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, convex Homotopy perturbation method (HPM) is presented for the approximate solution of the linear Fredholm-Volterra integral and integro-differential equation. Convergence and rate of convergence of the HPM are proved for both equations. Five numerical examples are provided to verify the validity and accuracy of the proposed method. Example reveals that HPM is very accurate and simple to implement for integral and integrodifferential equations.

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.

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.

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).

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).

The present work is devoted to the proof of uniqueness of the solution of the finite elements scheme in the case of variable coefficients. Finite elements method is applied for the numerical solution of the mixed problem for symmetric hyperbolic systems with variable coefficients. Moreover, dissipative boundary conditions and its stability are proved. Finally, numerical example is provided for the two dimensional mixed problem in simply connected region on the regular lattice. Coding is done by DELPHI7.