Quadrature Formulas Applicable to the Solution of Integral Equations on a Semi-Infinite Interval by the Mechanical Quadrature Method

Technical Summary: Quadrature Formulas for Integral Equations on a Semi-Infinite Interval

Problem Statement
The paper addresses the challenge of numerically solving integral equations defined on a semi-infinite interval $(0, \infty)$, specifically those arising in the mathematical theory of elasticity. While Gaussian-type quadrature formulas are well-established for finite intervals, their application to semi-infinite domains is complicated by the behavior of the solution at infinity. Existing methods based on Laguerre polynomials impose an exponential decay condition ($x^\alpha e^{-x}$) on the solution, which is often physically inaccurate for elasticity problems where the solution typically decays as a power function. The authors aim to construct a mechanical quadrature method capable of handling integrals with power-law weight functions, including regular integrals, Cauchy-type singular integrals, and integrals with logarithmic and step-function kernels.

Methodology
The core of the methodology involves constructing a new family of rational functions, $R^{(p,q)}_n(x)$, orthogonal on the ray $(0, \infty)$ with respect to the weight function $(1+x)^{-p}x^{-q}$. These functions are derived from Jacobi polynomials $P^{(p+q-2, -q)}_n$ via a linear fractional transformation mapping the interval $(-1, 1)$ to $(0, \infty)$.

The authors proceed through the following steps:

1. Interpolation: A continuous, bounded function $f(x)$ is interpolated using the rational functions $R^{(p,q)}_n(x)$.
2. Quadrature Derivation:
   • Regular Integrals: By substituting the interpolation formula into the integral and utilizing orthogonality conditions, a quadrature formula with nodes $z_i$ (roots of the rational polynomial) and specific weights $w_i$ is derived. This formula is exact for polynomials in $\frac{x-1}{x+1}$ of degree up to $2n-1$.
   • Cauchy-Type Integrals: Using the identity for partial fractions and the properties of the rational functions, a quadrature formula for the Cauchy principal value integral is constructed.
   • Second-Kind Singular Operators: The method is adapted to the characteristic part of second-kind singular integral equations. The authors determine the exponents $p$ and $q$ based on the boundedness of the solution near the origin and the specific kernel structure, allowing the singular operator to be discretized.
   • Weak Singularities: Formulas for integrals involving logarithmic kernels ($\ln|x-y|$) and step functions ($\text{sign}(x-y)$) are derived using the Christoffel–Darboux formula and specific integral representations.

Application and Validation
To validate the derived formulas, the authors apply the mechanical quadrature method to a two-dimensional elasticity problem: the deformation of an elastic half-plane.

• Physical Setup: The half-plane occupies $y \le 0$ and is rigidly fixed along the semi-infinite boundary $x > 0$. A uniformly distributed normal load acts on a finite segment of the free boundary.
• Mathematical Reduction: The boundary conditions reduce the problem to a second-kind singular integral equation on the semi-infinite interval. The unknown contact stresses are sought in a form consistent with the derived weight functions.
• Discretization: The integral equation is converted into a system of linear algebraic equations using the proposed quadrature nodes and weights. The system is solved for the unknown stress coefficients.
• Post-Processing: Once the stress distribution is determined, the formulas for integrals with logarithmic and step kernels are used to calculate the displacement components and reconstruct the deformed shape of the boundary.

Key Results

• Convergence: Numerical analysis demonstrates excellent convergence of the algorithm. For a Poisson ratio $\nu = 0.27$ and a load parameter $a=1$, the solution stabilizes rapidly, with $n=6$ nodes often being sufficient for high accuracy. Even for small values of $a$ (where the load is close to the fixed region), increasing the order to $n=20$ yields stable results.
• Accuracy: The computed values of the stress function $\phi^*(x)$ at various points show remarkable stability as the approximation order $n$ increases, confirming the high algebraic degree of precision of the proposed quadrature rules.
• Physical Insight: The method successfully determines the contact stresses and the deformed boundary shape of the elastic half-plane, illustrating the effectiveness of the approach for practical elasticity problems.

Significance and Claims
The authors claim that this work presents, "apparently for the first time," quadrature formulas specifically designed for evaluating integrals with power-law weight functions on a semi-infinite interval. Unlike previous approaches relying on Laguerre polynomials, this method accommodates the power-law decay typical of elasticity problems without imposing unrealistic exponential decay. The paper establishes a unified framework where the same set of nodes and weights can be applied to regular integrals, Cauchy-type singular integrals, and integrals with weak singularities (logarithmic and step kernels). The successful application to the contact problem of a fixed elastic half-plane serves as a rigorous validation of the theoretical derivations.