Abstract
In this study we present a new approach based on a quadratic approximation for singular integrals of Cauchy type, by using a small technical we arrive to eliminate completely the singularity of this integral. Noting that, this approximation is destined to solve numerically all singular integral equations with Cauchy kernel type on an oriented smooth contour.
Highlights
T0 )φ(t)dt where, Γ designates an oriented smooth contour, the points t and t0 are on Γ
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: ∫ ∫ a(t0 )φ(t0 ) + b(t0 πi )φ(t) dt + Γ t − t0 k(t, Γ t0 )φ(t)dt = f (t0 (1)where, Γ designates an oriented smooth contour, the points t and t0 are on Γ
For an arbitrary number σ = 0,1,2,...N-1 we define the spline function S2(φ,t,σ) dependents of φ, t and σ which represents the quadratic approximation of the function density φ(t) on the subinterval [tσ, tσ+1] of the curve Γ
Summary
T0 )φ(t)dt where, Γ designates an oriented smooth contour, the points t and t0 are on Γ. Fixing a natural number M>1 and divide each of segments [sσ, sσ+1] by the equidistant points: by a sequence of numerical integration operators. The function φ(t) will be said to satisfy a Hölder condition on Γ if for any two points t1 and t2 of Γ: sσk
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