On the theory of convolution equations of the third kind

The autoconvolution equation of the third kind with coefficient of general power type is dealt with by the method of weighted norms developed for equations with coefficients of linear and integer power type in recent joint work of the author with L. Berg, J. Janno, and B. Hofmann. For this equation two existence theorems and a uniqueness theorem are proved. Further, as an auxiliary equation a linear singular integral equation of Abel is treated anew and the existence of solutions to a related class of linear Volterra equations of the third kind is derived.     

Autoconvolution equations of the third kind with power-logarithmic coefficients

Existence and uniqueness theorems for a basic class of autoconvolution equations of the third kind with power-logarithmic coefficeint and free term are derived. Under suitable assumptions the existence of a solitary solution and of a one-parametric family of solutions is proved.     

Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions

In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0,1)$[0,1)$ to solve Volterra integral and integro-differential equations of convolution type without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.     

The boundaries of the solutions of the linear Volterra integral equations with convolution kernel

Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002.

In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used.

     

Applications of integral equations in particle-size statistics

We discuss the application of integral equations techniques to two broad areas of particle statistics, namely, stereology and packing. Problems in stereology lead to the inversion of Abel-type integral equations; and we present a brief survey of existing methods, analytical and numerical, for doing this. Packing problems lead to Volterra equations which, in simple cases, can be solved exactly and, in other cases, need to be solved numerically. Methods for doing this are presented along with some numerical results.     

Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations

In this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra integral equations of Hammerstein type are constructed. The coefficients of the methods are expressed in terms of the values of the Laplace transform of the kernel. These methods have been suitably constructed in order to be implemented in an efficient way, thus leading to a very low computational cost both in time and in space. The order of convergence of the constructed methods is studied. The numerical experiments confirm the expected accuracy and computational cost. AMS subject classification (2000)  65R20, 45D05, 44A35, 44A10     

A third order,semi-explicit method in the numerical solution of first kind Volterra integral equations

We present a new numerical method for the solution of nonsingular Volterra integral equations of the first kind. It belongs to a new class of methods that are semi-explicit, provide self-starting algorithms and possess favourable stability properties. The third order convergence of the particular method exhibited is established, under suitable conditions, and numerical results are illustrated.     

On solvability of singular integral-differential equations with convolution

In this paper, we study a class of singular integral-different equations of convolution type with Cauchy kernel. By means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation, we transform the equations into the Riemann-Hilbert problems with discontinuous coefficients and obtain the general solutions and conditions of solvability in class $\{0\}$. Thus, the result in this paper generalizes the classical theory of integral equations and boundary value problems.     

Numerical solution of linear Volterra integral equations of the second kind with sharp gradients

Collocation methods are a well-developed approach for the numerical solution of smooth and weakly singular Volterra integral equations. In this paper, we extend these methods through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case, the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous-time random walk in three dimensions that may be killed at a fixed lattice site. To demonstrate how separating the solution approximation from quadrature approximation may improve computational performance, we also compare our new method to several existing Gregory, Sinc, and global spectral methods, where quadrature approximation and solution approximation are coupled.     

On some properties of systems of Volterra integral equations of the fourth kind with kernel of convolution type

We consider the system of integral equations of the form Ax +V x = Ψ, where V is the Volterra operator with kernel of convolution type and A is a constant matrix, det A = 0. We prove an existence theorem and establish necessary and sufficient conditions for the kernel of the operator of the system to be trivial.     

A note on solving Cauchy integral equations of the first kind by iterations

The successive approximation method was applied for the first time by N.I. Ioakimidis to solve practical cases of a Cauchy singular integral equation: the airfoil one. In this paper we study a more general case. We prove the convergence of the method in this general case. The proposed method has been tested for two kernels which are particularly important in practice. Finally, some numerical examples illustrate the accuracy of the method.     

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.     

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.     

On oscillatory third order difference equations

In this paper, sufficient conditions have been obtained for oscillation/nonoscillation of a class of homogeneous/nonhomogeneous linear difference equations of third order.     

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.     

Initial-layer theory and model equations of Volterra type

^** Email: amb16{at}nyu.edu It is demonstrated here that there exist initial layers to singularlyperturbed Volterra equations whose thicknesses are not of theorder of magnitude of O(), 0. It is also shown that the initial-layertheory is extremely useful because it allows one to constructthe approximate solution to an equation, which is almost identicalto the exact solution.     

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation x?x−1, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.