A Pseudo Spline Methods for Solving an Initial Value Problem of Ordinary Differential Equation

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1 Joural of Matematics ad Statistics 4 (: 7-, 008 ISSN Sciece Publicatios A Pseudo Splie Metods for Solvig a Iitial Value Problem of Ordiary Differetial Equatio B.S. Ogudare ad G.E. Okeca Departmet of Pure ad Applied Matematics, Uiversity of Fort Hare, Alice, 5700 RSA Abstract: New sceme for solvig iitial value problem of ordiary differetial equatio was derived. Startig from te geeral metod of derivig te splie fuctio, te sceme was developed based o iterpolatio ad collocatio. Key words: Iitial value problems, ordiary differetial equatios, collocatio, pseudo splie fuctio INTRODUCTION Cosider te Iitial Value Problem (IVP: y'(t = f(t,y(t y(t = y 0 0 ( were, a t b, a = t 0 < t < t < < t N = b, (b a N =, N = 0,,, N- ad = t + is called te step legt. Te coditios o te fuctio f(t, y(t are suc tat existece ad uiqueess of solutio is guarateed []. Te umerical solutio of te Eq. ad received lots of attetio ad it is still receivig suc due to te fact tat may pysical (Egieerig, Medical, fiacial, populatio dyamics ad Biological scieces problems formulated ito matematical equatio results ito te above type. Te solutio is geerated i a step-by-step fasio by a formula wic is regarded as discrete replacemet of te Eq. [-5]. I te class of metods available i solvig te problem umerically, te most celebrated metods are te sigle-step ad te multisteps metods. I a sigle step, a iformatio at just oe poit is eoug to advace te solutio to te ext poit wile for te multisteps (as te ame suggests, iformatio at more ta oe previous poits will be required to advace te solutio to te ext poit. Te Euler s metod (te pioeerig metod, wic is te oldest metod ad te Ruge-Kutta metods fall i te class of te sigle-step metods wile te Adams metods are i te class of te multistep metod [-5].Te Adams metod is divided ito two amely te Adams- Basfort (explicit ad Adams-Moulto (implicit. Tese two metods combied ca be used as a predictorcorrector metod. Tis class of metod as bee proved to be oe of te most efficiet metod to solve certai class of IVP (o-stiff. I te literature, te derivatio of te Adams metod ad bee extesively dealt wit usig te iterpolatory polyomial for te discretised problem. For te derivatio of liear multisteps metod troug iterpolatio ad collocatio [,3,4,6]. Omolei et al. [8] used te collocatio metod to derive a ew class of te Adams-Basfort scemes for ODE wile Oumayi et al. [9] also used te collocatio metod for derivig a cotiuous multisteps metod. Lie et al. [7] discussed te super covergece properties of te collocatio metods. I tis study, we cosider a ew class of metods for solvig ( based o iterpolatio ad collocatio. Our metod is based o a geeral metod for derivig te splie fuctios. Te study is orgaised i te followig order, deals wit descriptio of piecewise iterpolatio fuctios, te derivatio of our sceme features is 3. Te results from some umerical examples will be give to illustrate ad validate our sceme i 4. Furtermore, our results will be compared wit a already kow sceme of Omolei et al. [8] wile te coclusio is featured i te last Sectio. Piecewise-iterpolatio: Oe of te metods of derivig te multisteps metod is by polyomial iterpolatio for a set of discrete poit, owever, polyomial iterpolatio for a set of (N+ poits {t k, y k } is frequetly usatisfactory because te iterpolatio error is related to iger derivatives of te iterpolated fuctio. To circumvet tis, we discretise te iterpolatio domai ad iterpolate Correspodig Autor: B.S. Ogudare, Departmet of Matematics, Obafemi Awolowo Uiversity, Ile-Ife, Nigeria 7

2 J. Mat. & Stat., 4 (: 7-, 008 locally. Te overall accuracy may be sigificatly improved eve if te iterpolatio polyomial is of low order. Iterpolatio fuctios obtaied o tis priciple are piece-wise iterpolatio fuctios or splies. We defie a splie fuctio as follows: Defiitio: A fuctio S(t is called a splie of degree k if: Te domai of S is te iterval [a, b] S, S, S,..., S (k- are all cotiuous o [a, b] Tere are poits t i (called kots suc tat a = t < t <... < t = b ad suc tat S is a polyomial of degree k o eac sub-iterval [t i, t i+ ], i =,...,. subject to te iterpolatig coditios S(t i = y(t i t [t i, t i+ ] i =,... - S (j r (t i = S (j r + (t i, j =,, k-, r =, -, i =, - Coditio (iv is te collocatio wile (v is te cotiuity coditio, oly o iterior kots. We sall ow use te piece-wise liear ad cubic iterpolatio splie fuctios to derive our metods. Adams metods are recoverable from our metods Derivatio of te sceme: Pseudo Quadratic splie fuctio: Let S(t be te desired fuctio, te liear lagrage iterpolatio formula gives te followig represetatio for S (t at te give poits t - ad t, for all t [t, t ], as: A = S(t + (t t S'(t (t Substitute (5 ito (4 we ave, S(t = S(t + (t S'(t + (t (t t S'(t (t t S'(t (t S'(t S(t = S(t + (t t (t t { } (t S'(t + (t (t t If i (4 we evaluate S(t at t = t : A = S(t (t t S'(t (t If (5 is substituted ito (4 we ave: S'(t S(t = S(t {(t t (t t } (t S'(t (t t (t (5 (6 (7 (5 (8 Collocatig (7 ad (8 at t = t + ad usig te property tat S(t y(t ad tat = t t we ave te: y + = y +f (9 S'(t S'(t S'(t S'(t = (t t (t t Simplifyig ( we ave: S'(t = {(t t S'(t + (t ts'(t } (t Itegratig (3: ( (3 ad y+ = y + {3f f } (0 If we also collocate (7 at t = t ad simplify we ave: y = y + {f + f } ( S(t = (t t S'(t (t S'(t + A (t (4 were, A is te costat of itegratio to be determied. Sice S(t iterpolates te fuctio f at t = t, it implies tat S(t = f(t, y(t. Tus for t = t : 8 Equatio (9 ad (0 correspod to te mid-poit rule ad te Adams-Basfort of secod order wile ( is a implicit metod (te Implicit Trapezoidal Metod. Various multisteps of te Adams forms ca be derived from te Eq. (7 ad (8 at differet collocatio poits (say t = t +, t +3,...

3 J. Mat. & Stat., 4 (: 7-, 008 Te local trucatio error: Assume tat y C 3 [a, b] for all x i a x b. Due to a stadard approac by Lambert [5] we ave bee able to sow tat te local trucatio errors associated wit tese umerical algoritms ca be expressed respectively as: e9 = y'''( ζ, ζ (x,x e0 = y '''( ζ, ζ (x, x + e = y'''( ζ, ζ (x, x Usig well kow aalysis i Herici [] ad Lambert [5], it ca be sow tat tese metods are all cosistet ad zero stable. Cosistecy ad zero stability are ecessary ad sufficiet coditios for te covergece of metods of tis kid, ece te tree umerical scemes are coverget wit errors of order O(. Pseudo cubic splie fuctio: Sice we are cosiderig a piecewise cubic splie, its secod derivative is piecewise liear o [t, t ], te te liear lagrage iterpolatio formula gives te represetatio for S (t at te give poits t ad t : S''(t S(t = (t + B(t (6 6 From (5 ad (6 we ave tat: S''(t A = S(t (tt (t 6 ad S''(t B= S(t (t t (t 6 Substitute for A ad B i (4, we ave: S''(t 3 S''(t 3 S(t = (t + (t t (t 6 6 S''(t + S(t (t t (t t (7 (t 6 S''(t + S(t (tt (tt (t 6 Collocatig (7 at t = t + yields: S(t = S(t S(t + S''(t (8 + By collocatio property, we ave: S''(t S''(t S''(t S''(t = (t t (t t S''(t = {(t S''(t + (t ts''(t } (t Itegratig Eq. (3 twice we ave, S''(t 3 S''(t 3 S(t = (t + (t t (t A(t + B(t t ( (3 (4 were, A ad B are costats. To determie tese costats, (4 is collocated at two poits say t = t ad t = t, tis yield: ad S''(t S(t = (t + A(t (5 6 9 y = y y + y'' (9 + ad usig (, we ave tat te coefficiet of i te Eq. 9 ca be replaced by: y'' = f t (t,y(t + fyf t (t,y(t (0 were, ere f t ad f y are te first partial derivatives of f(t, y(t wit respect to t ad y respectively. Usig te f+ f approximatio relatios, ft ad f+ f fy we simplify (8 to give: y+ = y y + {( + f (f+ f } y+ = y y + {(f+ f + ff+ ff } ( Neglectig te oliear part i (, Eq. becomes:

4 y+ = y y + {f+ f } ( wic is a implicit -step metod. Te local trucatio error associated wit ( as outlied for te scemes (9-( ca be sow to be 3. Te sceme was observed to be cosistet but to our surprise te metod is ot zero stable accordig to [,5] yet it gives a coverget solutio of maximum error of order O( 3. RESULTS Te scemes derived i tis study are: y + = y +f y+ = y + {3f f } y = y + {f + f } y+ = y y + {f+ f } Numerical examples: We sall cosider te followig problems:.. y y' =,y(0 =,t [0,] (t + Te exact solutio is give as: y(t = y' = y +, y(0 = 0.5, t [0,] + t Te exact solutio is give as: y(t = ( + t 0.5*exp(t J. Mat. & Stat., 4 (: 7-, 008 Table : Error of y(t for example ( = 0. t Metod A Metod B e e e e e e e e e e e e e e e e-003 Table : Error of y(t for example ( = 0. t Metod A Metod B e e e e e e e e e e e e e e e e-003 Table 3: Error of y(t for example ( = 0. for metods A ad B [8] t Metod A Metod B [8] e e e e e e e e e e e e e e e-003 Table 4: Error of y(t for example ( = 0. t Metod C Metod D Metod E Metod F e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-004 Table 5: Error of y(t for example ( = 0. t Metod C Metod D Metod E Metod F e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-004 Te umerical solutio geerated by tese metods DISCUSSION are compared wit te tird order metod of [8] ad tis is sow i Table 3. Te metods described by Eq. 9 ad 0 are From te tables of results displayed it could be respectively represeted as metod A ad B, wile see tat oe of our metods wic is of order metod C, D, E ad F are te combiatios of Eq. 9 performs better ta te tird order metod of [8]. wit (, (0 wit (, (9 wit ( ad (0 wit We te explicit metods of tis work are ( as predictor-corrector metods respectively. combied to form a predictor-corrector metod, te Table ad sows te maximum error of te results as sow i Table 4 ad 5 reveal tat tese Metods A ad B for te examples wit = 0.. metods give a better accuracy. 0

5 J. Mat. & Stat., 4 (: 7-, 008 CONCLUSION I tis study, we ave derived ew scemes for solvig first order differetial equatios based o iterpolatio ad collocatio troug te geeral metod for derivig te splie fuctios. Altoug we poited out i 3 tat tere are oter explicit metods tat could be obtaied from collocatig Eq. 7 ad 8 from oter poits oter ta te oes used i tis work, everteless tese metods are far less accurate due to istability. As a compariso to [8], Eq. 7 ad 8 are te cotiuous form of our sceme of order two wit equivalet discrete forms (9 ad (0. Our metod is easier to derive ad more user friedly ta te metod of derivatio i [8]. REFERENCES. Atkiso, K.E., 989. A Itroductio to Numerical Aalysis. d Ed. Jo Wiley ad Sos, New York. ISBN: Herici, P., 96. Discrete Variable Metods i Ordiary Differetial Equatios. st Ed. Jo Wiley ad Sos, New York. ISBN-3: Hildebrad, F.B., 974. Itroductio to Numerical Aalysis. d Editio. McGraw-Hill, New York. ISBN-3: Lambert, J.D., 973. Computatioal Metods i Ordiary Differetial Equatios. 3 rd Editio, Jo Wiley ad Sos, New York. ISBN-3: Lambert, J.D., 99. Numerical Metods for Ordiary Differetial Systems. d Editio Jo Wiley ad Sos, New York. ISBN- 3: Matew, J.H., 987. Numerical aalysis for Computer Sciece, Egieerig ad Matematics. st Editio Pretice-Hall, Ic., New Jersey. ISBN Lie, I. ad S.P. Norsett, 989. Supercovergece for multistep collocatio. Mat. Comp., 5: ttp:// 8. Omolei, J.O., M.A. Ibiejugba, M.O. Alabi ad D.J. Evas, 003. A ew class of adams-basfort scemes for ODEs. It. J. Comp. Mat., 80: DOI: 0.080/ Oumayi, P., U.W. Sirisea ad J. Fatoku, 00. A cotiuous metod for improved performace i te solutios of first order system of ordiary differetial equatios: Proc. Natl. Mat. Cet. Abuja, Nigeria, (, Natl. Mat. Cet., Abuja,

On the convergence, consistence and stability of a standard finite difference scheme

On the convergence, consistence and stability of a standard finite difference scheme AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 2, Sciece Huβ, ttp://www.sciub.org/ajsir ISSN: 253-649X, doi:.525/ajsir.2.2.2.74.78 O te covergece, cosistece ad stabilit of a stadard fiite differece