469 research outputs found
High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels
The solution of the Volterra integral equation, where , and are smooth functions, can be represented as ,, where , are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate via , in a neighborhood of the origin and use (*) on the rest of the interval . In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson's rule are treated in detail. The schemes are shown to be convergent of order . Asymptotic error estimates are derived in order to examine the numerical stability of the methods
Solving Continuous and Weakly Singular Linear Volterra Integral Equations of the second kind by Laplace Transform Method
This work provides solutions to some continuous and weakly singular linear Volterra integral equations of the second kind by Laplace transform method. With the basic definition of convolution integral of two functions and Volterra fundamental theorems, the Laplace transform method gives an efficient and remarkable performance. Test problems are presented to show the efficiency and reliability of the method. Keywords: Volterra Integral equations, continuous and weakly singular Kernels , Laplace metho
A new approach to the numerical solution of weakly singular Volterra integral equations
AbstractWe consider linear weakly singular Volterra integral equations of the second kind, with kernels of the form k(x,v)=|x−v|−αK(x,v),0<α<1, or k(x,v)=log|x−v|K(x,v), K(x,v) being a smooth function. The solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration. By a transformation of the unknown function we obtain an equation which is still weakly singular, but whose solution is as smooth as we like. This resulting equation is then solved by standard product integration methods
A Product Integration type Method for solving Nonlinear Integral Equations in L
This paper deals with nonlinear Fredholm integral equations of the second
kind. We study the case of a weakly singular kernel and we set the problem in
the space L 1 ([a, b], C). As numerical method, we extend the product
integration scheme from C 0 ([a, b], C) to L 1 ([a, b], C)
An improvement of the product integration method for a weakly singular Hammerstein equation
We present a new method to solve nonlinear Hammerstein equations with weakly
singular kernels. The process to approximate the solution, followed usually,
consists in adapting the discretization scheme from the linear case in order to
obtain a nonlinear system in a finite dimensional space and solve it by any
linearization method. In this paper, we propose to first linearize, via Newton
method, the nonlinear operator equation and only then to discretize the
obtained linear equations by the product integration method. We prove that the
iterates, issued from our method, tends to the exact solution of the nonlinear
Hammerstein equation when the number of Newton iterations tends to infinity,
whatever the discretization parameter can be. This is not the case when the
discretization is done first: in this case, the accuracy of the approximation
is limited by the mesh size discretization. A Numerical example is given to
confirm the theorical result
Product integration methods for second-kind Abel integral equations
AbstractThe 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
Solution of a singular integral equation by a split-interval method
The article is available at http://www.math.ualberta.ca/ijnam/Volume-4-2007/No-1-07/2007-01-05.pdf. This article is not available through the Chester Digital RepositoryThis article discusses a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The split-interval method is discussed and examples demonstrate its effectiveness
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