In this paper, we develop a method for evaluating one dimensional singular integrals (weakly, strongly, and hyper-singular) that converge in the sense of Cauchy principal value and Hadamard finite part integrals. A proof of convergence of this method is also provided.
Many problems in engineering and science require evaluating singular integrals. The problem considered in this paper is of practical interest in many areas. For example, in electromagnetic and acoustic wave scattering, the boundary integral equations have singular kernels, see [
The one dimensional singular integrals are defined in the literature as follows
∫ a b u ( t ) ( t − s ) p d t , s ∈ ( a , b ) , p > 0 , (1)
in which u ( t ) is a continuous function. These integrals are classified by the order of singularity. If p < 1 , the integral is called weakly singular. If p = 1 , the integral is strongly singular. If p > 1 , the integral is called hyper-singular, see [
There are many special methods developed to treat singular integral problems since numerical integration routines often lead to inaccurate solutions. For example, to deal with the singularities in surface integral equations, the method of moments regularizes the singular integrals by sourcing them analytically for specific observation point [
transformation which cancels the singularity of type 1 t [
methods can be characterized in three categories: singularity subtraction, analytical transformation, and special purpose quadrature.
In this paper, we present an alternative approach for evaluating one dimensional singular integrals (weakly, strongly, and hyper-singular) which converge in the sense of Cauchy principal value and Hadamard finite part integrals. In addition, a proof of convergence of this method is outlined in Section 2 to serve as a theoretical basis for the method. In Section 3, the detailed implementations of our method are described for integrals over the standard interval [−1,1] and arbitrary interval [a,b].
Let S = ∫ D f ( x ) d x be a singular integral. Without loss of generality, the
singularity can be assumed to be at zero. We assume that the integral S has a finite value in the sense of principal value or Hadamard finite part integral, or in other words, this integral converges.
Let D ϵ : = D \ B ϵ ( 0 ) , where B ϵ ( 0 ) : = { x : | x | < ϵ } , where ϵ > 0 . Then there exists numbers η ( n , ϵ ) and a j ( ϵ ) ,0 ≤ j ≤ n , such that
| ∫ D ϵ f ( x ) d x − ∑ j = 0 n a j ( ϵ ) ∫ D ϵ U j ( x ) d x | ≤ η ( n , ϵ ) → 0 , as n → ∞ , (2)
where U j ,0 ≤ j ≤ n , are Chebyshev polynomials of second kind
U j ( x ) = sin ( ( j + 1 ) cos − 1 ( x ) ) sin ( cos − 1 ( x ) ) , 0 ≤ j ≤ n . (3)
Since the integral S converges, l i m ϵ ↓ 0 | ∫ D ϵ f ( x ) d x − S | = 0 . Fix ϵ 0 > 0 , there
exists a δ ( ϵ 0 ) such that
| ∫ D ϵ 0 f ( x ) d x − S | < δ ( ϵ 0 ) ∀ ϵ ≤ ϵ 0 . (4)
For a fixed ϵ = ϵ 0 , there exists an n such that
| ∫ D ϵ 0 f ( x ) d x − ∑ j = 0 n a j ( ϵ 0 ) ∫ D ϵ 0 U j ( x ) d x | < δ ( ϵ 0 ) . (5)
Thus,
| S − ∑ j = 0 n a j ( ϵ 0 ) ∫ D ϵ 0 U j ( x ) d x | < 2 δ ( ϵ 0 ) . (6)
That means S can be approximated by ∑ j = 0 n a j ( ϵ 0 ) ∫ D ϵ 0 U j ( x ) d x .
Let us summarize the result.
Theorem 1. Let S = ∫ D f ( x ) d x be a singular integral that converges. Suppose x 0 is its singularity in D . Then, for any ϵ > 0 , there exists δ ( ϵ ) > 0 , n , and a j ( ϵ ) ,0 ≤ j ≤ n , such that
| S − ∑ j = 0 n a j ( ϵ ) ∫ D ϵ U j ( x ) d x | < δ ( ϵ ) , D ϵ : = D \ B ϵ ( x 0 ) . (7)
In this section, we present a method for evaluating singular integral of the type
S = ∫ − 1 1 f ( x ) d x . (8)
For a more general integral, we will extend this method in Section 3.2.
From Section 2, we need to find the coefficient a j such that
S ≃ ∑ j = 0 n − 1 a j ∫ − 1 1 U j ( x ) d x , (9)
or
f ( x ) ≃ ∑ j = 0 n − 1 a j U j ( x ) . (10)
Since U j are Chebyshev polynomials of second kind, they admit some nice properties
1. ∫ − 1 1 U j ( x ) d x = 2 sin 2 ( j π 2 ) j + 1 , (11)
2. ∑ k = 1 n U j ( x k ) U i ( x k ) ( 1 − x k 2 ) = n 2 δ i j , ∀ i , j < n , (12)
where x k ,1 ≤ k ≤ n , are the roots of U n ( x ) in [ − 1,1 ]
x k = cos ( k π n + 1 ) , 1 ≤ k ≤ n . (13)
Now consider the following summation with 0 ≤ i ≤ n − 1
∑ k = 1 n f ( x k ) U i ( x k ) ( 1 − x k 2 ) = ∑ k = 1 n ∑ j = 0 n − 1 a j U j ( x k ) U i ( x k ) ( 1 − x k 2 ) (14)
= ∑ j = 0 n − 1 a j ∑ k = 1 n U j ( x k ) U i ( x k ) ( 1 − x k 2 ) (15)
= ∑ j = 0 n − 1 a j n 2 δ i j (16)
= n 2 a i (17)
Thus, the coefficient a i can be computed by
a i = 2 n ∑ k = 1 n f ( x k ) U i ( x k ) ( 1 − x k 2 ) , 0 ≤ i ≤ n − 1 , (18)
and
S ≃ ∑ i = 0 n − 1 a i ∫ − 1 1 U i ( x ) d x = ∑ i = 0 n − 1 a i 2 sin 2 ( i π 2 ) i + 1 . (19)
In this section, we extend the method presented in Section 3.1 to singular integrals over arbitrary interval [ a , b ]
S = ∫ a b f ( x ) d x . (20)
Note that
∫ a b U i ( x ) d x = b − a 2 ∫ − 1 1 U i ( y ) d y . (21)
Therefore, we just need to recompute the coefficients a i in this formula
S ≃ ∑ i = 0 n − 1 a i ∫ a b U i ( x ) d x = b − a 2 ∑ i = 0 n − 1 a i ∫ − 1 1 U i ( y ) d y . (22)
Let x ∈ [ a , b ] and y ∈ [ − 1,1 ] . Define
y k : = cos ( k π n + 1 ) , 1 ≤ k ≤ n , (23)
the roots of U n ( y ) in [ − 1,1 ] . Then, using this change of variable formula to find the corresponding collocation points x k in [ a , b ]
x k = b − a 2 y k + b + a 2 , 1 ≤ k ≤ n . (24)
Let
f ( x k ) ≃ ∑ j = 0 n − 1 a j U j ( y k ) , 1 ≤ k ≤ n . (25)
Then the coefficients a i can be computed as follows. Consider the following summation with 0 ≤ i ≤ n − 1
∑ k = 1 n f ( x k ) U i ( y k ) ( 1 − y k 2 ) = ∑ k = 1 n ∑ j = 0 n − 1 a j U j ( y k ) U i ( y k ) ( 1 − y k 2 ) (26)
= ∑ j = 0 n − 1 a j ∑ k = 1 n U j ( y k ) U i ( y k ) ( 1 − y k 2 ) (27)
= ∑ j = 0 n − 1 a j n 2 δ i j (28)
= n 2 a i (29)
Thus
a i = 2 n ∑ k = 1 n f ( x k ) U i ( y k ) ( 1 − y k 2 ) , 0 ≤ i ≤ n − 1. (30)
Finally,
S = ∫ a b f ( x ) d x (31)
≃ ∑ i = 0 n − 1 a i ∫ a b U i ( x ) d x (32)
= b − a 2 ∑ i = 0 n − 1 a i ∫ − 1 1 U i ( y ) d y (33)
= b − a 2 ∑ i = 0 n − 1 a i 2 sin 2 ( i π 2 ) i + 1 . (34)
In this paper, we investigated one dimensional singular integrals. A method is developed for evaluating weakly, strongly, and hyper singular integrals which converge in the sense of Cauchy principal value and Hadamard finite part integrals. The convergence of this method is proved and the detailed implementations are provided. One of the advantages of this method is that it is simple and relatively easy to implement. This method can serve as an alternative approach to other special methods in the literature.
Tran, N.T. (2017) Simple Method for Evaluating Singular Integrals. American Journal of Computational Mathematics, 7, 444-450. https://doi.org/10.4236/ajcm.2017.74032