Explore the vast range of titles available.

Purpose This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm. Design/methodology/approach The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations. Findings Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software. Research limitations/implications The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement. Practical implications There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively. Social implications This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution. Originality/value To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

This paper considers the size-dependent plane frictional contact problem of a homogeneous coated half-plane indented by a rigid punch based on the couple stress elasticity. Using the Fourier integral transform technique in addition to the boundary and compatibility conditions, the mixed-boundary value problem is converted into a singular integral equation of the second kind. The integral equation is further derived and normalized for the cylindrical and flat punch case profiles. Applying the Gauss–Jacobi integration formula, the resulting singular integral equation is reduced to a system of algebraic equations. The obtained results are first validated based on those published for the case of a frictionless contact problem of a half-plane indented by rigid cylindrical and flat punches and solved based on the couple stress theory. A detailed parametric study is then performed to investigate the effect of the characteristic material length, the friction coefficient, the layer height, the shear modulus, the indentation load and Poisson’s ratio on the contact and in-plane stresses.

This paper investigates the contact problem of an elastic layer that is perfectly attached to a porous half-space by two rigid flat punches with collinear symmetry. Using integral transformation, the problem is condensed to a singular integral equation of the Cauchy type. Then, the exact expressions for the surface contact stress and surface interface displacement are provided. By using the Gauss–Chebyshev technique, the integral equations are solved numerically, and the variations of the unknown contact stresses and deformations for different parameters are addressed. The results indicate that stress concentration is typically higher on the outer edge of the contact area compared to the inner edge. This also explains why surface damage is more likely to occur on the outer edge in elastic and poroelastic materials. Due to the interaction between the two punches, there will be a superposition of normal displacements at the center. The deformation or bulging at the center can be managed by adjusting the parameter values, allowing the engineered material to fulfill its intended purpose. The potential applications of these research findings encompass safeguarding porous structures against contact-related deformation and damage.

A nonaxisymmetric contact problem on a circular interfacial inclusion of arbitrary shape under mixed boundary conditions with a piecewise homogeneous transversely isotropic space is considered. One edge of the inclusion is in smooth contact (tangential stresses are zero), while the other is in full contact with the medium. The problem is reduced to a system of five two-dimensional singular integral equations. For the inclusion, which has an axisymmetric shape under arbitrary loads, an exact solution of the system is obtained and its asymptotic properties are determined. On the basis of the latter, an efficient numerical-analytical method for solving the system of two-dimensional singular integral equations is proposed. The stress distribution in the vicinity of a coin-shaped inclusion is obtained and its features are revealed.

The severe electric and thermal environments may cause localized deterioration of the contact behavior of thermoelectric devices. The contact responses of the thermoelectricity under the joint effect of the finite size and the friction are analyzed. The law of the Coulomb friction is adopted. The obtained Fredholm kernel functions reveal the influence of the thickness and the friction. The known Jacobi polynomials are employed to discretize the obtained singular integral equation. The effects of the thermoelectric loadings (the total electric current and the total energy flux), friction coefficient, the thermoelectric strip thickness, and the elastic and thermoelectric material constants on the distribution of the normal traction and the surface in-plane stress are demonstrated in detail. The smaller thermal expansion coefficient and shear modulus will contribute to the lower stress concentration at both contact edges. The surface in-plane tensile stress behind the trailing edge can be alleviated as the thermoelectric strip becomes thinner.

Thermal contact damage is inevitable under the influence of friction. The frictional continuous contact model of the thermoelectric (TE) layer loaded by a rigid indenter is considered. Considering the complexity of friction and size effect, the nonlinear contact problem is reducedis transformed into solving the Cauchy singular integral equation (SIE) of the second kind by using the convolution formula and the superposition principle. The contact stress between the rigid indenter and the TE layer were affected by the external load, friction coefficient, TE material parameters. The results show that the softer TE layer can relieve the contact stress of the upper interface of TE. The change of friction coefficient and TE layer thickness can reduce the pressure at the rear end of the indenter. Properly adjusting these parameters can prevent the destruction and contact damage of the TE layer during the friction contact process.

The goal of this study is to construct a nonlinear electromagnetic field inside a waveguide. We assume that a body is located in a semi-infinite rectangular waveguide and that an electromagnetic field propagates inside the body. Iterative algorithms based on solving a volumetric nonlinear singular integral equation are proposed and described. Numerical results are presented. The boundary value problem for the system of Maxwell’s equations is reduced to a volumetric singular integral equation. An iterative method for creating a nonlinear medium inside the body with a dielectric structure is constructed. The problem is solved numerically. The size of the matrix obtained in the calculation is about 15 000 elements. The internal convergence of the iterative methods are demonstrated. The curves illustrating the field distribution inside the nonlinear body are plotted. A numerical method for finding wavenumbers that make it possible to create a nonlinear field is proposed and implemented.

Two different direct methods are proposed to solve Cauchy singular integral equations on the real line. The aforementioned methods differ in order to be able to prove their convergence which depends on the smoothness of the known term function in the integral equation.

Multiple processes in physics and technology are simulated using singular integral equations in exceptional cases, which necessitates development of approximate methods for solving such equations. A computational scheme using the Fourier transform is proposed in this work.

In this article, we use a fuzzy number in its parametric form to solve a fuzzy nonlinear integral equation of the second kind in the crisp case. The main theme of this article is to find a semi-analytical solution of fuzzy nonlinear integral equations. A hybrid method of Laplace transform coupled with Adomian decomposition method is used to find the solution of the fuzzy nonlinear integral equations including fuzzy nonlinear Fredholm integral equation, fuzzy nonlinear Volterra integral equation, and fuzzy nonlinear singular integral equation of Abel type kernel. We also provide some suitable examples to better understand the proposed method.