The authors present a numerical approach for solving integral equations on non-smooth domains. They consider the biharmonic Dirichlet problem \[ \Delta^2U|_D= 0, \quad U|_\Gamma= f_{1}, \quad \frac{\partial U} {\partial n} \Biggl|_\Gamma= f_{2} ,\tag{1} \] where \(D\) is a two-dimensional domain with simple closed piecewise smooth contour \(\Gamma \) and \(\frac{\partial U}{\partial n}\) is the outward normal derivative to that contour [P. Grisvard, Elliptic problems in nonsmooth domains (1985; Zbl 0695.35060)]. The solution \(U\) is sought in \(W_p^1 (\overline{D})\cap W_p^4 (D)\), where \(W_p^k (X)\) is the Sobolev space of \(k\)-times differentiable functions on \(X\) the derivatives of which belong to \(L_{p}(X)\). Reducing the initial problem to a boundary value problem for analytic functions in the same domain \(D\), the authors arrive to the integral equation of N. I. Muskhelishvili [Singular integral equations. 3rd ed. (1968; Zbl 0174.16202)]. The Muskhelishvili equation is an integral equation without critical geometry and the integral operators of which can be represented as elements of an algebra of Mellin operators with conjugation. Since the corresponding integral operator is non-invertible on the space of work, the authors correct it in such a way that the newly obtained operator is invertible and the solution of the associated integral equation is a solution of the Muskhelishvili equation.
Finally, the paper presents an extensive investigation of spline Galerkin method for the auxiliary integral equation and gives a stable and convergent approximate solution for the Muskhelishvili equation.
For the entire collection see [Zbl 1005.00052].