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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

    π‘˜

    𝑗=0

    = β„Ž 𝛽𝑗𝑓𝑛+𝑗

    π‘˜

    𝑗=0

    (1)

    where𝛼𝑗 and 𝛽𝑗 are uniquely determined and 𝛼0 + 𝛽0 β‰  0, π›Όπ‘˜ = 1 [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 π‘₯ = 𝛼𝑗 π‘₯ 𝑦𝑛+𝑗

    π‘˜

    𝑗=0

    + β„Ž 𝛽𝑗 π‘₯ 𝑓𝑛+𝑗

    π‘˜

    𝑗=0

    (2)

    where𝛼𝑗 and 𝛽𝑗 are expressed as continuous functions of π‘₯ and are differentiable at least once [2]. 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: [3] derived continuous solvers

    of IVPs using Chebyshev polynomial in a multistep collocation technique; [4] proposed a two-step continuous

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

    technique; [5] 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; [6]

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

    order IVPs; [7] derived a three-step hybrid LMM for solving first order Initial Value Problems (IVPs) of ODEs

    based on collocation and interpolation technique; [8] 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; [9] adopted the method of collocation and interpolation of power series approximate solution

    to generate a continuous LMM; [10] derived continuous LMMs for solving first order ODEs by interpolation

    and collocation technique using Hermite polynomials as basis functions; [11] 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; [12] presented extrapolation-based implicit-explicit second derivative

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

    [13] to derive a hybrid LMM with multiple hybrid predictors for solving stiff IVPs of ODEs; [14] 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 [15] 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 π‘₯ = 𝛼𝑗π‘₯ 𝑗

    π‘˜

    𝑗=0

    . (3)

    [18]developed continuous LMMs using Chebyshev polynomial function of the form:

    𝑦 π‘₯ = 𝛼𝑗𝑇𝑗 π‘₯ βˆ’ π‘₯π‘˜ β„Ž

    𝑀

    𝑗=0

    ,

    where𝑇𝑗 π‘₯ are some Chebyshev function.

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

    𝑦 π‘₯ = 𝛼𝑗 π‘₯ βˆ’ π‘₯π‘˜ 𝑗

    π‘˜

    𝑗=0

    . (4)

    [10]developed continuous LMMs using the Probabilists’ Hermite polynomials of the form

    𝑦 π‘₯ = π‘Žπ‘—π»π‘— π‘₯ βˆ’ π‘₯π‘˜

    π‘˜

    𝑗=0

    where𝐻𝑗 (π‘₯) 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 π‘₯ = π‘Žπ‘—πΏπ‘— π‘₯ βˆ’ π‘₯π‘˜

    π‘˜

    𝑗=0

    where𝐿𝑗 (π‘₯) are Laguerre polynomials generated by the formula [19]:

    𝐿𝑛 π‘₯ = 𝑒π‘₯

    𝑛!

    𝑑𝑛

    𝑑π‘₯𝑛 π‘’βˆ’π‘₯π‘₯𝑛 =

    1

    𝑛! 𝑑

    𝑑π‘₯ βˆ’ 1

    𝑛

    π‘₯𝑛

    and satisfy the recursive relations

    (𝑛 + 1)𝐿𝑛+1 π‘₯ = (2𝑛 + 1 βˆ’ π‘₯)𝐿𝑛 π‘₯ βˆ’ π‘›πΏπ‘›βˆ’1 π‘₯ and

    π‘₯𝐿′ 𝑛 π‘₯ = 𝑛𝐿𝑛 π‘₯ βˆ’ π‘›πΏπ‘›βˆ’1 π‘₯ for the solution of first-order IVPs of ODEs of the form:

    𝑦 β€² = 𝑓 π‘₯, 𝑦 π‘₯ , 𝑦 π‘₯0 = 𝑦0 . (5) The first seven Laguerre polynomials generated from the recursive relation are:

    𝐿0 = 1, 𝐿1 = βˆ’π‘₯ + 1, 𝐿2 = 1

    2 π‘₯2 βˆ’ 4π‘₯ + 2 , 𝐿3 =

    1

    6 βˆ’π‘₯3 + 9π‘₯2 βˆ’ 18π‘₯ + 6 ,

    𝐿4 = 1

    24 π‘₯4 βˆ’ 16π‘₯3 + 72π‘₯2 βˆ’ 96π‘₯ + 24 ,

    𝐿5 = 1

    120 βˆ’π‘₯5 + 25π‘₯4 βˆ’ 200π‘₯3 + 600π‘₯2 βˆ’ 600π‘₯ + 120 ,

    𝐿6 = 1

    720 π‘₯6 βˆ’ 36π‘₯5 + 450π‘₯4 βˆ’ 2400π‘₯3 + 5400π‘₯2 βˆ’ 4320π‘₯ + 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(π‘₯) to the IVP in Equation (5) by a polynomial of degree 𝑛 of the form:

    𝑦 π‘₯ = π‘Žπ‘—πΏπ‘— π‘₯ βˆ’ π‘₯π‘˜

    𝑛

    𝑗=0

    , π‘₯π‘˜ ≀ π‘₯ ≀ π‘₯π‘˜+𝑝 (6)

    which satisfies the equations

    𝑦′ π‘₯ = 𝑓 π‘₯, 𝑦 π‘₯ , π‘₯π‘˜ ≀ π‘₯ ≀ π‘₯π‘˜+𝑝

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