Proc. of the 8th WSEAS Int. Conf. on Mathematical Methods and Computational Techniques in Electrical Engineering, Bucharest, October 16-17,

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1 Proc. of the 8th WSEAS Int. Conf. on Mtheticl Methods nd Coputtionl Techniques in Electricl Engineering, Buchrest, October 6-7, 006 Guss-Legendre Qudrture Forul in Runge-utt Method with Modified Model of Newton Cooling Lw Mitree Podisuk Deprtent of Mthetics nd Coputer Science ing Mongkut's Institute of Technology Chokhunthrn Ldkrbng Bngkok 00 Thilnd Sirirt huntidilokwongs Deprtent of Mthetics ing Mongkut's Institute University of Technology Thonburi Bngkok 00 Thilnd Witchy Rttnetwee Deprtent of Mthetics Mhsrkh University Mhsrkh 000 Thilnd Abstrct: In this pper we introduce, the so clled 'Open Forul', two points forul, three points forul, four points forul, five points forul nd six points forul of the Runge-utt ethod to solve the initil vlue proble of the ordinry differentil eqution. These foruls use the points nd weights fro the Guss-Legendre Qudrture foruls for finding the vlue of the definite integrl. We will use these foruls to copute the nuericl solution of the initil vlue proble of the ordinry differentil equtions fro the odels tht re odified fro The Newton Cooling Lw. eywords: Guss-Legendre, Runge-utt, Open, Qudrture Fehlberg Introduction The ost frequently used re the four points The initil vlue proble of the forul of Runge-utt-Fehlberg forul. ordinry differentil eqution is of the for ( y (x, x [, b] Forultion The ide of our new foruls is to use ( y ( c. the points ( x + αih's in our forul fro The prtition P over the closed intervl the points in the Guss-Legendre forul, i.e. [,b] is the finite sequence of rel nubers, the roots of the Legendre orthogonl polynoil, x, x,..., x i + ih nd nd the weights i 's in our forul fro the 0 x n h (b / n. The k-points Runge-utt forul, the explicit single step ethod, to find the vlue of function t the bove points is of the for k ( y + y + i i i weights in the Guss-Legendre Qudrture forul. Thus we hve the following five foruls which we shll cll the "the open forul". Two Point Forul The two points re x + α h, nd f (x + αh, y + βh,, k n k + α k h + h k j β n + (k (k / k(k /. n j nd j x h + α with α 0.( / nd α 0.( + /. The two weights re A A 0.. Thus our two points forul is of the for

2 Proc. of the 8th WSEAS Int. Conf. on Mtheticl Methods nd Coputtionl Techniques in Electricl Engineering, Buchrest, October 6-7, 006 ( h y + y + ( + + αh, y + α f (x + α h + hf (x h α. Three Point Forul The three points forul is of the for h ( y y + + ( hf (x 0 0 h h f(x + h,y h h. Four Point Forul The four points forul is of the for (6 y + y + h i i hf (x h h h 8 + i h h h Five Point Forul The five points forul is of the for (7 y + y + h i i , nd + ( + + ( + 70 hf (x + ( + ( 70 h i. 70 h 70 h h h + h ( + 70h h h + h 9 + ( h h+ h h h. 68 Six Point Forul The six points forul is of the for

3 Proc. of the 8th WSEAS Int. Conf. on Mtheticl Methods nd Coputtionl Techniques in Electricl Engineering, Buchrest, October 6-7, 006 (8 9 y y + h( hf (x 9 f(x h,y h h h h h + h h h h h h h h h 0 h 60 Exple There re three exples in this pper, the first two exples re the initil vlue proble of ordinry differentil equtions tht re odified fro the Newton Cooling Lw by letting the derivtive of the teperture of n object is in the for (9 T (t k (tt(t nd we re looking for the function k(t which k (0 0 nd li k(t, S log T t T0 is the initil 0 teperture of the object nd S is the teperture of the surrounding. We select the function k(t of the for t (0 k(t. b + t With bove ssuption, we obtin the differentil eqution bt(t ( T (t (b + t ( T (0 T0. Exple The initil teperture is 8. the teperture of the surrounding is nd the teperture t tie equls to unit is Fro these infortion, we obtin the equtions ( T (t T(t (t ( T (0 8.. The nlyticl solution of the equtions (-( is.777t t T (t 8.e. We use the forul (, (, (6, (7 nd (8 to copute the nuericl solution of the eqution (-( nd the results re in the following tbles. Forul h 0. T ( 70.8 ( ( ( ( ( Tble Forul h 0.0 T ( 70.8 ( ( (

4 Proc. of the 8th WSEAS Int. Conf. on Mtheticl Methods nd Coputtionl Techniques in Electricl Engineering, Buchrest, October 6-7, 006 ( ( ( ( Tble ( ( Forul h 0.00 T ( 70.8 ( Tble 6. ( ( ( ( ( Tble. Exple The initil teperture is 8 with the teperture of the surrounding is 8 nd the teperture t the tie unit is 67. Fro these infortion, we obtin the following differentil eqution ( T (t T(t (t (6 T (0 8. The nlyticl solution of the t t+ 0.6 T (t 8e equtions (-(6 is. We use the forul (, (, (6, (7 nd (8 to copute the nuericl solution of the eqution (-(6 nd the results re in the following tbles. Forul h 0. T ( 67 ( ( ( ( ( Tble Forul h 0. 0 T ( 67 ( ( ( ( ( Tble Forul h T ( 67 In exple, if we use the Newton Cooling Lw then we obtin the following differentil eqution T (t.0767 ( T(t (8 T (0 8.. We use the forul (, (, (6, (7 nd (8 to copute the nuericl solution of the eqution (7-(8 nd the results re in the following tbles. Forul h 0. T ( 70.8 ( ( ( ( ( Tble 7 Forul h 0.0 T ( 70.8 ( ( ( ( ( Tble 8 Forul h 0.00 T ( 70.8 ( ( ( ( ( Tble 9. In exple, if we use the Newton Cooling Lw then we obtin the following differentil eqution T (t.0767 ( T(t

5 Proc. of the 8th WSEAS Int. Conf. on Mtheticl Methods nd Coputtionl Techniques in Electricl Engineering, Buchrest, October 6-7, (8 T (0 8. We use the forul (, (, (6, (7 nd (8 to copute the nuericl solution of the eqution (7-(8 nd the results re in the following tbles. Forul h 0. T ( 67.0 ( ( ( ( ( Tble 0 Forul h 0. 0 T ( 67.0 ( ( ( ( ( Tble Forul h T ( 67.0 ( ( ( ( ( Tble. Exple Find the nuericl solution of the eqution sin x y (9 y (x x x (0 y (. The nlyticl solution of the equtions (9-(0 is + cos( cos(x y(x. x We use the forul (, (, (6, (7 nd (8 to copute the nuericl solution of the equtions (9-(0 nd the results re in the following tbles. Forul h 0. T ( ( ( ( ( ( Tble Forul h 0.0 T ( ( ( ( ( ( Tble Forul h 0.00 T ( ( ( ( ( ( Tble. Conclusion All bove five new foruls work s good s they re expected. So the new five foruls will give us ore freedo to select the wy to look for the nuericl solution of initil vlue proble of the ordinry differentil eqution. We strongly recoend the forul (, ( nd (6. Note tht in the exple which is the heting sitution, the Newton Lw of Cooling work quite different fro our new pproch. We will keep working on this kind of proble with other new function k(t. 6 References [] Brice Crnhn, H.A. Luther nd Jes O. Wilkes, Applied Nuericl Methods John Wiley & Sons, Inc., New York, 969 [] Mitree Podisuk nd Soskun Phurk, 'Newton-Cotes Forul in Runge-utt Method WSEAS Trnsction on Mthetics, Issue, Volue, Jnury 00 pp. 8- [] Mitree Podisuk nd Sirikul Bunditsovpk, Soe Orthogonl Polynoils,

6 Proc. of the 8th WSEAS Int. Conf. on Mtheticl Methods nd Coputtionl Techniques in Electricl Engineering, Buchrest, October 6-7, Proceedings of the 7 th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Cncun, Mexico, My -, 00, Pge 0-07 [] Mitree Podisuk, Witchy Rttnetwee nd Dech Sn, Applictions of Orthogonl Polynoils, Proceedings of the 7 th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Cncun, Mexico, My -, 00, Pge -0.

B.Sc. in Mathematics (Ordinary)

B.Sc. in Mathematics (Ordinary) R48/0 DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 B.Sc. in Mthemtics (Ordinry) SUPPLEMENTAL EXAMINATIONS 01 Numericl Methods Dr. D. Mckey Dr. C. Hills Dr. E.A. Cox Full mrks for complete nswers