On the Derivation and Implementation of a Four Stage Harmonic Explicit Runge-Kutta Method *

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1 Applied Mathematics, 05, 6, Published Olie April 05 i SciRes. O the Derivatio ad Implemetatio of a Four Stage Harmoic Explicit Ruge-Kutta Method * Ashiribo Seapo Wusu, Moses Adebowale Akabi, Bakre Omolara Fatimah Departmet of Mathematics, Lagos State Uiversit, Lagos, Nigeria Departmet of Mathematics, Federal College of Educatio (Techical), Lagos, Nigeria wuss_ash@ahoo.com, akabima@gmail.com, larabakre@gmail.com Received 9 Jauar 05; accepted April 05; published April 05 Copright 05 b authors ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). Abstract I recet times, the derivatio of Ruge-Kutta methods based o averages other tha the arithmetic mea is o the rise. I this paper, the authors propose a ew versio of explicit Ruge-Kutta method, b itroducig the harmoic mea as agaist the usual arithmetic averages i stadard Ruge-Kutta schemes. Kewords Explicit, Harmoic, Ruge-Kutta, Autoomous. Itroductio Durig the last few decades, there has bee a growig iterest i problem solvig sstems based o the Ruge- Kutta methods. Several methods have bee developed usig the idea differet meas such as the geometric mea, cetroidal mea, harmoic mea, cotra-harmoic mea ad the heroia mea. I previous papers [] ad [], the authors preseted a three stage method based o the harmoic mea ad a multi-derivative method usig the usual arithmetic mea respectivel. Akabi [] developed a third-order method based o the geometric mea. I [4] ad [5], the cocept of the heroia mea was itroduced. Evas ad Yaacob [6] itroduced a fourth-order method based o the harmoic mea while Yaacob ad Saugi [7] also developed a fourth-order method which is a embedded method based o the arithmetic ad harmoic mea. Wazwaz [8] preseted a compariso of modified Ruge-Kutta methods based o varieties of meas. Usig the * Four Stage Harmoic Ruge-Kutta Method. How to cite this paper: Wusu, A.S., Akabi, M.A. ad Fatimah, B.O. (05) O the Derivatio ad Implemetatio of a Four Stage Harmoic Explicit Ruge-Kutta Method. Applied Mathematics, 6,

2 defiitio of the harmoic mea, a fourth-order Ruge-Kutta method is developed ad implemeted.. Derivatio of the 4sHERK Method The schemes itroduced b [7] ad [9] respectivel are ad HM kk kk kk 4 = h k k k k k k 4 k f x =, k f = x h, hk 5 k f = x h, hk hk k4 f = x h, hk hk hk AHM kk kk 4 = h k k 6 6 k k k k4 k = f k = f hk 5 k f = hk hk k4 f = hk hk hk Scheme () was referred to as RK-HM-AM. Usig the defiitio of harmoic mea, the followig scheme is proposed i this paper:,,,, 4, 4 = h Φ h () H ; ( ; 4kkkk 4 Φ H h) = kkk kkk kkk kkk k = f k = f b hk (5) k = f b hk b hk (6) k = f b hk b hk b hk (7) b b b b b ad b 4 are costats to be determied. The expasio of k, k ad k 4 as defied above give () () (4) 695

3 4 k = f fhb f f h b f f h b f O( h ) (8) 6 k f fh( b b ) f h fb b f f ( b b ) f = h f b b b b b b f f f b b f O h 6 4 ( k = f fh b b b f h f b b b b b b f b4 b4 b 4 f b4b4 b4b4 b4b4 f h ( fb b b f f b b b b b b b b b b b b b b b b ) ( 4 4 b4 b4 f f f b4 b4 b4 f O h ) bb b b 4 Substitutig (8), (9) ad (0) ito (4) ad simplifig the resultig expressio usig MATHEMATICA (versio 8.0.) package, the coefficiets of the powers of h i (4) are compared with that of the Talors expasio Φ h ad upo solvig the resultig sstem of o-liear equatios we have of H ; b = ; b = 0; b = ; b4 = 0; b4 = 0; b4 = ; () Thus, the icremetal fuctio (4) of the proposed scheme is ff f f 5 5 f f Φ H ( ; h) = f fhf h h f f f () ad the proposed scheme () is 6 kkkk 4 = 4h kkk kkk 4 kkk 4 kkk 4 k = f k f = hk,, (9) (0) () (4) k = f hk (5) k4 f = hk (6). Stabilit of the 4sHERK Method For the aalsis of the absolute stabilit of the proposed 4sHERK scheme, the scalar test problem = λ with λ solutio = e is used, λ is a complex variable (see [0]). With the above test problem, we have k = f = λ (7) hλ k = f hk= λ (8) 696

4 h λ k = f ( hk) = λ hλ hλ h λ h λ k4 = f hk= λ 4 Substitutig (7)-(0) i () ad simplifig the resultig expressio results i, Lettig z = λh ad evaluatig (9) (0) 5 5 = hλ h λ h λ h λ 8 64 () from (), the stabilit polomial of the proposed scheme is obtaied as 5 6 R( z) = = z z z z O( z ) () 8 64 The absolute stabilit regio of the 4sHERK scheme is give i Figure. 4. Estimatio Defiitio: The local trucatio error at x of the explicit oe step method () is defied to be T ( ) T = x x hφ x h H ; Ad ( x ) is the theoretical solutio (See [0]). Usig the above defiitio together with (), the local trucatio error (LTE) of the proposed scheme is give as LTE ( 4sHERK) = ( x h) 7 f f f 7 f f = h ff f f f f O h , () Figure. Absolute stabilit regio of the 4sHERK method. 697

5 ( x h) is obtaied b Talor series expasio. 5. Numerical Experimets Cosider the IVP with the theoretical solutio =, = (4) = x (5) Table. h =0.5, ( x) =, ( 0) =, exact solutio: ( x) x ( x) =. x Exact Sol. RK-4 RK-HM-AM [4] RK-HM [6] 4sHERK RK-4 RK-HM-AM [4] RK-HM [6] 4sHERK 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 e e e e e e e e e e e e e e e e 8 Table. h = 0., ( x) =, ( 0) =, exact solutio: ( x) x ( x) =. x Exact Sol. RK-4 RK-HM-AM [4] RK-HM [6] 4sHERK RK-4 RK-HM-AM [4] RK-HM [6] 4sHERK 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 e e e e e e e e 9 698

6 Table. h = 0.0, ( x) =, ( 0) =, exact solutio: ( x) x ( x) =. x Exact Sol. RK-4 RK-HM-AM [4] RK-HM [6] 4sHERK RK-4 RK-HM-AM [4] RK-HM [6] 4sHERK 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 e e e e e e e e 5 We appl the ew 4sHERK method () to the above IVP ad the results obtaied are compared with the classical 4-stage fourth-order Ruge-Kutta method ad the methods of [6] ad [4]. The results geerated b the ewl derived scheme i this paper evidetl proved the extet of accurac of the scheme i compariso with the other methods. 6. Coclusio Evidetl, the ewl derived scheme is more accurate as see from the computatioal results preseted i Table, Table ad Table, sice its absolute error is the least of all the methods preseted i this paper. It therefore follows that the scheme is quite efficiet. We therefore coclude that the 4sHERK method proposed is reliable, stable ad with high accurac i computatio. Refereces [] Wusu, A.S., Okuuga, S.A. ad Sofoluwe, A.B. (0) A Third-Order Harmoic Explicit Ruge-Kutta Method for Autoomous Iitial Value Problems. Global Joural of Pure ad Applied Mathematics, 8, [] Wusu, A.S. ad Akabi, M.A. (0) A Three-Stage Multiderivative Explicit Ruge-Kutta Method. America Joural of Computatioal Mathematics,, -6. [] Akabi, M.A. (0) O -Stage Geometric Explicit Ruge-Kutta Method for Sigular Autoomous Iitial Value Problems i Ordiar Differetial Equatios. Computig, 9, 4-6. [4] Evas, D.J. ad Yaacob, N.B. (995) A Fourth Order Ruge-Kutta Method Based o the Heroia Mea Formula. Iteratioal Joural of Computer Mathematics, 58, 0-5. [5] Evas, D.J. ad Yaacob, N.B. (995) A Fourth Order Ruge-Kurla Method Based o the Heroia Mea. Iteratioal Joural of Computer Mathematics, 59, -. [6] Evas, D.J. ad Yaacob, N.B. (99) A New Fourth Order Ruge-Kutta Formula Based o Harmoic Mea. Departmet of Computer Studies, Loughborough Uiversit of Techolog, Loughborough. [7] Yaacob, N. ad Saugi, B. (998) A New Fourth-Order Embedded Method Based o the Harmoic Mea. Matematica, Jilid, 998, -6. [8] Wazwaz, A.M. (994) A Compariso of Modified Ruge-Kutta Formulas Based o Variet of Meas. Iteratioal Joural of Computer Mathematics, 50, 05-. [9] Saugi, B.B. ad Evas, D.J. (99) A New Fourth Order Ruge-Kutta Method Based o Harmoic Mea. Computer Studies Report, Louborough Uiversit of Techolog, UK. [0] Lambert, J.D. (97) Computatioal Methods i ODEs. Wile, New York. 699