An asymptotic model for solving mixed integral equation in some domains

In this paper, we discuss the solution of mixed integral equation with generalized potential function in position and the kernel of Volterra integral term in time. The solution will be discussed in the space $$L_{2} (\Omega ) \times C[0,T],$$ L 2 ( Ω ) × C [ 0 , T ] , $$0 \le t \le T < 1$$ 0 ≤ t ≤ T < 1 , where $$\Omega$$ Ω is the domain of position and $$t$$ t is the time. The mixed integral equation is established from the axisymmetric problems in the theory of elasticity. Many special cases when kernel takes the potential function, Carleman function, the elliptic function and logarithmic function will be established.

The Hodge Laplacian on axisymmetric domains and its discretization

We study the mixed formulation of the abstract Hodge Laplacian on axisymmetric domains with general data through Fourier finite element methods (Fourier-FEMs) in weighted function spaces. Closed Hilbert complexes and commuting projectors are used as in the work of Arnold, Falk & Winther, (2010, Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.), 47, 281–354), by using the new family of finite element spaces for general axisymmetric problems introduced in Oh, (2015, de Rham complexes arising from Fourier-FEMs in axisymmetric domains. Comput. Math. Appl., 70, 2063–2073). In order to get stability results and error estimates for the discrete mixed formulation, we construct commuting projectors that can be applied to functions with low regularity.

Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.

Annular and circular rigid inclusions planted into a penny-shaped crack and factorization of triangular matrices

Analytical solutions to two axisymmetric problems of a penny-shaped crack when an annulus-shaped (model 1) or a disc-shaped (model 2) rigid inclusion of arbitrary profile are embedded into the crack are derived. The problems are governed by integral equations with the Weber–Sonine kernel on two segments. By the Mellin convolution theorem, the integral equations associated with models 1 and 2 reduce to vector Riemann–Hilbert problems with 3 × 3 and 2 × 2 triangular matrix coefficients whose entries consist of meromorphic and plus or minus infinite indices exponential functions. Canonical matrices of factorization are derived and the partial indices are computed. Exact representation formulae for the normal stress, the stress intensity factors (SIFs) at the crack and inclusion edges, and the normal displacement are obtained and the results of numerical tests are reported. In addition, simple asymptotic formulae for the SIFs are derived.

Singular integrals in axisymmetric problems of elastostatics

Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper. These equations are received after applying the integral transformation and Gauss–Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium. Initially, three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate. So, the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only. The thorough analysis of both displacement and traction kernels is accomplished, and similarity in behavior of both kernels is established. The kernels are expressed in terms of complete elliptic integrals of first and second kinds. The second kind elliptic integrals are nonsingular, and standard Gaussian quadratures are applied for their numerical evaluation. Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities. The numerical treatment of these integrals takes into account the presence of this integrable singularity. The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity, Catalan’s constant, the Gaussian surface integral. The comparison between analytical and numerical data has proved high precision and availability of the proposed method.