Numerical computation of Muskhelishvili's integral equation in plane elasticity

Approximation methods for the Muskhelishvili equation on smooth curves

“…Then there exists an integer n 0 such that equations (20) are solvable for all n ≥ n 0 , and the sequence {x n } n≥n 0 converges to a solution of equation (3) in the norm of C (Γ).…”

“…42 -43], [17], [19,Chapter 4], that equation (3) is closely connected to different problems of elasticity theory and the theory of slow viscous flows. In fact, a number of boundary value problems for the biharmonic operator in the domain D can be reduced to equation (3). This makes equation (3) a very valuable object for application of different numerical procedures because it allows us to essentially reduce computational costs while considering the corresponding problems for partial differential equations.…”

“…Therefore, no popular projection method such as Galerkin, collocation, qualocation and so on can be applied to equation (3) directly. Let us mention here that the Muskhelishvili equation was conceived as a tool for studying the following boundary value problem, see [19,Chapter 4], (6) ψ (t) + φ (t) +tφ (t) = f (t), t ∈ Γ, for two analytic functions φ and ψ in D. Therefore most of the known numerical methods for (3) are concerned with the approximate solution of the boundary value problem (6), [1], [2], [17].…”