Convergence in the integral metric of the general projective method for solving singular integral equations of the first kind with the Cauchy kernel

We study a projective method for solving singular integral equations of the first kind with the Cauchy kernel. Depending on the index of the equation, we introduce pairs of weight spaces which represent a restriction of the space of summable functions. We prove the correctness of the stated problem. We obtain sufficient conditions for the convergence of the projective method in the integral metric.     

Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs

This paper presents the mechanical quadrature methods (MQMs) for solving boundary integral equations (BIEs) of the first kind on open arcs. The spectral condition number of MQMs is only O(h⁻¹), where h is the maximal mesh width. The errors of MQMs have multivariate asymptotic expansions, accompanied with for all mesh widths h_i. Hence, once discrete equations with coarse meshes are solved in parallel, the accuracy order of numerical approximations can be greatly improved by splitting extrapolation algorithms (SEAs). Moreover, a posteriori asymptotic error estimates are derived, which can be used to formulate self-adaptive algorithms. Numerical examples are also provided to support our algorithms and analysis. Furthermore, compared with the existing algorithms, such as Galerkin and collocation methods, the accuracy order of the MQMs is higher, and the discrete matrix entries are explicit, to prove that the MQMs in this paper are more promising and beneficial to practical applications.     

Abel differential equations admitting a certain first integral

In this work algebro-geometric conditions to have a certain first integral for an Abel differential equation are given. These conditions establish a bridge with classical Galois theory because we transform the differential problem of finding a first integral for an Abel equation into an algebraic problem.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method

In this paper, linear and nonlinear Abel integral equations are transformed in such a manner that the Adomian decomposition method can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.     

Product integration methods for second-kind Abel integral equations

The construction and convergence of high-order product integration methods for the second-kind Abel equation are discussed and the results of De Hoog and Weiss are generalised. Backward difference methods are introduced, and numerical results are presented which verify the theoretical rates of convergence.     

On the convergence of numerical methods for solving a Volterra bilinear equations of the first kind

A justification is given for the convergence of quadratures (namely, the right rectangle and midpoint rules) for the numerical solution of a Volterra bilinear equation of the first kind. Numerical results for some benchmark problems are presented.     

On integral representation of Bessel function of the first kind

A first kind Fredholm integral equation with nondegenerate kernel is given, which particular solution is the Bessel function of the first kind. This equation is solved by means of Mellin transform pair.     

An algorithm based on the regularization and integral mean value methods for the Fredholm integral equations of the first kind

In this paper, an algorithm based on the regularization and integral mean value methods, to handle the ill-posed multi-dimensional Fredholm equations, is introduced. The application of this algorithm is based on the transforming the first kind equation to a second kind equation by the regularization method. Then, by converting the first kind to a second kind, the integral mean value method is employed to handle the resulting Fredholm integral equations of the second kind. The efficiency of the approach will be shown by applying the procedure on some examples.     

On the stability of the spline collocation method for a class of integral equations of the first kind

This paper is concerned with the stability of the spline collocation method for a class of integral equations of the first kind with logarithmic kernels. It is shown that a proper choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence for the approximate solutions.     

Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods

The accuracy of numerical solutions near singular points is crucial for numerical methods. In this paper we develop an efficient mechanical quadrature method (MQM) with high accuracy. The following advantages of MQM show that it is very promising and beneficial for practical applications: (1) the O(h_max³) O(h_{\rm {max}}^{3}) convergence rate; (2) the O(h_max⁵)O(h_{\rm {max}}^{5}) convergence rate after splitting extrapolation; (3) Cond = O(h_min^-1)O(h_{\rm {min}}^{-1}); (4) the explicit discrete matrix entries. In this paper, the above theoretical results are briefly addressed and then verified by numerical experiments. The solutions of MQM are more accurate than those of other methods. Note that for the discontinuous model in Li et al. (Eng Anal Bound Elem 29:59–75, 2005), the highly accurate solutions of MQM may even compete with those of the collocation Trefftz method.     

On Taylor-series expansion methods for the second kind integral equations

In this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.     

第二类Volterra型积分方程的Chebyshev-Legendre谱配置方法

提出了一种新的求解第二类线性Volterra型积分方程的Chebyshev谱配置方法.该方法分别对方程中积分部分的核函数和未知函数在Chebyshev-Gauss-Lobatto点上进行插值,通过Chebyshev-Legendre变换,把插值多项式表示成Legendre级数形式,从而将积分转换为内积的形式,再利用Legendre多项式的正交性进行计算.利用Chebyshev插值算子在不带权范数意义下的逼近结果,对该方法在理论上给出了L∞范数意义下的误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

Reducible quadrature methods for Volterra integral equations of the first kind

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.     

Stability of quadrature rule methods for first kind volterra integral equations

Gladwin [4] proved that Newton-Gregory formulas of order larger than 2 produce unstable algorithms when applied to nonlinear Volterra integral equations of the first kind. It is shown that similar results are true for all interpolatory quadrature rules using equidistant nodes. Upper bounds for the error order of quadrature rules, which lead to stable methods are given. Some higher order stable methods are indicated.     

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

Based on a new generalization of discrete Gronwall inequality in [L. Tao, H. Yong, A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equality of the second kind, J. Math. Anal. Appl. 282 (2003) 56-62], Navot's quadrature rule for computing integrals with the end point singularity in [I. Navot, A further extension of Euler-Maclaurin summation formula, J. Math. Phys. 41 (1962) 155-184] and a transformation in [P. Baratella, A. Palamara Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004) 401-418], a new quadrature method for solving nonlinear weakly singular Volterra integral equations of the second kind is presented. The convergence of the approximation solution and the asymptotic expansion of the error are proved, so by means of the extrapolation technique we not only obtain a higher accuracy order of the approximation but also get a posteriori estimate of the error.     

Numerical methods and asymptotic error expansions for the Emden-Fowler equations

In the present paper we analyse a numerical method for computing the solution of some boundary-value problems for the Emden-Fowler equations. The differential equations are discretized by a finite-difference method and we derive asymptotic expansions for the discretization error. Based on these asymptotic expansions, we use an extrapolation algorithm to accelerate the convergence of the numerical method.     

Solution of certain weakly singular integral equations of the first kind with indefinite parameters

In this paper we study a two-dimensional weakly singular integral equation of the first kind with logarithmic kernel. We construct a pair of spaces of the desired elements and the right-hand sides, where we prove the correctness of the problem under consideration and obtain inversion formulas for the integral operator.