A Family of Implicit Higher Order Methods for the Numerical Integration of Second Order Differential Equations

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  Mathematical Theory and Modelingwww.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 201267 A Family of Implicit Higher Order Methods for the NumericalIntegration of Second Order Differential Equations Owolabi Kolade MatthewDepartment of Mathematics, University of Western Cape, 7535, Bellville, South Africa*E-mail of the corresponding author:kowolabi@uwc.ac.za;kmowolabi2@gmail.com  Abstract A family of higher order implicit methods with k  steps is constructed, which exactly integrate the initialvalue problems of second order ordinary differential equations directly without reformulation to first ordersystems.   Implicit methods with step numbers }6,...,3,2{ ∈ k  are considered. For these methods, astudy of local truncation error is made with their basic properties. Error and step length control based onRichardson extrapolation technique is carried out. Illustrative examples are solved with the aid of MATLAB package. Findings from the analysis of the basic properties of the methods show that they areconsistent, symmetric and zero-stable. The results obtained from numerical examples show that thesemethods are much more efficient and accurate on comparison.   These methods are preferable to someexisting methods owing to the fact that they are efficient and simple in terms of derivation and computation Keywords: Error constant, implicit methods, Order of accuracy, Zero-Stability, Symmetry 1. Introduction In the last decade, there has been much research activity in the area of numerical solution of higher orderlinear and nonlinear initial value problems of ordinary differential equations of the form 10)1()( )(,0)...,,,,( −− == mmm t  y y y yt  f  η    n  yt m ℜ∈= },{,...,2,1  (1)   which are of great interest to Scientists and Engineers. The result of this activity are methods which can beapplied to many problems in celestial and quantum mechanics, nuclear and theoretical physics, astrophysics,quantum chemistry, molecular dynamics and transverse motion to mention a few. In literature, most modelsencountered are often reduced to first order systems of the form ],[,)(),,( 00 bat  yt  y yt  f  y ∈==′  (2)before numerical solution is sought [see for instance, Abhulimen and Otunta (2006), Ademiluyi andKayode (2001), Awoyemi (2005), Chan et al. (2004)].In this study, our interest is to develop a class of k-steps linear multistep methods for integration of generalsecond order problems without reformulation to systems of first order. We shall be concerned primarilywith differential equations of the type  Mathematical Theory and Modelingwww.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 201268 1,0,),,(],,[,)(),,,( 0)( =ℜ∈′∈=′=′′ m y yt  f bat t  y y yt  f  y nmm η   (3)Theorem 1 If f(t,y), f:  Ʀ  x  Ʀ    →    Ʀ  is defined and continuous on all t  Є [ a, b ] and  ∞<<∞−  y and a constant  L exist such that  **),(),(  y y L yt  f  yt  f  −<−   (4)  for every pair (t, y) and (t, y*) in the quoted region then, for any y 0 Є  Ʀ  the stated initial value problemadmits a unique solution which is continuous and differentiable on [ a, b ].   Efforts are made to develop a class of implicit schemes of higher step-numbers with reduced functionsevaluation for direct integration of problem (3) for k=2,3,... ,6.The remainder of the paper is organized in the following way. Under materials and methods, constructionof the schemes for approximating the solutions of (3) is presented with the analysis of their basic propertiesfor proper implementation. Some sample problems coded in MATLAB are equally considered. Finally,some concluding comments are made to justify the obtainable results and suitability of the proposedschemes on comparisons. 2. Materials and methods2.1 Construction of the schemes : The proposed numerical method of consideration for direct integration of general second order differential equations of type (3) is of the form )...(... 110211110 k nk nnk nk nnk n  f  f  f h y y y y ++−+−++ +++++++= β  β  β α α α  ,   (5)taken from the classical K-step method with the algorithm ∑ ∑ = =++ == k  jk  j jn j jn j n f h y 002 ...,1,0,  β α   (6) where y n+j is an approximation to y(x n+j ) and fn+j =f(x n+j , y n+j , y’ n+j ). The coefficients α  j and  β  j are constantswhich do not depend on n subject to the conditions 0,1 00 ≠+= β α α  k   are determined to ensure that the methods are symmetric, consistent and zero stable. Also, method (4) isimplicit since β k  ≠ 0.The values of these coefficients are determined from the local truncation error ( lte ) )]...(...[ 110211110 k nk nnk nk nnk nk n  f  f  f h y y y yT  ++−+−+++ +++++++−= β  β  β α α α   (7)  Mathematical Theory and Modelingwww.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 201269 generated by one-step application of (5) for numerical solution of (3). Clearly, accuracy of theseschemes depend on the real constants α  j and  β  j . In attempt to obtain the numerical values of these constants,the following steps were adopted;Taylor series expansion of  k n  y + ,  121 .,..,, −+++ k nnn  y y y and k nnn  f  f  f  +++ .,..,, 21 about the point ),( nn  yt  yields )(0!)(...!2)()( )1()()2( 2)1( ++ +++++=  p pn pnnnk n h y pkh ykh ykh yT    })!1()(0!)(...!2)()({ )1(1)()2( 2)1(10 ++−= ++++++− ∑  pn p pn pnnnk  j j  y p jh y p jh y jh y jh y α     −+−++++− +−−= ∑ )1(1)(2)4(2)3()2( 02 )!1()(0)!2()(...!2)()(  pn p pn pnnnk  j j  y p jh y p jh y jh y jh yh β   (8)Terms in equal powers of  h are collected to have +      −−+      −+      −= ∑ ∑∑∑ −= =−=−=+ )2(2 10022)1(1010 !2)(!21 nk  jk  j j jnk  j jnk  j jk n  yh jk hy jk  yT  β α α α    ++      −− ∑∑ =−= ...!3)(!3 )3(3 01033 nk  j jk  j j  yh j jk   β α    ( ) 1)(0210 0)2(! )(! +=−−= +      −−− ∑∑  p pn pk  j j pk  j j p p h yh p j p j pk   β α   (9)Accuracy of order p is imposed on T n+k  to obtain C i =0, 0 ≤ i ≤ p . Setting k=2(3)6, j=0(1)6 in equation (9),the obtainable algebraic system of equations are solved with MATLAB in the form  AX=B for variousstep-numbers to obtain coefficients of the methods parameters displayed in Table 0.Using the information in Table 0 for k=2(3)6 in (5), we have the following implicit schemes )10(122 12212 nnnnn  f  f  f h y y y n +++−= ++++  (10)P=4, C p+2 ≈ -4.1667x10 -3  which coincides with Numerov’s method of Lambert (see for more details in [7]) )99( 1233 1232123 nnnnnnn  f  f  f  f  h y y y y n −−+++−= ++++++    Mathematical Theory and Modelingwww.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 201270 (11)P=5, C p+2 ≈ -4.1667x10 -3   )8188( 12464 1234 21234 nnnnnnnnn  f  f  f  f  f  h y y y y y n −+−++−+−= ++++++++  (12)P=6, C p+2 ≈ -4.1667x10 -3 )726267( 12510105 12345 212345 nnnnnn nnnnn  f  f  f  f  f  f  h y y y y y y n −−+−++ +−+−= +++++ +++++  (13)P=7, C p+2 ≈ -4.1667x10 -3 )63352336( 1261520156 123456 2123456 nnnnnnn nnnnnn  f  f  f  f  f  f  f  h y y y y y y y n ++−+−++ −+−+−= ++++++ ++++++  (14)P=7, C p+2 ≈ -4.1667x10 -3  2.2 Analysis of the basic properties of methods (10),...,(14). To justify the accuracy and applicability of our proposed methods, we need to examine their basicproperties which include order of accuracy, error constant, symmetry, consistency and zero stability. Order of accuracy and error constant  :Definition1. Linear multistep methods (10)-(14) are said to be of order p, if p is the largest positive integer  for which C  0 =C  1 = ... =C   p =C   p+1 =0 but C   p+2   ≠ 0 . Hence, our methods are of orders p =4(5)8 with principaltruncation error C p+2 ≈ -4.1667x10 -3 . Symmetry : According to Lambert (1976), a class of linear multistep methods (10)-(14) is symmetric if   jk  j − = α α     jk  j − = β  β  , j=0(1) k/2, for even k (15)  jk  j − −= α α     jk  j − −= β  β  , j=0(1) k, for odd k (16)  Consistency  Deinition2:  A linear multistep method is consistent if;a). It has order p ≥  1
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