Methods For Volterra Integral Equations Research Articles

Convergence and stability results in Runge-Kutta type methods for Volterra integral equations of the second kind

Runge-Kutta type methods for Volterra integral equations of the second kind are introduced which contain additional terms in order to improve the stability behaviour. The order of convergence is given and the stability region is derived for the basic test kernelK=af.

The repetition factor and numerical stability of volterra integral equations

In this paper we prove that direct linear multistep methods for Volterra integral equations of the second kind with repetition factor equal to one are always stable. We show trivially that this result is not true for first kind equations. We also demonstrate constructively that direct linear multistep methods for both first and second kind Volterra integral equations can have repetition factors greater than one, and indeed of arbitrary high order, and be numerically stable. Finally we explain why the first form of Simpson's rule for second kind equations is stable while the second form is unstable.

A note on collocation methods for Volterra integral equations of the first kind

It has been shown that if a Volterra integral equation of the first kind with continuous kernel is solved numerically in a given intervalI by collocation in the space of piecewise polynomials of degreem≧0 and possessing finite discontinuities at their knotsZ N then a careful choice of the collocation points yields convergence of orderp=m+2 on a certain finite subset ofI (while the global convergence order ism+1; this subset does not contain the knotsZ N . In this note it will be shown that superconvergence onZ N can be attained only if some of the collocation points coalesce (Hermite-type collocation).

Quadrature Rule Methods for Volterra Integral Equations of the First Kind

A new class of quadrature rule methods for solving nonsingular Volterra integral equations of the first kind are introduced; these methods are based on an appropriate modification of the higher-order Newton-Gregory methods which are known to be divergent. Methods up to order six are constructed explicitly and illustrated with numerical examples.

Quadrature rule methods for Volterra integral equations of the first kind

A new class of quadrature rule methods for solving nonsingular Volterra integral equations of the first kind are introduced; these methods are based on an appropriate modification of the higher-order Newton-Gregory methods which are known to be divergent. Methods up to order six are constructed explicitly and illustrated with numerical examples.

Remarks on frequency domain methods for Volterra integral equations

The equation u( t) = − ∫ 0 t A( t − τ) g( u( τ)) dτ + h( t), t ⩾ 0 is studied on a Hilbert space H. A( t) is a family of bounded linear operators and g can be unbounded and nonlinear. Stability and asymptotic stability of solutions are studied. Frequency domain conditions are statements about the Laplace transform of A. An extension of the frequency domain method of Popov for H = R 1 is given. Here it is assumed that g is the gradient of a functional G. The frequency domain conditions are related to monotonicity and convexity conditions on A thus connecting Popov's result with work of Levin and London on equations in R 1. A second result is given in which g is not assumed to be a gradient. This extends a result of Levin in R 1. The ideas are illustrated by an example of a nonlinear partial differential functional equation.

High Order Methods for Volterra Integral Equations of the First Kind

The application of block by block methods for the numerical solution of Volterra integral equations of the first kind is considered. Two classes of numerical schemes are developed. A necessary and sufficient condition for convergence and a sufficient condition for numerical stability are given.

Improved Numerical Methods for Volterra Integral Equations of the First Kind

Improved Numerical Methods for Volterra Integral Equations of the First Kind

Product integration methods for Volterra integral equations of the first kind

The numerical solution of Volterra integral equations of the first kind can be accomplished if the integral is replaced by certain simple quadrature rules, such as the midpoint or the trapezoidal methods. When the kernel of the integral equation oscillates more rapidly than the solution one can use product integration techniques to increase the accuracy. Such an approach is investigated in this paper.

Improved numerical methods for Volterra integral equations of the first kind

Improved numerical methods for Volterra integral equations of the first kind

Numerial Methods for Volterra Integral Equations with Singular Kernels

Numerial Methods for Volterra Integral Equations with Singular Kernels

A Method for Solving Nonlinear Volterra Integral Equations of the Second Kind

The approach given in this paper leads to numerical methods for Volterra integral equations which avoid the need for special starting procedures. Formulae for a typical fourth-order method are derived and some numerical examples presented. A convergence theorem is given for the method described.

A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems

A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems

Numerical methods for Volterra integral equations of the first kind

Numerical methods for Volterra integral equations of the first kind

A method for solving nonlinear Volterra integral equations of the second kind

The approach given in this paper leads to numerical methods for Volterra integral equations which avoid the need for special starting procedures. Formulae for a typical fourth-order method are derived and some numerical examples presented. A convergence theorem is given for the method described.