This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems
Abstract
A linear first order ordinary differential equation (ODE) with a positive parameter ɛ and a multipoint nonlocal initial value condition (NLIVC) is considered. The existence of a classical solution of the multipoint nonlocal initial value problem (NLIVP) is proved. A uniform on ɛ a priori estimate and asymptotic expansion of smooth solution is obtained. The differential problem with integral kind of NLIVC is considered and reduced to appropriate multipoint NLIVP.
The approximations of the nonlinear heat transport problem are based on the finite volume (FM) and averaging (AM) methods [1,2]. This procedures allows reduce the nonlinear 2‐D problem for partial differential equation (PDE) to a initial‐value problem for a system of 2 nonlinear ordinary differential equations(ODE) of first order in the time t or to a initial‐value problem for one nonlinear ODE of first order with two nonlinear algebraic equations.
In this paper, we propose a method for studying the initial value problem for a first-order nonlinear integro-differential equation. The initial problem is reduced by substitution to a nonlinear integral equation with the Urson operator. To construct a solution to a nonlinear integral equation, the Newton-Kantorovich method is used.
On a multipoint nonlocal initial value problem for a singularly-perturbed first-order ODE
