}$ (1)

i.e.

Obviously $K\left(P,Q\right)=K\left(Q,P\right)$ , i.e. the kernel is symmetric, and even positive definite, since $K\left(P,Q\right)>0$ , $\forall \left(P\mathrm{,}Q\right)\in \mathcal{A}$ .

Consider in $L_{dM}^2\left(\mathcal{A}\right)$ the eigenvalue problem

$\mathcal{K}\phi =\mu \phi$ (2)

$\mathcal{K}\phi \mathrm{<mo>\; :\; </mo>}={\displaystyle {\int }_{\mathcal{A}}}K\left(\cdot \mathrm{,}Q\right)\phi \left(Q\right)\text{d}{M}_Q$ (3)

where

$\begin{array}{c}\text{d}{M}_Q=\rho \left(Q\right)\text{d}Q+{\displaystyle \sum _{h=1}^r}{m}_h\delta \left(A_h\right)+{\displaystyle \sum _{k=1}^n}{\displaystyle {\int }_{{\gamma }_k}}{\sigma }_k\left(Q\left(s\right)\right)\delta \left(Q\left(s\right)\right)\text{d}s\\ +{\displaystyle \sum _{\mathcal{l}=1}^m}{\displaystyle {\int }_{{\Sigma }_{\mathcal{l}}}}{\tau }_{\mathcal{l}}\left(Q\left(u,v\right)\right)\delta \left(Q\left(u,v\right)\right)\text{d}\sigma \end{array}$

( $\delta$ 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 $\mathcal{A}$ .

Numerical Example 1

Let $a=b=c=1$ , and $\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z$ with $\rho \left(x,y,z\right)=1+2x+3y+4z$ (see Figure 1). Under this assumption, the apprpximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 1, whereas the relevant eigenfunctions are shown in Figure 2.

Numerical Example 2

Let $a=b=c=1$ , and $\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z$ with $\rho \left(x,y,z\right)=8{\text{π}}^3{\text{e}}^{-{\left(x+y+z\right)}^2}\sin \left(\text{π}xyz\right)$ (see Figure 3). Under this assumption, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 2, whereas the relevant eigenfunctions are shown in Figure 4.

Numerical Example 3

Let $a=b=c=1$ , and

$\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z+\delta \left(x-\frac{1}{2}\right)\delta \left(y-\frac{1}{2}\right)\delta \left(z-\frac{1}{2}\right)\text{d}x\text{d}y\text{d}z$

with $\rho \left(x,y,z\right)={\text{e}}^{-x-y-z}$ (see Figure 5). Under this assumption, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 3, whereas the relevant eigenfunctions are shown in Figure 6.

Numerical Example 4

Let $a=b=c=1$ , and $\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z+{\displaystyle {\sum }_h}{m}_h\delta \left(x-x_h\right)\delta \left(y-y_h\right)\delta \left(z-z_h\right)\text{d}x\text{d}y\text{d}z$ with $\rho \left(x,y,z\right)=2\log \left(1+z\left(1+y\left(1+x\right)\right)\right)$ , and $m_h=1/\left(x_h+y_h+z_h\right)$ ,

$x_h=y_h=z_h=\frac{1}{H}\left(h-\frac{1}{2}\right)$ for $h=1,2,3=H$ (see Figure 7). Under this assump-

tion, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 4, whereas the relevant eigenfunctions are shown in Figure 8.

Numerical Example 5

Let $a=b=c=1$ , and $\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z+{\displaystyle {\sum }_h}{m}_h\delta \left(x-x_h\right)\delta \left(y-y_h\right)\delta \left(z-z_h\right)\text{d}x\text{d}y\text{d}z$ with

$\rho \left(x,y,z\right)=1+\cos \left[\frac{\text{π}}{2}\left(1-2x-3y-4z\right)\right]$ , and $m_h=3x_h+2y_h+z_h$ ,

$x_1=x_2=x_5=x_6=y_1=y_3=y_5=y_7=z_1=z_2=z_3=z_4=\frac{1}{4}$ ,

$x_3=x_4=x_7=x_8=y_2=y_4=y_6=y_8=z_5=z_6=z_7=z_8=\frac{3}{4}$ for $h=1,2,\cdots ,8=H$

(see Figure 9). Under this assumption, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 5, whereas the relevant eigenfunctions are shown in Figure 10.

Numerical Example 6

Let $a=b=c=1$ , and $\begin{array}{c}\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z+\sigma \left(x,y,z\right)\delta \left(z-{\zeta }_s\right)\text{d}x\text{d}y\text{d}z\\ +{\displaystyle {\sum }_h}\delta \left(x-x_h\right)\delta \left(y-y_h\right)\delta (z-z_h)\text{d}x\text{d}y\text{d}z\end{array}$ with $\rho \left(x,y,z\right)=\left[\cosh \left(2y-x\right)+\sin \left(2x-y\right)\right]/\left(1+{\text{e}}^{x+y}\right)$ , $\sigma \left(x,y,z\right)={\text{e}}^{-2x-3y+4z}$ , ${\zeta }_s=\frac{1}{4}$ , and $x_h=1-y_h=\frac{1}{H}\left(h-\frac{1}{2}\right)$ , $z_h=\frac{4}{5}$ for $h=1,2=H$ (see Figure 11).

Under this assumption, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 6, whereas the relevant eigenfunctions are shown in Figure 12.

Numerical Example 7

Let $a=b=c=1$ , and $\begin{array}{c}\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z+\sigma \left(x,y,z\right)\delta \left(x-{\xi }_s\right)\text{d}x\text{d}y\text{d}z\\ +{\displaystyle {\sum }_h}{m}_h\delta \left(x-x_h\right)\delta \left(y-y_h\right)\delta \left(z-z_h\right)\text{d}x\text{d}y\text{d}z\end{array}$ with $\rho \left(x,y,z\right)=\text{sech}\left(4{\left[{\left(x-1/2\right)}^2+{\left(y-1/2\right)}^2+{\left(z-1/2\right)}^2\right]}^{1/2}\right)$ , $\sigma \left(x,y,z\right)=5x+\sin \left(2y\right)\cos \left(3z\right)$ , ${\xi }_s=\frac{1}{5}$ , and $m_h=y_h+z_h$ , $x_h=\frac{3}{4}$ ,

$y_h=z_h=\frac{1}{H}\left(h-\frac{1}{2}\right)$ for $h=1,2,3=H$ (see Figure 13). Under this assumption, the approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 7, whereas the relevant eigenfunctions are shown in Figure 14.

Numerical Example 8

Let $a=b=c=1$ , and

$\begin{array}{c}\text{d}M=\rho \left(x\mathrm{,}y\mathrm{,}z\right)\text{d}x\text{d}y\text{d}z+\sigma \left(x\mathrm{,}y\right)\delta \left(z-{\zeta }_s\left(x\mathrm{,}y\right)\right)\text{d}x\text{d}y\text{d}z\\ +\delta \left(x-\frac{3}{4}\right)\delta \left(y-\frac{3}{4}\right)\delta (z-\frac{3}{4})\text{d}x\text{d}y\text{d}z\end{array}$

with $\rho \left(x,y,z\right)=\left[3+2\cos \left(xz\right)+\sin \left(yz\right)\right]/\left(1+2x+3y+4z\right)$ , and

$\sigma \left(x,y\right)=\left(1+2x+3y\right)/\left[7+cos\left(6y\right)+sin\left(5x\right)\right]$ ,

${\zeta }_s\left(x,y\right)=\frac{1}{2}\left[1+\cos \left(x\right)-\sin \left(3y\right)\right]$ (see Figure 15). Under this assumption, the

approximate eigenvalues evaluated by using the Rayleigh-Ritz and inverse iteration methods are listed in Table 8, whereas the relevant eigenfunctions are shown in Figure 16.

Numerical Example 9

Let $a=b=c=1$ , and

$\begin{array}{c}\text{d}M=\rho \left(x,y,z\right)\text{d}x\text{d}y\text{d}z+\sigma \left(x,y\right)\delta \left(z-{\zeta }_s\left(x,y\right)\right)\text{d}x\text{d}y\text{d}z\\ +\tau \left(z\right)\delta \left(x-{\xi }_l\left(z\right)\right)\delta (y-{\eta }_l\left(z\right))\text{d}x\text{d}y\text{d}z\\ +{\displaystyle {\sum }_h}{m}_h\delta \left(x-x_h\right)\delta \left(y-y_h\right)\delta \left(z-z_h\right)\text{d}x\text{d}y\text{d}z\end{array}$

with $\rho \left(x,y,z\right)=4+3\cos \left(3x\right)+2\sin \left(2y\right)+\tan \left(z\right)$ , $\sigma \left(x,y\right)=2\log \left(1+y+xy\right)$ , ${\zeta }_s\left(x,y\right)=\frac{1}{4}\left(1-x+2y\right)$ , $\tau \left(z\right)=2z$ , ${\xi }_l\left(z\right)={\eta }_l\left(z\right)=4z-3$ , and

$m_h=x_h+y_h+z_h$ , $x_1=x_2=y_1=y_3=\frac{1}{4}$ , $x_3=x_4=y_2=y_4=\frac{3}{4}$ , $z_h=\frac{2}{3}$ for

$h=1,2,3,4=H$ (see Figure 17). Under this assumption, the approximate eigenvalues are evaluated as in Table 9, whereas the relevant eigenfunctions are shown in Figure 18.

Acknowledgements

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.

Cite this paper

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

References