Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

A recurrence formula for the direct solution of singular integral equations

An iterative method for the solution of singular integral equations is given in this paper by developing a recurrence formula. Discretizing the above formula, by using appropriate quadrature rules, the solution of the singular integral equation is given in an extremely simple form. The number of numerical operations required for such a solution is considerably reduced, when compared to the number of operations required for a classical type of solution. Illustrative examples are given, indicating the efficiency of the method. It is shown that the number of operations in this procedure is only half the number of the operations for a typical numerical method. The convergence of the method is studied in the space of Hölder continuous functions. In the particular case of plane elasticity more efficient bounds are given. In the same case it is proved that the procedure is equivalent to the Schwarz's alternating method and convergence is assured [18].     

The regularization method for Fredholm integral equations of the first kind

In this work, the Fredholm integral equations of the first kind will be examined. The regularization method combined with the existing techniques are applied to handle the ill-posed Fredholm problems. Examples will be used to highlight the reliability of the regularization method.     

A method for solving ill-posed integral equations of the first kind

Improperly-posed (or Hadamard-incorrect) problems may arise when numerical solutions are extremely sensitive to a discretization process. The nonconformal contact problem in three-dimensional elastostatics falls into this category. It is shown how such contact stress problems may be formulated and successfully solved using the “Functional Regularization Method” of Tychonov. The Functional Regularization Method requires the use of a parameter, called the Regularization Parameter. Although no general rules for the choice of such a parameter appear to exist, we have determined appropriate bounds on the parameter for a wide class of contact problems (including that of Hertz). The method developed should be capable of extension to more general ill-posed problems. It is also shown that refinements in the discretization process such as reduced mesh lengths or higher order quadrature formulas may postpone, but do not necessarily remove the numerical difficulties associated with the physics of the problem.     

The multi-projection method for weakly singular Fredholm integral equations of the second kind

In this paper, we propose a multi-projection method and its re-iterated algorithm for solving weakly singular Fredholm integral equations of the second kind. We apply our methods to Petrov–Galerkin versions to establish excellent superconvergence results, and we illustrate our theoretical results with a numerical example.     

Bounds of high quality for first kind Volterra integral equations

E-Methods for solving linear Volterra integral equations of the first kind with smooth kernels are considered.E-Methods are a new type of numerical algorithms computing numerical approximations together with mathematically guaranteed close error bounds. The basic concepts from verification theory are sketched and such self-validating numerics derived. Computational experiments show the efficiency of these procedures being an advance in numerical methods.     

On a special class of nonlinear fredholm integral equations of the first kind

In this paper a special class of nonlinear Fredholm integral equations of the first kind, the so-called Urysohn equation, is considered, where the kernel depends ont only via the unknown functionx(t). To overcome the ambiguity, a decreasing rearrangement approach is used. Moreover, a constrained least squares method helps regularizing the problem. As a specific property, the equation can be decomposed into a well-posed nonlinear part, the inversion of a function, and an ill-posed linear part, a linear Fredholm integral equation of the first kind. The linear part of this two-stage procedure was already discussed in [8]. In the present paper the two-stage procedure is compared with a one-stage nonlinear least squares approximation which is directly applied to the nonlinear original integral equation. The comparison is explained by means of a computational case study for a specific example arising in optics.     

Superconvergence of collocation methods for Volterra integral equations of the first kind

Collocation methods for solving first-kind Volterra equations in the space of piecewise polynomials possessing finite (jump) discontinuities at their knots and having degreem≧0 are known to have global order of convergencep=m+1. It is shown that a careful choice of the collocation points (characterized by the Lobatto points in (0, 1]) yields convergence of order (m+2) at the corresponding Legendre points.     

A Galerkin collocation method for some integral equations of the first kind

Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkin's method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.     

On the approximate solution of singular integral equations

A method for obtaining the approximate solution of singular integral equations of the first and second kinds is suggested. The solution is represented in the form of power series with undetermined coefficients multiplied by a function in which the essential features of the singularity of the solution are preserved. The method of collocations is used to determine the unknown coefficients. The examples show that the method suggested is more general and gives good results even in the case when the form of solution does not exactly preserve the essential features of singularity. The method is simpler than others which use the properties of orthogonal polynomials, and is applicable for the solution of single equations as well as systems of simultaneous equations.     

A note on collocation methods for Volterra integral equations of the first kind

It has been shown that if a Volterra integral equation of the first kind with continuous kernel is solved numerically in a given intervalI by collocation in the space of piecewise polynomials of degreem≧0 and possessing finite discontinuities at their knotsZ _N then a careful choice of the collocation points yields convergence of orderp=m+2 on a certain finite subset ofI (while the global convergence order ism+1; this subset does not contain the knotsZ _N . In this note it will be shown that superconvergence onZ _N can be attained only if some of the collocation points coalesce (Hermite-type collocation).     

On solving Fredholm integral equations of the first kind using the interval programming algorithm of Robers

A discretized version of Fredholm integral equation of the first kind is solved using an interval programming algorithm and the results are compared with an initial value method.     

Extrapolation for solving a system of weakly singular nonlinear Volterra integral equations of the second kind

This article discusses an extrapolation method for solving a system of weakly singular nonlinear Volterra integral equations of the second kind. Based on a generalization of the discrete Gronwall inequality and Navot's quadrature rule, the modified trapeziform quadrature algorithm is presented. The iterative algorithm for solving the discrete system possesses a high accuracy order O(h ^2+α). After the asymptotic expansion of errors is proved, we can obtain an approximation with a higher accuracy order using extrapolation. An a posteriori error estimation is provided. Some numerical results are presented to illustrate the efficiency of our methods.     

A weighted H1 seminorm regularization method for Fredholm integral equations of the first kind

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H¹ seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.     

A numerical treatment to the solution of certain first kind Fredholm integral equations

This paper presents results obtained by an implementation of the kernel basis functions method to the solution of a special Fredholm integral equation of the first kind. The approximate solution is expressed in the linear form u_k(t)=∑w_jK(s_j, t), j=1, 2,..., k, where the unknown parameters w_j and s_j are determined by solving two linear overdetermined systems and a polynomial equation of the k-th order. Test examples are used to show that the numerical solution is comparable to the exact one.