The paper studied the integration of logarithmic singularity problem J(y)=???(y)log|y-y0?|dA, where y=(?,?), y0=(?0,?0) the domain ? is rectangle ? = [r1, r2] � [r3, r4], the arbitrary point y? and the fixed point y0?. The given density function ?(y), is smooth on the rectangular domain ? and is in the functions class C2,? (?). Cubature formula (CF) for double integration with logarithmic singularities (LS) on a rectangle ? is constructed by applying type (0, 2) modified spline function D?(P). The results obtained by testing the density functions ?(y) as linear and absolute value functions shows that the constructed CF is highly accurate. � 2016 Author(s).

In this note, a general form of Fredholm-Volterra integro-differential equation order one is considered. The truncated Legendre series is used as bases function to approximate unknown function and Gauss-Legendre quadrature formula is applied for kernel integrals. Reduced algebraic equations are solved by using collocation method with roots of Legendre polynomials as collocation points. Three numerical examples with the comparisons are provided to show the validity and accuracy of the suggested method. Numerical results reveal that proposed method is dominated with repeated trapezoidal rule and differential transform method. � 2018 Author(s).

In this note, singular integration problems of the form H? (h) = ??? h(x,y)/|-x0|2-? dA, 0 ? ? ? 1, where ? = [x0,y0] � [b1, b2], x= (x,y) ? ? and fixed point x 0 = (x0,y0) ? ? is considered. The density function h(x, y) is assumed given, continuous and smooth on the rectangle ? and belong to the class of functions C2,?(?). Cubature formula for double integrals with algebraic singularity on a rectangle is constructed using the modified spline function S?(P) of type (0, 2). Highly accurate numerical results for the proposed method is given for both tested density function h(x, y) as linear, quadratic and absolute value functions. The results are in line with the theoretical findings. � 2015 IEEE.

In the paper, we propose a systematic approach to design and investigate the adequacy of the computational models for a mixed dissipative boundary value problem posed for the symmetric t-hyperbolic systems. We consider a two-dimensional linear hyperbolic system with variable coefficients and with the lower order term in dissipative boundary conditions. We construct the difference splitting scheme for the numerical calculation of stable solutions for this system. A discrete analogue of the Lyapunov�s function is constructed for the numerical verification o f stability o f solutions for the considered problem. A priori estimate is obtained for the discrete analogue of the Lyapunov�s function. This estimate allows us to assert the exponential stability of the numerical solution. A theorem on the exponential stability of the solution of the boundary value problem for linear hyperbolic system and on stability of difference splitting scheme in the Sobolev spaces was proved. These stability theorems give us the opportunity to prove the convergence of the numerical solution. � 2019 by authors, all rights reserved.

An efficient approximate method for solving Fredholm-Volterra integral equations of the third kind is presented. As a basis functions truncated Legendre series is used for unknown function and Gauss-Legendre quadrature formula with collocation method are applied to reduce problem into linear algebraic equations. The existence and uniqueness solution of the integral equation of the 3rd kind are shown as well as rate of convergence is obtained. Illustrative examples revels that the proposed method is very efficient and accurate. Finally, comparison results with the previous work are also given. � 2017, American Institute of Mathematical Sciences. All rights reserved.

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed. � 2017 Author(s).

In this note, we have developed new classes of one dimensional orthogonal polynomials Zk(i,n)(x),i = {1,2},n = 0,1,2,..., namely extended Chebyshev polynomials (ECPs) of the first and second kinds, which are an extension of the Chebyshev polynomials of the first and second kinds respectively. For non-homogeneous SIEs (bounded and unbounded case) truncated series of the first and second kind of ECPs are used to find approximate solution. It is found that first and second kinds of ECPs Zk(in)(x),i = {1,2} are orthogonal with weights w(1,k)(x)= xk-1/1-x2k and w(2,k)(x) = xk-11-x2k, where k is positive odd integer. Spectral properties of first and second kind of ECPs are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method.

In this paper, the problem of arc cracks that lie in the boundary of half circle in an elastic half plane is investigated. The complex potential variables with free traction boundary condition is used to formulate the problem into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function with the help of curve length coordinate method. The numerical results have shown that our results are in good agreement with the previous works. Stress intensity factors for different cracks position are presented.

Simple and efficient convex homotopy perturbation method (HPM) is presented to obtain an approximate solution of hyper-singular integral equations of the first kind. Convergence and error estimate of HPM are obtained. Three numerical examples were provided to verify the effectiveness of the HPM. Comparisons with reproducing kernel method (Chen et al., 2011) for the same number of iteration is also presented. Numerical examples reveal that the convergence of HPM can still be achieved for some problems even if the condition of convergence of HPM is not satisfied.

This paper deals with the interaction between two inclined cracks in the upper part of bonded dissimilar materials subjected to various stresses which is normal stress (Mode I), shear stress (Mode II), tearing stress (Mode III) and mixed stress. This problem is formulated into hypersingular integral equations (HSIE) by using modified complex potentials (MCP) with the help of continuity conditions of the resultant force and displacement functions where the unknown is the crack opening displacement (COD) function and the tractions along the crack as the right hand terms. Then, the curve length coordinate method and appropriate quadrature formulas are used to solve numerically the obtained HSIE to compute the stress intensity factors (SIF) in order to determine the stability behavior of materials containing cracks. Numerical results showed the behavior of the nondimensionalSIF at the cracks tips. It is observed that the various stresses and the elastic constants ratio are influences to the value of nondimensional SIF at the crack tips. � Published under licence by IOP Publishing Ltd.

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods. � 2017 Author(s).

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [?1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707?1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265�274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636�641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points. � 2016, The Author(s).

Modified homotopy perturbation method (HPM) is used to solve the hypersingular integral equations (HSIEs) of the second kind on the interval [-1, 1] with the assumption that the kernel in the form K(x, t)(x-t)-c0 where K(x, t) is a constant on the diagonal of the domain. This method introduced selective functions as Chebyshev polynomials of second kind and unknown parameters that leads to two step iterations and gives exact solution. Example are presented to prove the efficiency and realiability of the method. � 2016 Author(s).

In this note, we propose a new class of orthogonal polynomials (named Bachok–Hasham polynomials H1nk(x)), which is an extension of the Chebyshev polynomials. Eigenfunctions and corresponding eigenvalues are found for the homogeneous second kind of Logarithmic Singular Integral Equations (LogSIEs). For non-homogeneous LogSIEs truncated series of the first kind Bachok–Hasham polynomials are used to find approximate solution. It is found that first kind of Bachok–Hasham polynomials (H1nk(x)) are orthogonal with weight [Formula presented], where k is positive odd integer. Properties of first kind of Bachok–Hasham polynomials are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method. © 2019 THE AUTHORS

Numerical solutions for an elastic half plane with circular arc cracks subjected to uniaxial tension ??x = p is presented. The free traction on the boundary of the half plane is assumed. Based on the modified complex potential and superposition method, the problem is formulated into a singular integral equation with the distribution dislocation function as unknown. Numerical examples exhibit the behavior of the stress intensity factor at the cracks tips for various positions. Our numerical results are in agreement with the existence one. � 2019, Akademi Sains Malaysia.

We develop the Newton-Kantorovich method to solve the system of 2 × 2 nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

Modified complex potential with free traction boundary condition is used to formulate the curved crack problem in a half plane elasticity into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function. Numerical examples exhibit the stress intensity factor increases as the cracks getting close to each other, and close to the boundary of the half plane. � 2018 Elsevier Inc.

The new hypersingular integral equations (HSIEs) for the multiple cracks problems in both upper and lower parts of the bonded dissimilar materials are formulated using the modified complex potential method, and with the help of the continuity conditions of the resultant force function and displacement. The crack opening displacement is the unknown and the traction along the crack as the right term of the equations. The appropriate quadrature formulas are used in solving the obtained HSIEs for the unknown coefficients. Numerical results for the multiple inclined or circular arc cracks subjected to the remote shear stress are presented.

The new hypersingular integral equations (HSIEs) for the multiple cracks problems in both upper and lower parts of the bonded dissimilar materials are formulated using the modified complex potential method, and with the help of the continuity conditions of the resultant force function and displacement. The crack opening displacement is the unknown and the traction along the crack as the right term of the equations. The appropriate quadrature formulas are used in solving the obtained HSIEs for the unknown coefficients. Numerical results for the multiple inclined or circular arc cracks subjected to the remote shear stress are presented. � 2019 Elsevier Inc.

The multiple cracks problem in an elastic half-plane is formulated into singular integral equation using the modified complex potential with free traction boundary condition. A system of singular integral equations is obtained with the distribution dislocation function as unknown, and the traction applied on the crack faces as the right hand terms. With the help of the curved length coordinate method and Gauss quadrature rule, the resulting system is solved numerically. The stress intensity factor (SIF) can be obtained from the unknown coefficients. Numerical examples exhibit that our results are in good agreement with the previous works, and it is found that the SIF increase as the cracks approaches the boundary of half plane. � 2017 Author(s).