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• IOSR Journal of Mathematics (IOSR-JM)

e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 15, Issue 6 Ser. IV (Nov – Dec 2019), PP 51-64 www.iosrjournals.org

DOI: 10.9790/5728-1506045164 www.iosrjournals.org 51 | Page

Continuous Implicit Linear Multistep Methods for the Solution of

Initial Value Problems of First-Order Ordinary Differential

Equations

B.V. Iyorter 1 , T. Luga

2 andS.S. Isah

3

1 Department of Mathematics and Computer Science, College of Science and Education, University of Mkar,

Mkar, Nigeria

2, 3 Department of Mathematics/Statistics/Computer Science,College of Science, Federal University of

Agriculture, Makurdi, Nigeria.

Corresponding Author:B.V. Iyorter

Abstract: Continuous implicit linear multistep methods are developed for the solution of initial value problems of first-order ordinary differential equations. The linear multistep methods are derived via interpolation and

collocation approach using Laguerre polynomials as basis functions. The discrete form of the continuous

methods are evaluated and applied to solve first-order ordinary differential equations. The proposed optimal

order methods producedbetter approximations than the Adams-Moulton methods. Keywords:Linear Multistep Methods, Laguerre Polynomial, Collocation, Interpolation, Optimal Order Scheme. ----------------------------------------------------------------------------------------------------------------------------- ----------

Date of Submission:13-11-2019 Date of Acceptance:27-11-2019

----------------------------------------------------------------------------------------------------------------------------- ----------

I. Introduction Many phenomena in real life are described in terms of differential equations. Most of these equations

are difficult to solve using analytic techniques, hence, numerical approximations are obtained. Some of the

numerical methods for solving ordinary differential equations are the Linear Multistep Methods (LMMs). They

are not self-starting, hence requires single-step methods like the Runge-Kutta family of methods for starting

values.

Linear multistep methods are generally defined as

𝛼𝑗𝑦𝑛+𝑗

𝑘

𝑗=0

= ℎ 𝛽𝑗𝑓𝑛+𝑗

𝑘

𝑗=0

(1)

where𝛼𝑗 and 𝛽𝑗 are uniquely determined and 𝛼0 + 𝛽0 ≠ 0, 𝛼𝑘 = 1 .

Discrete schemes are generated from Equation (1) and applied to solve first order Ordinary Differential

Equations (ODEs). Continuous collocation and interpolation technique is applied in the development of LMMs

of the form

𝑦 𝑥 = 𝛼𝑗 𝑥 𝑦𝑛+𝑗

𝑘

𝑗=0

+ ℎ 𝛽𝑗 𝑥 𝑓𝑛+𝑗

𝑘

𝑗=0

(2)

where𝛼𝑗 and 𝛽𝑗 are expressed as continuous functions of 𝑥 and are differentiable at least once . Continuous

collocation and interpolation technique can also be applied in the derivation of block and hybrid methods. Other

techniques for derivation of LMMs include interpolation, numerical integration and Taylor series expansion.

Several researchers have applied different techniques to derive LMMs:  derived continuous solvers

of IVPs using Chebyshev polynomial in a multistep collocation technique;  proposed a two-step continuous

multistep method of hybrid type for the direct integration of second order ODEs in a multistep collocation

technique;  developed a new class of continuous implicit hybrid one-step methods for the solution of second

order ODEs using the interpolation and collocation techniques of the power series approximate solution; 

developed a four step continuous block hybrid method with four non-step points for the direct solution of first-

order IVPs;  derived a three-step hybrid LMM for solving first order Initial Value Problems (IVPs) of ODEs

based on collocation and interpolation technique;  adopted the method of collocation of the differential

system and interpolation of the approximate solution at grid and off grid points to yield a continuous LMM with

constant stepsize;  adopted the method of collocation and interpolation of power series approximate solution

to generate a continuous LMM;  derived continuous LMMs for solving first order ODEs by interpolation

and collocation technique using Hermite polynomials as basis functions;  proposed a family of single-step,

• Continuous Implicit Linear Multistep Methods for the Solution of Initial Value Problems of First-

DOI: 10.9790/5728-1506045164 www.iosrjournals.org 52 | Page

fifth and fourth order continuous hybrid linear multistep methods for solving first order ODEs derived via

interpolation and collocation procedure;  presented extrapolation-based implicit-explicit second derivative

linear multistep methods for solving stiff ODEs.The interpolation and collocation approach was also adopted by

 to derive a hybrid LMM with multiple hybrid predictors for solving stiff IVPs of ODEs;  derived

continuous linear multistep methods for solving Volterraintegro-differential equations of second order by

interpolation and collocation technique using the shifted Legendre polynomial as basis function with

Trapezoidal quadrature and  derived a family of hybrid linear multi-step methods type for solution of third

order ODEs. In this paper, continuous implicit LMMs for the solution of IVPs of first order ODEs are

developedvia collocation and interpolation approach using the Laguerre polynomials as basis functions. The

corresponding discrete schemes areevaluated.

II. Methods Many researchers have proposed LMMs in form of different polynomial functions. [16, 17] proposed

continuous LMM in form of a powers series polynomial function of the form

𝑦 𝑥 = 𝛼𝑗𝑥 𝑗

𝑘

𝑗=0

. (3)

developed continuous LMMs using Chebyshev polynomial function of the form:

𝑦 𝑥 = 𝛼𝑗𝑇𝑗 𝑥 − 𝑥𝑘 ℎ

𝑀

𝑗=0

,

where𝑇𝑗 𝑥 are some Chebyshev function.

proposed a polynomial function similar to the one in Equation (3) but presented as

𝑦 𝑥 = 𝛼𝑗 𝑥 − 𝑥𝑘 𝑗

𝑘

𝑗=0

. (4)

developed continuous LMMs using the Probabilists’ Hermite polynomials of the form

𝑦 𝑥 = 𝑎𝑗𝐻𝑗 𝑥 − 𝑥𝑘

𝑘

𝑗=0

where𝐻𝑗 (𝑥) are probabilists’ Hermite polynomials.

Several other polynomial functions have been used for the derivation of linear multistep methods, however, in

this paper, we will develop continuous LMMs and obtain their corresponding discrete forms using the Laguerre

polynomials of the form:

𝑦 𝑥 = 𝑎𝑗𝐿𝑗 𝑥 − 𝑥𝑘

𝑘

𝑗=0

where𝐿𝑗 (𝑥) are Laguerre polynomials generated by the formula :

𝐿𝑛 𝑥 = 𝑒𝑥

𝑛!

𝑑𝑛

𝑑𝑥𝑛 𝑒−𝑥𝑥𝑛 =

1

𝑛! 𝑑

𝑑𝑥 − 1

𝑛

𝑥𝑛

and satisfy the recursive relations

(𝑛 + 1)𝐿𝑛+1 𝑥 = (2𝑛 + 1 − 𝑥)𝐿𝑛 𝑥 − 𝑛𝐿𝑛−1 𝑥 and

𝑥𝐿′ 𝑛 𝑥 = 𝑛𝐿𝑛 𝑥 − 𝑛𝐿𝑛−1 𝑥 for the solution of first-order IVPs of ODEs of the form:

𝑦 ′ = 𝑓 𝑥, 𝑦 𝑥 , 𝑦 𝑥0 = 𝑦0 . (5) The first seven Laguerre polynomials generated from the recursive relation are:

𝐿0 = 1, 𝐿1 = −𝑥 + 1, 𝐿2 = 1

2 𝑥2 − 4𝑥 + 2 , 𝐿3 =

1

6 −𝑥3 + 9𝑥2 − 18𝑥 + 6 ,

𝐿4 = 1

24 𝑥4 − 16𝑥3 + 72𝑥2 − 96𝑥 + 24 ,

𝐿5 = 1

120 −𝑥5 + 25𝑥4 − 200𝑥3 + 600𝑥2 − 600𝑥 + 120 ,

𝐿6 = 1

720 𝑥6 − 36𝑥5 + 450𝑥4 − 2400𝑥3 + 5400𝑥2 − 4320𝑥 + 720 .

• Continuous Implicit Linear Multistep Methods for the Solution of Initial Value Problems of First-

DOI: 10.9790/5728-1506045164 www.iosrjournals.org 53 | Page

Derivation of the Linear Multistep Methods

We wish to approximate the exact solution 𝑦(𝑥) to the IVP in Equation (5) by a polynomial of degree 𝑛 of the form:

𝑦 𝑥 = 𝑎𝑗𝐿𝑗 𝑥 − 𝑥𝑘

𝑛

𝑗=0

, 𝑥𝑘 ≤ 𝑥 ≤ 𝑥𝑘+𝑝 (6)

which satisfies the equations

𝑦′ 𝑥 = 𝑓 𝑥, 𝑦 𝑥 , 𝑥𝑘 ≤ 𝑥 ≤ 𝑥𝑘+𝑝

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