Application of Jacobi polynomials to approximate solution of a complete singular integral equation with Cauchy kernel on the real half-line

Abstract

SummaryJacobi polynomials are used to derive approximate solutions of the complete singular integral equation with Cauchy-type kernel defined on the real half-line in the case of constant complex coefficients. Moreover, estimations of errors of the approximated solutions are presented and proved.

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“…the class of Hölder continuous functions on (−1, 1), bounded in a neighbourhood of the point z = 1, and admitting an integrable singularity at the point z = −1), then for 0 < θ < π we have (cf. [34]) (8) Z (t) = a 2 − b 2 (1 − t) α (1 + t) β , α = ω 1 + iω 2 , β = −ω 1 − iω 2 , and for −π < θ < 0 we obtain 1), bounded at neighbourhoods of points z = ±1), then for 0 < θ < π we have (cf. [34]) (10) Z (t) = a 2 − b 2 (1 − t) α (1 + t) β , α = ω 1 + iω 2 , β = 1 − ω 1 − iω 2 , and for −π < θ < 0 we obtain (11) Z (t) = − a 2 − b 2 (1 − t) α (1 + t) β , α = 1 + ω 1 + iω 2 , β = −ω 1 − iω 2 .…”

“…(6) to the right-hand side and then solving it as a dominant equation in h(1) class (cf. [34]), after a few elementary transformations Eq. (6) can be reduced to the Fredholm equation of the form (12) u(t) + (12) is equal to (13) u(t) = F (t) − where Γ(t, τ ) is the resolvent of the kernel N (t, τ ).…”

“…in [9,34]. Moreover, in [34] Jacobi polynomials were applied to derive approximate solutions of (3).…”

“…in [9,34]. Moreover, in [34] Jacobi polynomials were applied to derive approximate solutions of (3). The main objective of the present paper is to build the approximate solutions of (2) in the h(∞) and h(0, ∞) function classes.…”