Finite Difference Formulae for Unequal Sub- Intervals Using Lagrange s Interpolation Formula
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1 Int. Journal o Mat. Analyi, Vol., 9, no. 7, Finite Dierence Formulae or Unequal Sub- Interval Uing Lagrange Interpolation Formula Aok K. Sing a and B. S. Badauria b Department o Matematic, Faculty o Science, Banara indu Univerity, Varanai-5, India a aok@bu.ac.in, b drbbadauria@yaoo.com Abtract General inite dierence ormulae and te correponding error term ave been derived conidering unequally paced grid point, and uing Lagrange interpolation ormula. Furter te inite dierence ormulae and te error term or equally paced ub-interval ave alo been obtained a pecial cae o te preent tudy. Matematic Subject Claiication: Primary 5D, Secondary 5D5 Keyword: Finite dierence ormulae, Lagrange interpolation ormula, Error term, Clamped Simpon rule. Introduction Finite dierence metod i one o te very eective metod ued or olving te dierential equation ordinary or partial numerically. It involve replacing te derivative appearing in te dierential equation and boundary condition by uitable inite dierence approimation. Te accuracy o te olution depend upon te number o grid point, coen. By increaing te number o grid point one can increae te accuracy o te olution to a deire degree, owever it involve increaingly tediou matematical analyi. Baed on Tayor erie, Kan and Oba [-8] preented ome new dierence ceme or inite dierence approimation. Tey obtained cloed-
2 8 Aok K. Sing and B. S. Badauria orm epreion o tee new dierence ormulae, wic can give approimation o arbitrary order. Recently, uing Lagrange interpolation ormula, Sing and Torpe [9] ave given a general metod rom wic variou type o inite dierence ormulae can be obtained by aigning te uitable value to te parameter. Furter te metod alo acilitate te generation o inite dierence ormulae or iger derivative by dierentiation. owever te applicability o te above metod appear to be limited a teir metod old only wen te grid point are equally paced. Anoter cla o problem concerned wit te inite dierence ormulae in te numerical analyi i to ind tem in cae o unequal ubinterval. Ti ituation pecially arie in te matematical modelling wen te unction a been obtained eperimentally and te independent variable i not under te control o te eperimenter, tu one need to ind te inite dierence ormulae or te obervational data. To tackle ti ituation, by introducing generalized Vandermonde determinant, Li [] preented general eplicit dierence ormulae wit arbitrary order accuracy or approimating irt and iger order derivative, wic can be ued or bot equally and unequally paced data. owever, we ue te available Lagrange interpolation ormula to obtain te inite dierence ormulae or unequally paced ubinterval.. Analyi Uing te Lagrange interpolation ormula, unction can be epreed a[] were tand or j j n l j j j, wile l j i a given below l j π π j j. In te above equation, prime denote dierentiation wit repect to and.... π. Te truncation error, in te evaluation o i a given below: n
3 Finite dierence ormulae 87 E n π n! n ξ n were ξ denote n t derivative o ξ interval[, ]. Te interval [, ] n widt,,, n, wile ξ lie between te be divided into n ubinterval o unequal n n n i i..., uc tat. a Tree Point Finite dierence ormulae: For ti cae n, and ence etting, and 5 in equation we ave epreion or a. Dierentiating equation wit repect to, we get. 7 Ten by putting repectively,, derivative o and /, te ormulae or te irt order at te point,, can be obtained a, 8a, 8b and. 8c
4 88 Aok K. Sing and B. S. Badauria Uing 5 in equation, te error term ave been calculated a ollow E ξ 9. and E ξ Te correponding error term in te ormulae 8a, b, c or, and /, ξ ξ ξ. reult in repectively a, and Dierentiation o equation 7 produce dierence ormula or econd derivative a [ ], and te correponding error term can be obtained by dierentiating equation. For, we obtain te inite dierence ormulae and te correponding truncation error or equal ub-interval a obtained by Sing and Torpe [9]. b Four point inite dierence ormulae: In ti cae n, and ubtituting Sing and Torpe[],, and in te correponding equation o, we get were. Dierentiation o produce
5 Finite dierence ormulae 89. For /,, and /, te repective inite dierence ormulae are obtained rom a ollow:, 5a, 5b, 5c. 5d Te aociated error term or our point ormula i obtained rom equation a
6 8 Aok K. Sing and B. S. Badauria { } [ ] ξ iv E. Te truncation error correponding to equation 5a, b, c, d are derived repectively, rom equation a [,,, ] / ξ iv. Dierentiation o equation give dierence ormulae correponding to te econd derivative a. 7 For /,, and /, epreion 7 reult in, 8a, 8b, 8c and
7 Finite dierence ormulae 8, 8d repectively. Truncation error concerned wit equation 7 i obtained rom dierentiation o equation a { } [ ] ξ iv E 5, 9 rom wic repective truncation error correponding to equation 8 can be derived. Dierentiation o equation 7 give wic i te dierence ormula or te tird derivative in term o te unction at te our point ituated at unequal interval. Te correponding truncation error can be obtained by dierentiating equation 9. Putting in above epreion, we obtained te reult or equal ub-interval a obtained by Sing and Torpe [9]. c Five point inite dierence ormulae: Te value o n i our or ive point dierence ormulae. Subtituting,,,, in te correponding equation obtained rom, and ten dierentiating te reulting equation, we get
8 8 Aok K. Sing and B. S. Badauria Te epreion or te above derivative at and, are obtained by etting repectively, /, and / : and, 5
9 Finite dierence ormulae 8 were. Alo te truncation error correponding to - 5 are a given below: ξ v E ξ v E 7 and ξ v E 8 Te econd and tird order derivative o te unction at, and are, 9
10 8 Aok K. Sing and B. S. Badauria,,,.
11 Finite dierence ormulae 85 Equation 9- repreent te orward, central and backward dierence ormulae wic are widely ued to approimate te derivative in practice. Alo te ourt order derivative i given by iv. 5 Te truncation error correponding to 9-5 can be obtained by dierentiating, a obtained in -8 correponding to -5. A particular cae, i we put in te above epreion 9-5, we obtained te reult or equal ub-interval a given below 5 5 a b 5 5 c 8 5 d e 5 8 and iv, g
12 8 Aok K. Sing and B. S. Badauria wic are eactly ame a obtained by Sing and Torpe [9]. Formulation o te above inite dierence approimation clearly ugget tat te inite dierence ormulae in term o any number o point can be obtained wit te elp o equation and. Te matematical epreion reulting due to urter dierentiation o equation will give inite dierence ormulae or iger derivative.. Concluion ere we ave preented,, and 5 point eplicit inite dierence ormulae along wit te error term, or approimating te irt and iger order derivative or unequally paced data. By dierentiating te equation, inite dierence ormulae or iger order derivative can alo be obtained. Tee ormulae can be ued directly to olve te ordinary and partial dierential equation and will erve to approimate te derivative at te unequally paced grid point. A cientiic program o te above ormulae will acilitate te ue o any inite dierence ormulae o deired derivative. Acknowledgment. Autor would like to tank te Centre or Interdiciplinary Matematical Science, Banara indu Univerity, Varanai-5, or providing te inancial aitance and oter acilitie. Reerence [] F. B. ildebrand, Introduction to Numerical Analyi, Mc Graw ill, Inc, New York, 97. [] Jianping Li, General eplicit dierence ormula or numerical dierentiation, J. Comput. Appl. Mat., 8 5, 9 5. [] I. R. Kan and R. Oba, Cloed-orm epreion or te inite dierence approimation o irt and iger derivative baed on Taylor erie, J. Comput. Appl.Mat., 7 999a, [] I. R. Kan and R. Oba, Digital dierentiator baed on Taylor erie, IEICE Tran. Fund. E8-A, 999b, 8 8.
13 Finite dierence ormulae 87 [5] I. R. Kan and R. Oba, New inite dierence ormula or numerical dierentiation, J. Comput. Appl. Mat.,, 9 7. [] I. R. Kan and R. Oba, Matematical proo o eplicit ormula or tapcoeicient o Taylor erie baed FIR digital dierentiator, IEICE Tran.Fund.E8- A,, [7] I. R. Kan and R. Oba, Taylor erie baed inite dierence approimation o iger-degree derivative, J. Comput. Appl. Mat., 5 a, 5. [8] I. R. Kan, R. Oba and N. ozumi, Matematical proo o cloed orm epreion or inite dierence approimation baed on Taylor erie, J. Comput. Appl. Mat., 5 b, -9. [9] A. K. Sing and G. R. Torpe, Finite dierence ormulae rom Lagrange interpolation ormula, J. Scientiic Reearc, 5 8, -7. [] A. K. Sing and G. R. Torpe, Simpon /-rule o integration or unequal diviion o integration domain, J. Concrete Applicable Mat.,, 7-5. Received: September, 8
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