On a method for solving Prandtl's integro-differential equation applied to problems of continuum mechanics using polynomial approximations

Abstract

In the present paper, using the ideas of the well-known method for solving singular integral equations (SIE), based on Gaussian quadrature formulas for ordinary integrals and singular integral with Cauchy kernel, a somewhat different approach is proposed for solving the Prandtl integro-differential equation (IDE). This approach in accordance with a polynomial method includes different approximation methods for a continuous component of Prandtl's IDE solution, namely an approximation in the form of Bernstein's polynomials, Chebyshev's polynomials of the first kind and approximation in the form of Lagrange interpolation polynomial at the Chebyshev knots. As a result, the Prandtl IDE is reduced to the system of linear algebraic equations (SLAE) of the rather simple structure. These different approximation methods will be illustrated by the example of the problem on contact interaction between a stringer of finite length, having a variable along the length rigidity in tension-compression or a variable cross-section, and an elastic semi-infinite plate. This interaction is described by the Prandtl IDE. The comparative numerical analysis of the results obtained by different methods is carried out.