Applied Mathematics, 05, 6, 694-699 Published Olie April 05 i SciRes. http://www.scirp.org/joural/am http://dx.doi.org/0.46/am.05.64064 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 Email: 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). http://creativecommos.org/liceses/b/4.0/ 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, 694-699. http://dx.doi.org/0.46/am.05.64064
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 8 7 9 k4 f = x h, hk hk hk 4 0 0 AHM kk kk 4 = h k k 6 6 k k k k4 k = f k = f hk 5 k f = hk hk 8 7 9 k4 f = hk hk hk 4 0 0 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 4 4 4 k = f k = f b hk (5) k = f b hk b hk (6) k = f b hk b hk b hk (7) 4 4 4 4 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
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 4 4 4 4 4 4 4 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 4 4 4 4 4 4 4 4 4 4 ) ( 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 () 8 6 96 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
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 64 6 56 6 56 4 4 4 5, () Figure. Absolute stabilit regio of the 4sHERK method. 697
( 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 0.5.8099.8044.8065.8047.8099 0.40847e 6 0.80746e 6 0.56807e 6 0.7569e 8 0.5.474487.47454.47444.47449.474487 0.55688e 6 0.4459504e 6 0.68606666e 6 0.440687e 8 0.75.87566.8765.8758.8749.87565 0.59089e 6 0.47774995e 6 0.787e 6 0.44098678e 8 0.5.4456.4445.4408.448.4456 0.5949e 6 0.48040e 6 0.764467e 6 0.45594e 8 0.65.5.50000058.4999995.4999998.5 0.58978e 6 0.47970e 6 0.765e 6 0.4069489e 8 0.75.5888.5894.5887.588.5888 0.5656855e 6 0.4605098e 6 0.7055079e 6 0.888494e 8 0.875.6584.65894.65895.6587.6589 0.5475559e 6 0.4466e 6 0.689059e 6 0.7954e 8.70508.7054.70507.70505.70508 0.599864e 6 0.460686e 6 0.660089e 6 0.57476e 8 0.5.8077564.807765.80775.80775.807756 0.56e 6 0.490784e 6 0.696096e 6 0.459544e 8 0.5.8708869.870899.870889.8708807.8708869 0.497869e 6 0.406709e 6 0.69668e 6 0.7697e 8 0.75.964967.96496.96498.964907.964967 0.4878e 6 0.94668e 6 0.6000e 6 0.065e 8 0.5.0.00000047.9999996.9999994.0 0.4685586e 6 0.8054e 6 0.58404588e 6 0.05875e 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 0..095445.0954456.09544499.0954449.095445 0.497954e 6 0.84e 6 0.8686867e 6 0.89740e 9 0..8596.866.8578.857.8596 0.080908e 6 0.77578e 6 0.60694e 6 0.77e 8 0..64906.6499.649087.649077.64906 0.0744e 6 0.9857e 6 0.890566e 6 0.6688e 8 0.4.464079.4640.464059.464049.464079 0.78774e 6 0.9765556e 6 0.9840694e 6 0.5506e 8 0.5.4456.448.446.446.4456 0.7988e 6 0.9807497e 6 0.987049e 6 0.75e 8 0.6.4897.4899.4895.4894.4897 0.4689e 6 0.9559589e 6 0.94788e 6 0.078778e 8 0.7.54994.549957.54995.549905.54994 0.976557e 6 0.97475e 6 0.8874859e 6 0.094097e 8 0.8.64555.64577.6456.6457.64555 0.4674e 6 0.87047e 6 0.8997e 6 0.00779e 8 0.9.67005.6707.67987.67978.67005 0.85889e 6 0.86709e 6 0.7485859e 6 0.9688008e 9 0.0.70508.7050.70506.705054.70508 0.9686e 6 0.7805796e 6 0.678445e 6 0.9755e 9 698
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 0.0.00995049.00995049.00995049.00995049.00995049 0.068e 0.86459e 0.654554e 0.0446e 5 0.0.09809.09809.09809.09809.09809 0.84488e 0.490985e 0.50055e 0.444089e 5 0.0.09560.09560.09560.09560.09560 0.5486944e 0.4995786e 0.760050e 0.444089e 5 0.04.09048.09048.09048.09048.09048 0.69868555e 0.66466e 0.98097e 0.6668e 5 0.05.04880885.04880885.04880885.04880885.04880885 0.849765e 0.760466e 0.08978e 0 0.6668e 5 0.06.058005.058005.058005.058005.058005 0.95894404e 0.87445e 0.55544e 0 0.6668e 5 0.07.0677078.0677078.0677078.0677078.0677078 0.07787e 0 0.97665e 0.988588e 0 0.6668e 5 0.08.077096.077096.077096.077096.077096 0.74576e 0 0.069966e 0 0.5066e 0 0.6668e 5 0.09.0867805.0867805.0867805.0867805.0867805 0.685e 0 0.55869e 0 0.655869e 0 0.6668e 5 0..095445.095445.095445.095445.095445 0.5677e 0 0.46e 0 0.7669644e 0 0.6668e 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, 44-45. [] 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