In this paper, numerical solutions of multiple cracks problems in an infinite plate are studied. Hypersingular integral equations (hieq) for the cracks are formulated using the complex potential method. For all kernels such as regular or Hypersingular kernels, we are using the appropriate quadrature formulas to solve and evaluate the unknown functions numerically. Furthermore, by using this equation the stress intensity factor (SIF) was calculated for crack tips. For two serial cracks (horizontal) and two dissimilar cracks (horizontal and inclined), our numerical results agree with the previous works.

In this article we investigate the two-term Abel’s integral equations. We will do this in two different ways and show that such equation is reducible to an integro-differential equation of Volterra type.

A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we , rst formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.

In this paper, we compute the approximate numerical solution for the Volterra-Fredholm integral equation (VFIE) by using the shifted Jacobi collocation (SJC) method which depends on the operational matrices. Some properties of the shifted Jacobi polynomials are introduced. These properties allow us to transform the VolterraFredholm integral equation into a system of algebraic equations in a nice form with the expansion coefficients of the solution. Also, the convergence and error analysis are studied extensively. Finally, some examples which verify the efficiency of the given method are supplied and compared with other methods.

In this paper, using orthogonally of Tchebychev polynomials, we present an orthonormal wavelet basis for L2[0, 1]. We use this basis for solving Neumann problems with Galerkin method. The property of this basis is that a variety of integral operators is represented in this basis as sparse matrices, to high precision. Some examples are solved to illustrate the efficiency and accuracy of this method.

New sharp Kolmogorov type inequalities for hyper singular integrals with homogeneous characteristic of multivariate functions from Holder spaces are obtained. Proved inequalities are used to solve the Stechkin’s problem on the best approximation of unbounded Hypersingular integral operator by bounded ones on functional classes which are defined by a majorant of the modulus of continuity.