Abstract:
We used function theoretic method to solve a singular integral equation with logarithmic kernel in two disjoint finite intervals where the unknown function satisfying the integral equation may be bounded or unbounded at the nonzero finite endpoints of the interval concerned. An appropriate solution of this integral equation is then applied to solve the problem of scattering of time harmonic surface water waves by a fully submerged thin vertical barrier with a single gap.

Abstract:
In this study, we introduce an efficient method to find and discuss an approximate solution of the integral equation of type Volterra-Fredholm in the space L2[a, b] x [0, T]. The kernel of Fredholm is considered in position and represented in a logarithmic form, while the kernel of Volterra is taken in time as a continuous function. Using a numerical method we obtain a linear system of Fredholm integral equations is position which will be solved.

Abstract:
In this article Chebyshev and trigonometric polynomials are used to construct an approximate solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane.

Abstract:
In this article, we present approximate solution of the two-dimensional singular nonlinear mixed Volterra-Fredholm integral equations (V-FIE), which is deduced by using new strategy (combined Laplace homotopy perturbation method (LHPM)). Here we consider the V-FIE with Cauchy kernel. Solved examples illustrate that the proposed strategy is powerful, effective and very simple.

Abstract:
In the
paper, the approximate solution for the two-dimensional linear and nonlinear
Volterra-Fredholm integral equation (V-FIE) with singular kernel by utilizing
the combined Laplace-Adomian decomposition method (LADM) was studied. This
technique is a convergent series from easily computable components. Four
examples are exhibited, when the kernel takes Carleman and logarithmic forms.
Numerical results uncover that the method is efficient and high accurate.

Abstract:
In this paper, under certain conditions, the solution of mixed type of Fredholm-Volterra integral equation is discussed and obtained in the space L_2 ( 1, 1) × C[0, T ], T < ∞. Here, the singular part of kernel of Fredholm-Volterra integral term is established in a logarithmic form, while the kernel of Fredholm-Volterra integral term is a positive continuous function in time and belongs to the class C[0, T ], T < ∞. The solution, when the mixed type integral, takes a system form of Fredholm integral equation of the first or second kind are discussed.

Abstract:
In this paper an explicit solution of a generalized singular integral equation with a Hilbert kernel depending on indices of characteristic operators is presented.

Abstract:
Boundary integral methods for the solution of boundary value PDEs are an alternative to `interior' methods, such as finite difference and finite element methods. They are attractive on domains with corners, particularly when the solution has singularities at these corners. In these cases, interior methods can become excessively expensive, as they require a finely discretised 2D mesh in the vicinity of corners, whilst boundary integral methods typically require a mesh discretised in only one dimension, that of arc length. Consider the Dirichlet problem. Traditional boundary integral methods applied to problems with corner singularities involve a (real) boundary integral equation with a kernel containing a logarithmic singularity. This is both tedious to code and computationally inefficient. The CBIEM is different in that it involves a complex boundary integral equation with a smooth kernel. The boundary integral equation is approximated using a collocation technique, and the interior solution is then approximated using a discretisation of Cauchy's integral formula, combined with singularity subtraction. A high order quadrature rule is required for the solution of the integral equation. Typical corner singularities are of square root type, and a `geometrically graded h-p' composite quadrature rule is used. This yields efficient, high order solution of the integral equation, and thence the Dirichlet problem. Implementation and experimental results in \textsc{matlab} code are presented.

Abstract:
A plane rectangular bar of conducting and permeable material is placed in an external low-frequency magnetic field. The shielding properties of this object are investigated by solving the given plane eddy current problem for the vector potential with the boundary integral equation method. The vector potential inside the rectangle is governed by Helmholtz' equation, which in our case is solved by separation. The solution is inserted into the remaining boundary integral equation for the exterior vector potential in the domain surrounding the bar. By expressing its logarithmic kernel as a Fourier integral the overall solution inside and outside the bar is calculated using analytical means only.

Abstract:
在Bernt利用Picard迭代给出的随机积分方程解的存在唯一性定理基础上, 通过定义本性有界可测函数作为核函数并对核函数的积分进行限制, 给出了带核函数随机积分方程解的存在唯一性定理 Bernt uses Picard iteration to give the existence and uniqueness theorem about the solution of stochastic integral equations. On this basis, we define the essential bounded function as the kernel function and limit the integral of kernel function, then we give the existence and uniqueness theorem of the solution of stochastic integral equations with kernel function