Orthogonal Approximate Solution of Cauchy-type Singular Integral Equations

Abstract

-Chebyshev polynomials of the first and second kind are used to derive approximate solutions of the Cauchy-type singular integral equations of the formThis result extends the well-known numerical solution by Jacobi polynomials of L.N.

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“…As follows from the above-performed considerations, the latter is equivalent to the solvability of the equation (40) (1 − t) α u n (t) + N n (t, τ )(1 − τ ) α u n (τ ) dτ = F n (t), Re α > 0, where Since Z (t) = ± √ a 2 − b 2 (1 − t) α (1 + t) β , 0 < Reα, Reβ < 1, in the h(−1, 1) class, it follows from [39] that N * (t, τ ) − N * n−1 (t, τ ) ≤ M 2 ln 3 n n µ . Thus, for sufficiently large n, the equation Z(t)u n−1 (t) + N * (t, τ )−N * n−1 (t, τ ) dτ ≤ M ln 3 n n µ , ε 3 = F * (t) − F * n (t) ∞ M ln 2 n n µ , we finish the proof of estimate (41).…”

“…Then the system of linear equations (35) is solvable for sufficiently large n and the estimate (41) Z(t)(u(t) − u n−1 (t)) ∞ ≤ M ln 3 n n µ , holds, where M is a constant independent of n.…”

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

“…N * (t, τ )−N * n−1 (t, τ ) dτ ≤ M ln 3 n n µ , ε 3 = F * (t) − F * n (t) ∞ M ln 2 n n µ , we finish the proof of estimate (41).…”

“…Then the system of linear equations (35) is solvable for sufficiently large n and the estimate (41) Z(t)(u(t) − u n−1 (t)) ∞ ≤ M ln 3 n n µ , holds, where M is a constant independent of n. Since Z (t) = ± √ a 2 − b 2 (1 − t) α (1 + t) β , 0 < Reα, Reβ < 1, in the h(−1, 1) class, it follows from [39] that N * (t, τ ) − N * n−1 (t, τ ) ≤ M 2 ln 3 n n µ . Thus, for sufficiently large n, the equation Z(t)u n−1 (t) + 1 −1 N * n−1 (t, τ )Z(τ )u n−1 (τ ) dτ = F * n (t), −1 < t < 1,…”

A Singular Integral Equation with a Cauchy Kernel on the Real Half-Line

“…By substituting (3.10) into (3.7) and by taking into account formula (2.2) in [6], namely, the formula…”