Quadratic Approximation for Singular Integrals

In this study we present a new approach based on a quadratic approximation for singular integrals of Cauchy type, by using a small technica l we arrive to eliminate completely the singularity of this integral. Noting that, this approximation is d estined to solve numerically all singular integral equations with Cauchy kernel type on an oriented sm ooth contour.


INTRODUCTION
Many problems of mathematical physics, engineering and contact problems in the theory of elasticity lead to singular integral equations with Cauchy kernel type: Eq. 1: where, Γ designates an oriented smooth contour, the points t and t 0 are on Γ. The Eq. 1 plays an important role in modern numerical computations in the applied sciences, in particular in the applied mathematical.
Our schemes describe the quadratic method for the approximation of singular integral operator with Cauchy kernel Eq. 2: by a sequence of numerical integration operators. Noting that, for the existence of the principal value of the integral (2) for a given density ϕ(t) we will need more than mere continuity. In other words, the density ϕ(t) has to satisfy the Hölder condition H(µ) (Muskhelishvili, 2008).
The function ϕ(t) will be said to satisfy a Hölder condition on Γ if for any two points t 1 and t 2 of Γ: where, A is positive constant, called the Holder constant and µ the Holder index.  designates the smallest arc with ends t a and t a+1 (Nadir, 1985;1998;Nadir and Antidze, 2004;Nadir and Lakehali, 2007;Sanikidze, 1970;Antidze, 1975).
For t ok ≤t≤ σ(k+2) Eq. 3: This spline function exists and is unique also called a quadratic interpolating polynomial.
Define for an arbitrary numbers σ and v such that 0≤σ,v≤N-1 the function β σv (ϕ,t,t 0 ) dependents of ϕ,t and t 0 by Eq. 4: where, t∈[t σ , t σ+1 ] and t 0 ∈ [t v ,t v+1 ] and noting by Ψ σv (ϕ,t,t 0 ) the quadratic approximation of the density ϕ(t) at the point t∈[t σ ,t σ+1 ] Eq. 5: Replacing the density ϕ(t) by this last quadratic approximation Ψ σv (ϕ, t, t 0 ) in the singular integral (2): In order to obtain the following approximation Eq. 6: Theorem: Let Γ be an oriented smooth contour and let ϕ be a density function defined on Γ and satisfies the Hölder condition H(µ) then, the following estimation: Holds for all t and t 0 on the contour Γ. The constant C depends only of Γ.
Proof: Choosing the points t∈[t σ ,t σ+1 ] and t 0 ∈[t v ,t v+1 ] then for t σk ≤t≤t σ(k+2) and t ik ≤ t 0 ≤ t v(k+2) we have Eq. 7: Taking into account the expression (7) we get Eq. 8: Hence: It can be seen that, the equality t-t 0 = 0 is possible only when σ = v-1,v+1 and v For the two first cases the integral (8) exists when t tends to t 0 . The last case, if σ = v we can easily seeing that, the function β σσ (ϕ, t,t 0 ) contains (t-t 0 ) as factor this means, for the points t,t 0 ∈ [t σ ,t σ+1 ] we write Eq. 9: Hence, for t σk ≤t,t 0 ≤ t σ(k+2) : Passing now to the estimation of the expression (8): For t∈[t σ ,t σ+1 ] and t 0 ∈ [t v , t v+1 ] with the conditions σ≠v-1,v+1 and v we obtain: Indeed, it is clear that: ( 2 k 2 ) Further, for the three cases mentioned above σ = v-1,v+1 and v using the smoothness of the contour Γ and the function ϕ (t) in the Hölder space H (µ) we get: Example 1: Consider the singular integral: where, Γ designates the circle centered at the origin point 0 with a unit radius and the density function ϕ is given by the following expression: where, Γ designates the circle centered at the origin point 0 with a unit radius and the density function ϕ is given by the following expression: 2 t 1 (t) t 5t 6 + ϕ = + + Numerical experiments: Using our approximation, we apply the algorithms to singular integrals and we present results concerning the accuracy of the calculations, in these numerical experiments each Table  1 and 2 represent the exact principal value of the singular integral and I p corresponds to the approximate calculation produced by our approximation at interpolation point's values.

CONCLUSION
This approximation can be used to remove integrable singularity. It was tested for the numerical calculus of many singular integrals of Cauchy type, where it gave good results.