Inverse Hybrid Linear Multistep Methods for Solving the Second Order Initial Value Problems in Ordinary Differential Equ
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Inverse Hybrid Linear Multistep Methods for Solving the Second Order Initial Value Problems in Ordinary Differential Equations Oluwasegun M. Ibrahim1
· Monday N. O. Ikhile2
Accepted: 7 October 2020 © Springer Nature India Private Limited 2020
Abstract Inverse linear multistep methods (ILMMs) for first and second order differential equations have been proved to be suitable numerical methods for the solution of inverse initial value problems (IVPs). This paper presents the hybrid version of the ILMMs for the numerical solution of second order inverse IVPs. The stability of the proposed methods is represented in the boundary locus graph. The applicability of the schemes is demonstrated herein for the solution of linear and nonlinear problems. Computational results on the problems are compared with those from the existing method and ode45 (the explicit Runge–Kutta method). Keywords Inverse LMM · Second order inverse IVPs · Hybrid LMM Mathematics Subject Classifications 34M50 · 65L05 · 65L06 · 65L20
Introduction The modelling of real-life phenomenon may result in second order differential equations of the form F x, y, y , y = 0, y(a) = y0 , y (a) = y0 , a ≤ x ≤ b, F ∈ Rm . (1) In this regard, we consider the two special classes of (1) which are:
and
B
y = f (x, y) ,
y(a) = y0 ,
y (a) = y0 , a ≤ x ≤ b,
f ∈ Rm
(2)
y = η x, y ,
y(a) = y0 ,
y (a) = y0 , a ≤ x ≤ b, η ∈ Rm .
(3)
Oluwasegun M. Ibrahim [email protected] Monday N. O. Ikhile [email protected]
1
African Institute for Mathematical Sciences, Kigali, Rwanda
2
Advanced Research Laboratory, Department of Mathematics, University of Benin, Benin City, Nigeria 0123456789().: V,-vol
123
158
Page 2 of 17
Int. J. Appl. Comput. Math
(2020) 6:158
The initial value problem (IVP) (2) often occur in physical phenomenon such as mechanical systems without dissipation, satalite tracking and celestial mechanics [1]. Several numerical techniques have been used to solve the special class of the IVP (2). See for example, the works presented in [1,7,8,19,21,22,24] and other references herein. An example of a realistic problem emanating naturally from the theory of viscoelasticity that gives rise to an IVP of the form (3) is described in [2,3]. However, the IVP (2) sometimes may be converted easily into the IVP (3). For instance, the situation when f (x, y) = ωy, ω = 0, ω ∈ R. The equivalent IVP (3) is expressed as η x, y = ω−1 f (x, y). That is, y = ω−1 y . The IVP (2) of the form
f x, y = φ f (x, y)
with function φ sometimes may be converted into the integration problem (3). As in [4–6], the LMM for solving the second order IVP (2) is given by k
α¯ j yn+ j = h 2
j=0
k
β¯ j f n+ j , k ≥ 2.
(4)
j=0
There is a vast literature on the development of efficient numerical methods from the LMM (4) for the solution of special second order IVP (for which y is explicitly missing) (2). See the works in [1,7–12,19,22]. The direct application of the LMM (4) on IVP (2) has been found to be more advantag
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