A collocation procedure is developed for the linear and nonlinear Fredholm and Volterraintegro-differential equations, using the globally defined B-spline and auxiliary basis functions.The solutionis collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula.The error analysis of proposed numerical method is studied theoretically. Numerical results are given toil...

The purpose of this article is to explain how some apparently simple problems from numerical linear algebra are in fact extremely difficult, so that we cannot hope to solve them effectively in general. However, if we build and analyze algorithms to solve them in special cases of interest for the numerical analysis of partial differential equations, we find that the theory needed to validate the...

Sketching has emerged as a powerful technique for speeding up problems in numerical linear algebra, such as regression. In the overconstrained regression problem, one is given an n × d matrix A, with n d, as well as an n × 1 vector b, and one wants to find a vector x̂ so as to minimize the residual error ‖Ax− b‖2. Using the sketch and solve paradigm, one first computes S · A and S · b for a rand...

We use standard ideas from numerical linear algebra plus orthogonal bases for polynomials to compute abstract finite elements. We express nodal bases in terms of orthogonal ones. This paradigm allows us to differentiate polynomials, evaluate local bilinear forms, and compute with affine equivalent elements. It applies not only to Lagrangian elements, but also to Hermite elements, H(div) and H(c...

11 1951, Paul S. Dwyer pnblishcd I,inenr Comptntions, perhaps the first hook dcvoted cntirely to numerical linear algebra.' Digital conipiiting was in its infancy, and Dwyer focused on computation with tncchanical calculators. Nonctheless, the hook was state of the art. Figurc 1 reproduces a pagc of thc book dealing with Gaussian elimination. In 1954, Alston S. Householder published Prim+ p b n...

We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in i-'(R) to within 0(N~2) by a piecewise linear function with O(N) nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approximations, is accurate to 0(N~l). These numerical met...

This chapter discusses the design of modern numerical linear alge

In this paper, a numerical method is proposed to solve FredholmVolterra fractional integro-differential equation with nonlocal boundary conditions. For this purpose, the Chebyshev wavelets of second kind are used in collocation method. It reduces the given fractional integro-differential equation (FIDE) with nonlocal boundary conditions in a linear system of equations which one can solve easily...

The fundamental ideas of wavelets are introduced within the context of mathematical physics. We present essential background notions of mathematical bases, and discuss Fourier, polynomial, and wavelets bases in this light. We construct several types of wavelets and wavelet-like bases and illustrate their use in algorithms for the solution of a variety of integral and di erential equations.

This paper deals with the asymptotic stability of exact and discrete solutions of neutral multidelay-integro-differential equations. First, sufficient conditions are derived that guarantee the asymptotic stability of the continuous solutions. Then, adaptations of classical Runge-Kutta and linear multistep methods are suggested for solving these systems. Stability criteria are constructed for th...