Kind Volterra Equations Research Articles

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill‐posed problems that are obtained by a system of linear two‐dimensional Volterra integral equations with a singular matrix in the leading part. These problems may arise in the modeling of certain heat conduction processes as well as in the dynamic simulation packages such as compressible flow through a plant piping network. Owing to the ill‐posed nature of the first kind Volterra equation that appears in the system, we will focus on the two families of regularization algorithms, ie, the Landweber and Lavrentiev type methods, where we treat both the exact and perturbed data. Our aim is to work directly with the original Volterra equations without any kind of reduction. Two fast iterative algorithms with reasonable computational complexity are developed. Numerical experiments on a few test problems are used to illustrate the validity and efficiency of the proposed iterative methods in comparison with the classical regularization methods.

Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels

We discuss the numerical solution to a class of weakly singular Volterra integral equations in this paper. Firstly, the fractional Lagrange interpolation is applied to deal with the singularity of the solution, and efficient fractional collocation boundary value methods are developed. Secondly, local convergence estimates are derived from examining the asymptotic property of the solution and the interpolation remainder. We find that the second kind Volterra integral equation with a weakly singular kernel can be efficiently solved on a uniform grid. Finally, several numerical examples are given to illustrate the performance of fractional collocation boundary value methods.

An adaptive Huber method for non-linear systems of weakly singular second kind Volterra integral equations

Numerical methods for systems of weakly singular Volterra integral equations are rarely considered in the literature, especially if the equations involve non-linear dependencies between unknowns and their integrals. In the present work an adaptive Huber method for such systems is proposed, by extending the method previously formulated for single weakly singular second kind Volterra equations. The method is tested on example systems of integral equations involving integrals with kernels K(t,τ)=(t−τ)−1/2, K(t,τ)=exp[−λ(t−τ)](t−τ)−1/2 (where λ>0), and K(t,τ)=1. The magnitude of the errors, and practical accuracy orders, observed for IE systems, are comparable to those for single IEs. In cases when the solution vector is not differentiable at t=0, the estimation of errors at t=0 is found somewhat less reliable for IE systems, than it was for single IEs. The stability of the IE systems solved appears to be sufficient, in practice, for the numerical stability of the method.

Heat flux determination from measured heating rates using thermographic phosphors

A new method for measuring the heating rate (defined as the time rate of change of temperature) and estimating heat flux from the heating rate is proposed. The example problem involves analytic heat conduction in a one-dimensional slab, where the measurement location of temperature or heating rate coincides with the location of the estimated heat flux. The new method involves the solution to a Volterra equation of the second kind, which is inherently more stable than Volterra equations of the first kind. The solution for heat flux from a measured temperature is generally a first kind Volterra equation. Estimates from the new approach are compared to estimates from measured temperatures. The heating rate measurements are accomplished by leveraging the temperature dependent decay rate of thermographic phosphors (TGP). Results indicate that the new data-reduction method is far more stable than the usual minimization of temperature residuals, which results in errors that are 1.5–12 times larger than those of the new approach. Furthermore, noise in TGP measurements was found to give an uncertainty of 4% in the heating rate measurement, which is comparable to the noise introduced in the test case data. Results of the simulations and the level of noise in TGP measurements suggest that this novel approach to heat flux determination is viable.

Optimal control problem for nonlinear stochastic difference second kind volterra equations

The problems of stability and optimal control for stochastic difference equations are receiving important attention now (see for example [1–6]). In this paper, the necessary optimality condition for nonlinear stochastic difference second kind Volterra equation are constructed. For stochastic integral-functional equations analogous results were obtained in [7].

Boundedness of Discrete Volterra Equations

Investigation of the boundedness of the solutions of Volterra equations with discrete time by means of the direct Liapunov method and comparison theorems is performed. Boundedness conditions of solutions of some particular classes of equations are derived. Using such conditions, stability of Direct Quadrature methods for second kind Volterra equations is analyzed.

Continuous collocation approximations to solutions of first kind Volterra equations

In this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind Volterra integral equations. The results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations. The convergence analysis is based on a Runge-Kutta or ODE approach.

AN ANALYTICAL SOLUTION OF A SECOND KIND VOLTERRA INTEGRAL EQUATION REPRESENTING THE CONTRIBUTION OF GRAVITATIONAL WAVES TO THE POLARIZATION OF THE COSMIC MICROWAVE BACKGROUND

Volterra integral equations of the second kind have a remarkable importance in many physical problems. In this paper, I shall discuss the particular case of a Volterra equation whose solution represents the polarization of the cosmic microwave background (CMB) induced by cosmological gravitational waves. The analytical study of the radiative transfer equation, written in the Chandrasekhar formalism, is performed after a change of variables that follows the formalism developed by Polnarev in 1980. Then, the problem turns into the solution of a second kind Volterra equation, containing the gravitational wave spectrum in its source term. An analytical solution for this Volterra equation can be found if its kernel has a polynomial or an exponential form: in these cases, respectively, the methods of the resolvent and of the iterated kernels can be successfully applied; the solution thus determined represents the polarization of the CMB photons induced by cosmological gravitational waves. The calculated polarization degree for the CMB is plotted versus the wave number k of the gravitational waves inducing the polarization. In the case of an exponential kernel, i.e. for a Universe fast reionized at small redshift (z≤20), I obtain a remarkable polarization peak at small k (k≤4); such a degree of polarization for the CMB should be observable at angular scales near 9°, thus providing also a test for many reionization models. On the contrary, for a standard ionization history, a CMB polarization comparable in magnitude to the one predicted above could only be observed at small angular scales (θ≤2°).

Measurement of plasma current distribution using magnetic probes

Magnetic probes are commonly used for measuring current distributions in plasmas. Unfortunately they must be placed inside the plasma, perturbing the current distribution under measurement. Other authors have studied the influence of a magnetic probe on a current sheet for both thin and thick current distributions. For the case of thick current sheets (as happens in plasma focus discharges), the measured voltage can be written as a first kind convolution of a Volterra equation. In order to get the real current distribution, an inversion of this equation is needed. First kind Volterra equations (Abel integrals included) are in some sense not a well-posed problem and, for numerical inversion, the input data must be smoothed. Further, the magnetic probe problem represents a particular case among first kind Volterra equations because the kernel is singular and the illness of the problem becomes more severe. On the other hand, we are interested in inverting experimental data with errors, so it is mandatory to evaluate the error in the inverted profile. For Volterra integral equations, error propagation has not been widely discussed in the literature. In this work a direct method, the matrix method plus an explicit filter, are used to obtain the solution. One of the main advantages of the method is the easy calculation of the error in the inverted signal from the errors of the parameters and the input data. In order to minimize error magnification, some limitations to the geometrical configuration of the probe are found. Also, it is shown that this method is a good compromise between accuracy and computational speed (it can be used on line working with thousands of data points).

Inversion of First Kind Volterra Equations: Back to Direct Methods?

Abstract Numerical inversion of the first kind Volterra equation (the Abel inversion included) has been extensively studied. Direct methods were probably the first methods used to attempt the inversion. Together with the computer hardware evolution, new methods were devised in order to deal with the inherent problem of this kind of equation, that is, error magnification. Using a large number of data points (several thousands) most methods are difficult to use, specially when the inversion and its error are required on line, that is, while performing the experiments. Further, error propagation (coming from the input data and from the parameters of the problem) is, usually, a difficult task and has not been extensively studied. On the other hand, direct methods together with an adequate filter give good resolution, are fast, and error propagation is easily performed. In this work we used the so called Matrix Method for inverting three different equations, showing how to build the resolvent nucleus and how errors propagate through the solution.

Roots of the discrete solution of first kind volterra equations

The problem is to find a point z such that y(z) = M where M is a given real number and y(x) is the unique solution of a Volterra integral equation of the first kind. A fourth order method illustrates the algorithm to be described.

Simple roots of the solution of a second kind volterra equation

A point δ such that y (δ)= k is detimined a fourth order quadrature method to approximate the integral and an interative methe to find δ. The method is easily generalized to higher order quardrature and more efficient root finding methods.

The nonlinear Volterra equation of Abel's kind and its numerical treatment

Starting from an existence and uniqueness theorem for a generalized nonsingular second kind Volterra equation existence and uniqueness for the solution of the nonlinear, weakly singular first kind Volterra equation is examined. A new type of numerical method is developed. A basic lemma concerning the boundedness of a special difference inequality is given and order two or three convergence of the method is shown. Two numerical examples illustrate the theoretical results.

Solving Nonlinear Second Kind Volterra Equations by Modified Increment Methods

One step procedures for solving Volterra integral equations have appeared in the literature but have generally been ignored because of their complicated nature as well as their inefficiency. In this paper, a general class of methods are modified to give increased efficiency. Error bounds are provided, a class of A-stable methods are presented and numerical examples follow.

On the numerical solution of Brownian motion processes

A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.

On the numerical solution of Brownian motion processes

A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.

A product integration method for a class of singular first kind Volterra equations

Convergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t?s) ?? , 0<?<1, wherek(t, s) is continuous, is examined. It is shown that convergence of order one holds if the solution of the Volterra equation has a Lipschitz continuous first derivative andk(t, s) is suitably smooth. In addition, convergence is shown to hold when the solution has only Lipschitz continuity and the same conditions onk(t, s) apply. An existence theorem of Kowalewski is used to relate these conditions on the solution to conditions on the data andk(t, s).