Adomian's method for Hammerstein integral equations arising from chemical reactor theory

An ordinary differential equation with a parameter in the boundary conditions describes the steady state in an adiabatic tubular chemical reactor. In this paper, the problem is considered as a Hammerstein integral equation and solutions are obtained using Adomian's decomposition method.     

Solution of the integral equations of hereditary theory

An approximate method is proposed for calculating the product of the Rabotnov operator and a continuous function of time. A collocation method is used to construct an approximate solution of the Volterra equation with a nondifference kernel in the form of the product of a fractional-exponential function and an arbitrary function of time. The error of the approximate solution is estimated.Moscow Institute of Electronic Machine Building. Georgian Polytechnic Institute, Tbilisi. Translated from Mekhanika Polimerov, No. 1, pp. 77–81, January–February, 1973.     

Team decision theory and integral equations

The coordination of decisions under uncertainty in a team leads to optimality conditions that are integral equations. A specific example of a two-division firm is developed to illustrate these conditions. Numerical imbedding techniques are used to solve the firm's decision problem. Extensions toward more general techniques and applications are indicated.     

On a class of nonlinear integral equations arising in neutron transport

We apply the method of continuation to study the structure of the solutions of quadratic integral equations for a certain class of kernel functions.     

On integral equations of axisymmetric potential theory

It is shown that two integral equations of the first kind, muchused in, respectively, axisymmetric electrostatics and hydrodynamics,are wrong in the sense that they do not in general possess solutions.A theorem is established giving the precise conditions necessaryfor solutions to exist, but perhaps more important practicallyis the fact, brought out by examples, that the necessary conditionsare far from sufficient. An alternative integral equation inthe electrostatic case is proposed and justified, one havinga similar form and the same computational advantages, but freefrom existence difficulties. The apparent paradox that ‘solutions’are found to problems when the governing equations may not possesssolutions is explained by the fact that these purported solutionsare obtained by numerical or asymptotic analysis, when an approximatingequation possesses a solution, but one which cannot be saidto approximate to the solution of the problem if the equationto which, formally, it approximates cannot, through being meaningless,represent the problem. The arguments are given mainly in theelectrostatic context, but it is shown how they are modifiedto carry over to the hydrodynamical one.     

Study of Volterra integro-differential equations arising in viscoelasticity theory

Volterra integrodifferential equations with unbounded operator coefficients in a Hilbert space that are operator models of integrodifferential equations arising in viscoelasticity theory are studied. These equations are shown to be well-posed in Sobolev spaces of vector functions, and spectral analysis is applied to the operator functions that are the symbols of the given equations.     

Initial value methods in the theory of Fredholm integral equations

This paper deals with the problem of establishing the equivalence of a family of integral equations of Fredholm type with kernels that depend on a parameter and a related Cauchy system of integrodifferential equations. We also show how the Cauchy problem can be given an abstract formulation as an initial value problem in a complex Banach space.This research was supported by the University of Nevada at Las Vegas, Research Grant No. 4503.     

Joseph Liouville's contribution to the theory of integral equations

It is well known that Liouville did pioneering work on the application of specific integral equations in different parts of mathematics and mathematical physics. However, his short paper on spectral theory of Hilbert-Schmidt like operators has been neglected. With that paper Liouville initiated the general theory of integral equations.     

Numerical solution of a class of integral equations arising in two-dimensional aerodynamics

We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This enables us to study the problem in terms of an equivalent integral equation of the second kind. Using Galerkin's method, we are able to derive a convergent numerical algorithm for its solution. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as our method of computation. Extensive numerical calculations are presented establishing the validity of the theory.This paper was prepared with support of the National Aeronautics and Space Administration, Grant No. NSG-2140.The authors would like to acknowledge the help of Messrs. Tuli Haromy, Charles Doughty, Karl Kuopus, and Steven Sedlacek in the preparation of this paper.