Advances on Collocation Based Numerical Methods for Ordinary Differential Equations and Volterra Integral Equations

Solution of two point boundary value problems, a numerical approach: parametric difference method

“…Following the ideas in [4], re-formulate the problem (1) either by introducing a physical parameter or identify a physical parameter that appears either in the differential equation or in the boundary conditions as independent variable. Let us introduce λ as physical parameter in differential equation 1, so we have obtained, We wish to determine the approximate numerical solution u(x) of the problem (2) for different values of λ and λ 0 ≤ λ ≤ λ M . We define M − 1 numbers of nodal points in [λ 0 , λ M ], as λ 0 < λ < λ 2 < ...... < λ M using a uniform step length H such that λ i = λ 0 + jH, j = 0, 1, .., M. Also, we define N − 1 numbers of nodal points in [a,b], the domain in which the solution of the problem (1) is desired, as a ≤ x 0 < x 1 < x 2 < ...... < x N = b using a uniform step length h such that x i = x 0 + ih, i = 0, 1, .., N. To simplify the expression we denote the numerical approximation of u(x) at the node x = x i as u i .…”

“…So it is essential to consider approximate solutions by means of numerical techniques. To solve these problems numerically, we have many accurate numerical methods for instance, shooting-projection [1], collocation [2], finite difference methods [3] available in the literature. However the relatively less advanced method, the method of parameter differentiation is known in the literature and some historical development can be found in [4].…”

Strong Stability Preserving Multistage Integration Methods

“…. , p, and four integers: the order of the method p, the stage order q, the number of external approximations r, and the number of internal approximations or stages s. The framework of general linear methods have been used recently to analyze the numerical stability of several class of methods [4,5,9,10,11,13,14,15]. General linear methods for second order equations were considered in [12,16,17] and for VIEs and VIDEs in [38,54].…”