The paper studied the integration of logarithmic singularity problem J ((y) over bar) = integral integral(del) zeta((y) over bar) log vertical bar(y) over bar = (y) over bar (0)*vertical bar dA, where (y) over bar = (alpha, beta), (y) over bar (0) = (alpha(0), beta(0)), the domain del is rectangle (y) over bar [r(1), r(2)] x [r(3), r(4)]; the arbitrary point (y) over bar is an element of del and the fixed point (y) over bar (0) is an element of del. The given density function zeta((y) over bar), is smooth on the rectangular domain del and is in the functions class C-2,C-tau (del). Cubature formula (CF) for double integration with logarithmic singularities (LS) on a rectangle del is constructed by applying type (0, 2) modified spline function D-Gamma(P). The results obtained by testing the density functions zeta((y) over bar) as linear and absolute value functions shows that the constructed C-F is highly accurate.

In this note, singular integration problems of the form H alpha(h) = integral integral(Omega) h(x,y)/vertical bar(x) over bar - (x) over bar (0)vertical bar(2-alpha)dA, 0 <= alpha <= 1, where Omega = [a(1), a(2)] x [b(1), b(2)], (x) over bar = (x, y) is an element of Omega and fixed point (x) over bar (0) = (x(0), y(0)) is an element of Omega; is considered. The density function h (x, y) is assumed given, continuous and smooth on the rectangle Omega and belong to the class of functions C-2,C-alpha(Omega). Cubature formula for double integrals with algebraic singularity on a rectangle is constructed using the modified spline function S-Omega( 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.

In this paper, modified homotopy perturbation method (MHPM) is applied to solve the general Fredholm-Volterra integro-differential equations (FV-IDEs) of order m with initial conditions. Selective functions and unknown parameters allowed us to obtain two step iterations. It is found that MHPM is a semi-analytical method for FV-IDEs and could avoid complex computations. Numerical examples are given to show the efficiency and reliability of the method. Proof of the convergence of the proposed method is also given.

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.

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.

The formulation for the curved crack in a finite plate is established. The technique is the curved crack in a finite plate is divided into two sub-problems i.e. the curved crack problem in an infinite plate and the finite plate without crack. For the first problem, the curved problem is formulated into Fredholm integral equation, where as for the second problem the complex boundary integral equations based on complex variables are considered. The solution of the coupled boundary integral equations gives the solution on the domain of the boundary.

In this note, we consider a general 2 x 2 system of nonlinear Volterra type integral equations. The modified Newton method (modified NM) is used to reduce the nonlinear problems into 2 x 2 linear system of algebraic integral equations of Volterra type. The latter equation is solved by discretization method. Nystrom method with Gauss-Legendre quadrature is applied for the kernel integrals and Newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points. Existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained. Finally, numerical examples are provided to show the validity and efficiency of the method presented. Numerical results reveal that the proposed methods is efficient and accurate. Comparisons with other methods for the same problem are also presented. (C) 2019 Elsevier B.V. All rights reserved.

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. (C) 2017 Ain Shams University.

Using the strain space thermoplasticity theory, proposed by the first author, the coupled dynamic thermomechanical boundary value problems are formulated. The strain space thermoplasticity theory, in contrast to the existing one, allows to formulate the coupled thermoplastic boundary value problems for the displacement and temperature increments. The explicit and implicit finite difference equations for two dimensions case of the boundary value problems are constructed. The numerical solution of the explicit finite difference equations reduced to the application of the recurrent formulas, whereas the implicit scheme reduced to the application of the elimination method. Comparison shows that the numerical results obtained using the explicit and implicit schemes for aforementioned methods are coincides.

The interaction between the inclined and curved cracks is studied. Using the complex variable function method, the formulation in hypersingular integral equations is obtained. The curved length coordinate method and suitable quadrature rule are used to solve the integral equations numerically for the unknown function, which are later used to evaluate the stress intensity factor. There are four cases of the mode stresses; Mode I, Mode II, Mode III, and Mix Mode are presented as the numerical examples.

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.

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) - c(0) 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.

The behavior of the stress intensity factor at the tips of cracks subjected to uniaxial tension sigma x=p with traction-free boundary condition in half-plane elasticity is investigated. The problem is formulated into singular integral equations with the distribution dislocation function as unknown. In the formulation, we make used of a modified complex potential. Based on the appropriate quadrature formulas together with a suitable choice of collocation points, the singular integral equations are reduced to a system of linear equations for the unknown coefficients. Numerical examples show that the values of the stress intensity factor are influenced by the distance from the cracks to the boundary of the half-plane and the configuration of the cracks.

We consider an n x n system of nonlinear integral equations of Volterra type (nonlinear VIEs) arising from an economic model. By applying the Newton-Kantorovich method to the nonlinear VIEs we linearize them into linear Volterra type integral equations (linear VIEs). Uniqueness of the solution of the system is shown. An idea has been proposed to find the approximate solution by transforming the system of linear VIEs into a system of linear Fredholm integral equations by using sub-collocation points. Then the backward Newton interpolation formula is used to find the approximate solution at the collocation points. Each iteration is solved by the Nystrom type Gauss-Legendre quadrature formula (QF). It is found that by increasing the number of collocation points of QF with fewer iterations, a high accurate approximate solution can be obtained. Finally, an illustrative example is demonstrated to validate the accuracy of the method.

The Newton-Kantorovich method (NKM) is widely used to find approximate solutions for nonlinear problems that occur in many fields of applied mathematics. This method linearizes the problems and then attempts to solve the linear problems by generating a sequence of functions. In this study, we have applied NKM to Volterra-type nonlinear integral equations then the method of Nystrom type Gauss-Legendre quadrature formula (QF) was used to find the approximate solution of a linear Fredholm integral equation. New concept of determining the solution based on subcollocation points is proposed. The existence and uniqueness of the approximated method are proven. In addition, the convergence rate is established in Banach space. Finally illustrative examples are provided to validate the accuracy of the presented method. (C) 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University.

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. (C) 2018 Elsevier Inc. All rights reserved.

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. (C) 2019 Elsevier Inc. All rights reserved.

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. (C) 2019 Elsevier Inc. All rights reserved.

This paper deals with the multiple inclined or circular arc cracks in the upper half of bonded dissimilar materials subjected to shear stress. Using the complex variable function method, and with the help of the continuity conditions of the traction and displacement, the problem is formulated into the hypersingular integral equation (HSIE) with the crack opening displacement function as the unknown and the tractions along the crack as the right term. The obtained HSIE are solved numerically by utilising the appropriate quadrature formulas. Numerical results for multiple inclined or circular arc cracks problems in the upper half of bonded dissimilar materials are presented. It is found that the nondimensional stress intensity factors at the crack tips strongly depends on the elastic constants ratio, crack geometries, the distance between each crack and the distance between the crack and boundary. (C) 2019 Elsevier Inc. All rights reserved.