Zainidin K. Eshkuvatov

Computation of Energy Release Rates for a Nearly Circular Crack

This paper deals with a nearly circular crack, Ω in the plane elasticity. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over a considered domain, Ω and it is then transformed into a similar equation over a circular region, 𝐷, using conformal mapping. Appropriate collocation points are chosen on the region 𝐷 to reduce the hypersingular integral equation into a system of linear equations with (2𝑁

Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

Approximate solution of the system of nonlinear integral equation by Newton–Kantorovich method

The Newton–Kantorovich method is developed for solving the system of nonlinear integral equations. The existence and uniqueness of the solution are proved, and the rate of convergence of the approximate solution is established. Finally, numerical examples are provided to show the validity and the efficiency of the method presented.

Mode Stresses for the Interaction between an Inclined Crack and a Curved Crack in Plane Elasticity

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.

On the Solution of Singular Ordinary Differential Equations Using a Composite Chebyshev Finite Difference Method

In this paper, a numerical algorithm based upon a hybrid of Chebyshev polynomials and block-pulse functions is proposed for solving both linear and nonlinear singular boundary value problems. Composite Chebyshev finite difference method is indeed an extension of the well-known Chebyshev finite difference method. We take advantage of the useful properties of Chebyshev polynomials and finite difference method to reduce the computation of the problem to a set of algebraic equations simplifying the problem. Several examples are included to illustrate the applicability and accuracy of the introduced method. Convergence analysis is presented.

On the number of eigenvalues of a model operator in fermionic Fock space

We consider a model describing a truncated operator H (truncated with respect to the number of particles) acting in the direct sum of zero-, one-, and two-particle subspaces of a fermionic Fock space a(L2(3)) over L2(3). We admit a general form for the kinetic part of the hamiltonian H, which contains a parameter γ to distinguish the two identical particles from the third one. In this note: (i) We find a critical value γ* for the parameter γ that allows or forbids the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model has a threshold resonance) and we prove that only for γ γ*. (ii) In the case γ > γ* we also establish the following asymptotics for the number N(z) of eigenvalues z below Emin, the lower limit of the essential spectrum of H:

Polynomial Spline Approach for Double Integrals with Algebraic Singularity

In this note, cubature formulas are constructed to evaluate the double integrals on the rectangle with algebraic singularity by replacing the density function f(x,y) with the S?(P) modified spline function interpolation of type (0,2). Rate of convergence are obtained in the classes of function f(x,y) C2,?(D).

On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

We develop the Newton-Kantorovich method to solve the system of 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.

An Automatic Quadrature Schemes and Error Estimates for Semibounded Weighted Hadamard Type Hypersingular Integrals

The approximate solutions for the semibounded Hadamard type hypersingular integrals (HSIs) for smooth density function are investigated. The automatic quadrature schemes (AQSs) are constructed by approximating the density function using the third and fourth kinds of Chebyshev polynomials. Error estimates for the semibounded solutions are obtained in the class of . Numerical results for the obtained quadrature schemes revealed that the proposed methods are highly accurate when the density function is any polynomial or rational functions. The results are in line with the theoretical findings.

Effective quadrature formula in solving linear integro-differential equations of order two

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.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.