Legendre spectral collocation method for Fredholm integro-differential-difference equation with variable coefficients and mixed conditions

A priori Nyström-method error bounds in approximate solutions of 1-D Fredholm integro-differential equations

“…Many authors [18,4,27,15,16,7,28,5,3,19,22] have developed a diverse range of numerical methods for obtaining the solution of not only (1) but also its extension to higher differential orders. The primary aim of the present paper is to develop and to implement a computational method that yields explicitly computable and spectrally accurate a priori error predictions for the numerical solution of (1) using proves a convergence theorem for the iterative method, but does not explicitly analyse errors.…”

Error analysis of a spectrally accurate Volterra-transformation method for solving 1-D Fredholm integro-differential equations

“…from which||r N || can be computed directly; similarly,||s|| can be computed directly from (48). Finally, (35), (37) and (46) give the case-1 theoretical bound…”

“…In the substantial literature on the approximation of solutions of one-dimensional Fredholm integrodifferential equations (FIDEs), corresponding error analyses are notably scarce. For example, though the independent studies (in chronological order) [38,4,39,28,5,15,34,40,8,31,2,1,22,35] present diverse FIDE-solution techniques of varying degrees of efficiency and (disparate) accuracy, only [28,40,31,1] include a discussion of errors and, in even these cases, error analyses are limited (see summary in [21, §1]) to estimates of convergence rates: that is, the direct computation of theoretically predicted error bounds is almost entirely absent.…”

A new class of operational matrices method for solving fractional neutral pantograph differential equations

“…Fractional calculus is a generalization of calculus to an arbitrary order. In recent years, it is extensively applied to various fields such as viscous elastic mechanics, power fractal networks, electronic circuits [1][2][3]. Due to the memory and non-local characters of fractional derivative, many scholars use fractional differential equations to simulate complex phenomenon in order to make it closer to the real problem.…”

“…With the help of these matrices and orthogonal polynomials, we can reduce a fractional differential or integral equation to algebraic equations, and get the approximate solution. Much research in this field has emerged, such as Legendre operational matrices [1][2][3], Chebyshev operational matrices [4], block pulse operational matrices [5].…”