1. Introduction. 2. Numerical Methods

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1 America Joural o Computatioal ad Applied Matematics, (5: 9- DOI:.59/j.ajcam.5. A Stud o Numerical Solutios o Secod Order Iitial Value Problems (IVP or Ordiar Dieretial Equatios wit Fourt Order ad Butcer s Fit Order Ruge-Kutta Metods Md. Jaagir Hossai, Md. Sa Alam, Md. Babul Hossai,* Departmet o Basic Sciece, World Uiversit o Baglades, Daa, Baglades Departmet o Matematics, Mawlaa Basai Sciece ad Tecolog Uiversit, Tagail, Baglades Abstract I tis paper, we mail preset ourt order Ruge-Kutta (RK ad Butcer s it order Ruge-Kutta (RK5 Metods or solvig secod order iitial value problems (IVP or ordiar dieretial equatios (ODE. Tese two proposed metods are quite proiciet ad practicall well suited or solvig egieerig problems based o suc problems. To obtai te accurac o te umerical solutios or tis stud, we ave compared te appromate solutios wit te exact solutios ad origiate a good agreemet. Numerical ad grapical comparisos betwee ourt order Ruge-Kutta metod ad Butcer s it order Ruge-Kutta metod ave bee preseted. I order to, acieve more accurac i te solutio; te step size eeds to be ver small. Moreover, te error terms ave bee aalzed o tese two proposed metods or dieret step sizes to scrutiize supremac. A umerical example is give to exibit te reliabilit ad eiciec o tese two metods. Kewords Dieretial Equatio, Iitial Value Problem, Error aalsis, Butcer s Metod, Ruge-Kutta Metod. Itroductio Most o te problems i sciece, matematical psics ad egieerig are ormulated b dieretial equatios. Te solutio o dieretial equatios is a sigiicat part to develop te various modelig i sciece ad egieerig. Tere are ma aaltical metods or idig te solutio o ordiar dieretial equatios. But a ew umbers o dieretial equatios ave aaltic solutios were a large umbers o dieretial equatios ave o aaltic solutios. I tis case we use te umerical metods to get te appromate solutio o a dieretial equatio uder te prescribed iitial coditio or coditios. Numerical metods are widel used or solvig dieretial equatios were it is diicult to obtai te exact solutios. Tere are ma tpes o umerical metods or solvig iitial value problems or ordiar dieretial equatios suc as Eulers metod, Ruge-Kutta ourt order metod (RK. Ruge-Kutta metod is te powerul umerical tecique to solve te iitial value problems (IVP. Tis metod widel used oe sice it gives reliable startig values ad is particularl suitable we te computatio o iger * Correspodig autor: babulossai@aoo.com (Md. Babul Hossai Publised olie at ttp://joural.sapub.org/ajcam Coprigt Scietiic & Academic Publisig. All Rigts Reserved derivatives is complicated. Also tis metod gives more accurac o te umerical results ad used i most computer programs or a dieretial equatio. Ma autors tr to spread tese metods to get more accurate solutio o Iitial Value Problems (IVP. I [] te autor preseted it order improved Ruge-Kutta metod or solvig ordiar dieretial equatio, i [] te autor preseted o it order Ruge-Kutta metods, also i [] te autor preseted a comparative stud o umerical solutios o Iitial Value Problems (IVP or ordiar dieretial equatios (ODE wit Euler ad Ruge-Kutta metods. Also i [-5] preseted various umerical metods or idig appromatios o Iitial Value Problems (IVP i ordiar dieretial equatios (ODE. I tis paper, we coverted te coeiciets o Butcer s RK5 table to te geeral it order Ruge-Kutta metod. Also we use ourt order Ruge-Kutta (RK ad it order Ruge-Kutta (RK5 metods or solvig secod order iitial value problems i ordiar dieretial equatios. Appromate solutios are compared to te exact solutios b calculate te error terms.. Numerical Metods I tis sectio we will represet two metods Ruge-Kutta ourt order (RK ad Butcer s it order Ruge-Kutta (RK5 metods or solvig iitial value problem (IVP or

2 Md. Jaagir Hossai et al.: A Stud o Numerical Solutios o Secod Order Iitial Value Problems (IVP or Ordiar Dieretial Equatios wit Fourt Order ad Butcer s Fit Order Ruge-Kutta Metods ordiar dieretial equatios (ODEs. Tese metods are distiguised b teir order i te sese tat agrees wit Talor s series solutio up to terms o rr were rr is te order o te metod. Next we will appl tese two metods or solvig secod order iitial value problem... Ruge-Kutta Fourt Order Metod We cosider te ollowig iitial value problem: d = ( x, ( dx wit iitial coditio ( x = ( Now te metod is based o computig + as ollows: + = + ( ( were x + = x + = ( x, = ( x +, + = ( x +, + = ( x + For =,,, Fourt Order Ruge-Kutta Metod or Solvig Iitial Value Problem (IVP or te Sstem o Two Dieretial Equatio We cosider te ollowig sstem o dieretial equatios dx = p( t, x, ( d = Q( t, x, (5 wit te iitial coditios x ( t = x, ( t = ( Te ourt order RugeKutta metod will become as: + = + ( ( + = + ( l + l + l + l (8 were ti + = ti + = P( t, x, i i i l = Q( t, x, i i i = P( ti +, +, + l l = Q( ti +, +, + l = P( ti +, +, + l l = Q( ti +, +, + l = P( ti + l l = Q( t x l i i i +.. Butcer s Fit Order Ruge-Kutta Metod We cosider te ollowig iitial value problem d = ( x, (9 dx Wit iitial coditio ( x = ( Now te metod is based o computig + as ollows + = + ( ( 9 were x + = x + = ( x, = x + = x = + + x, 8 = ( x +, =,, Butcer s Fit Order Ruge-Kutta Metod or Solvig Iitial Value Problem (IVP or te Sstem o Two Dieretial Equatios We cosider te ollowig sstem o dieretial equatios:

3 America Joural o Computatioal ad Applied Matematics, (5: 9- wit te iitial coditios dx = P( t, x, ( d = Q( t, x, ( x ( t = x, ( t = ( Te Butcer s it order Ruge-Kutta metod will become as: + = + ( (5 9 + = + (l + l + l + l5 + l ( 9 Were t = t i + i + = P( t, x, i i i l = Q( t, x, i i i l l P ti + l = i l = Q ti + l = P ti + +, + l + l = Q ti + +, + l + l = P ti l + l = Q ti l + l = P( ti + + l+ l 9 9 l5 = Q( ti + + l+ l l = Q( t i x 8 + 5, 8 + l5 i i l For i =,,, l + + l l. Error Aalsis Our mai goal is to id te more accurate results i umerical solutios o ordiar dieretial equatios. Tere are two tpes o errors occurs (Trucatio errors ad Roud-o errors. Trucatio error i umerical aalsis arises we appromatios are used to estimate some quatit. Roudig errors origiate rom te act tat computer ca ol represet a limited umber o sigiicat igures. Tus, suc umbers caot be represeted exactl i computer memor. Te discrepac itroduced b tis limitatio is called Roud-o error. Te accurac o te solutio will deped o ow small we tae te step size,. A umerical metod is said to be coverget i te umerical solutio approaces te exact solutio as te step size teds to zero. Te covergece o iitial value problem is calculated b e = t ( < δ were t ( deotes te appromate solutio ad deotes te exact solutio 5 ad δδ depeds o te problem wic varies rom. Te errors or tese two ormulas are deied b errors = ( t.. Illustrative Example I tis sectio, we illustrate a secod order iitial value problem as umerical example to compare betwee tese two metods. All te computatios are perormed b MATLAB sotware. Numerical results ad errors are computed ad te outcomes are represeted b grapicall. Example: Cosider te iitial value problem d + = si t ( dx wit te iitial coditios ( =, ( = o te iterval t. Te exact solutio o tis iitial value problem is give b ( t = cos t + si t si t. 5 Now tis secod order iitial value problem ca be writte i te orm o sstem o irst order dieretial equatios b d puttig x = as d = x (8 dx = si t (9 wit te iitial coditios ( =, x( = ( Te appromate results, exact results ad mamum errors or step sizes.,.5,.5 ad.5 are sow i

4 Md. Jaagir Hossai et al.: A Stud o Numerical Solutios o Secod Order Iitial Value Problems (IVP or Ordiar Dieretial Equatios wit Fourt Order ad Butcer s Fit Order Ruge-Kutta Metods Tables - ad te graps o te umerical solutios are displaed i Figures -8. Table. Numerical appromatios ad mamum errors or step size =. tt RK Metod =. RK5 Metod =. Exact value t Errors ( t Errors ( E+.988E+.5E+.858E+.9E+.9E+.8E+.59E E-.99E-.955E-.98E- -.9E- -.8E E E E+ -.89E+ -.E E+.E-.5E-.E-.5E-.E- 8.E-.5E-5.5E-5.E-5.E-5.88E-5.59E-5.9E-5.585E-5.58E-5.E-5.98E-5.5E-5.85E-5.55E-5.95E+.E+.59E+.8E+.5E+.9889E+.895E+.55E+ 9.5E-.55E-.998E-.9E E- -.58E- -.9E E E E+ -.59E+ -.58E+.85E-8.59E-8 5.9E-8 8.E-8.88E-.E-.88E-.88E-.8E-.5E-.E-.E- 5.99E-8.8E-8 8.E-8.E-.5E-.E-.895E-.8E-.9E+.E+.58E+.88E+.5589E+.99E+.8999E+.555E E-.58E-.995E-.958E E- -.59E- -.E E E E+ -.58E E+ Table. Numerical appromatios ad mamum errors or step size =.5 tt RK Metod =.5 RK5 Metod =.5 Exact value t Errors ( t Errors ( E+.9E+.58E+.899E+.59E+.995E+.889E+.559E+ 9.55E-.588E-.9958E-.985E E- -.8E- -.8E E- -.88E+ -.85E+ -.E E+.8E E-8.99E-8.589E-.5E- 5.98E- 8.E-.E-.9E-.E-.E-.E-.5E-.5E-.585E-.555E-.E-.99E-.5E- 9.59E-.9E+.E+.5E+.888E+.5585E+.99E+.899E+.555E+ 9.58E-.58E-.99E-.9558E E E E E E+ -.88E E E+.E-.9E-9.88E-9.9E-9.8E-9.5E-9.99E-9 5.E-9 5.E-9.59E-9.9E-9.E-9.5E-.8E-9.9E-9.8E E-9.5E-8.5E-8.E-8.9E+.E+.58E+.88E+.5589E+.99E+.8999E+.555E E-.58E-.995E-.958E E- -.59E- -.E E E E+ -.58E E+

5 America Joural o Computatioal ad Applied Matematics, (5: 9- Table. Numerical appromatios ad mamum errors or step size =.5 RK Metod =.5 RK5 Metod =.5 Exact value tt ( t Errors ( t Errors...9E+.8E-9.9E+.8E-.9E+..E+.9E-9.E+.E-.E+..55E+.9E-9.58E+.E-.58E+..88E+.8E-8.88E+ 8.E-.88E+.5.5E+.8E E+.E-.5589E+..998E+.9E-8.99E+.E-.99E+..899E+ 5.E E+.55E-.8999E+.8.55E+.55E E+.555E-.555E E- 9.E E-.59E E-..599E-.E-.58E-.8E-.58E E-.89E-.99E- 9.58E-.995E-..95E-.8E-.958E-.55E-.958E E-.5E E-.99E E-. -.E-.58E E-.88E- -.59E E-.588E- -.E-.98E- -.E E-.55E E-.9E E E+.88E E+.9E E E+.E E+.55E E E+ 8.5E E+.E- -.58E E+.9E E+.5E E+ Table. Numerical appromatios ad mamum errors or step size =.5 RK Metod =.5 RK5 Metod =.5 Exact value tt ( t Errors t ( Errors...9E+.98E-.9E+.9E-.9E+..E+.E-.E+.E-.E+..59E+.9E-.58E+.898E-.58E+..88E+.9895E-.88E+.5E-.88E E+.595E E+.5E-.5589E+..99E+.5E-9.99E+.E-.99E+..899E+.89E E+.9E-.8999E E+.85E-9.555E+.8E-.555E E E E-.E E-..588E-.85E-9.58E-.E-.58E-..995E- 8.9E-9.995E-.8E-.995E-..958E- 8.9E-9.958E-.9E-.958E E- 9.58E E-.E E E- 9.8E E-.88E- -.59E E- 9.8E-9 -.E- 5.5E- -.E E- 9.9E E-.5E E E+ 8.9E E+ 9.59E E E+.89E E+.E E E+ 5.E E+.E- -.58E E+.859E E+.5E E+

6 Md. Jaagir Hossai et al.: A Stud o Numerical Solutios o Secod Order Iitial Value Problems (IVP or Ordiar Dieretial Equatios wit Fourt Order ad Butcer s Fit Order Ruge-Kutta Metods Figure. Exact umerical solutios Figure. Appromate umerical solutios Figure. Error or step size =.

7 America Joural o Computatioal ad Applied Matematics, (5: 9-5 Figure. Error or step size =.5 Figure 5. Error or step size =.5 Figure. Error or step size =.5

8 Md. Jaagir Hossai et al.: A Stud o Numerical Solutios o Secod Order Iitial Value Problems (IVP or Ordiar Dieretial Equatios wit Fourt Order ad Butcer s Fit Order Ruge-Kutta Metods Figure. Error or dieret step sizes usig RK metod Figure 8. Error or dieret step sizes usig RK5 metod 5. Discussio Te acquired results are displaed i Tables (- ad grapicall preseted i Figures (-8. Te appromate solutios ad errors are calculated wit te step sizes.,.5,.5 ad.5 b usig MATLAB. Te appromate solutios are compared to te exact solutios. Te Ruge-Kutta it order (RK5 appromatios or same step size coverge to exact solutio. It is also metio tat te small step size provides te better appromatio. We observe tat rom Tables (- Ruge-Kutta it order (RK5 metod gives more accurate results ad better ta Ruge-Kutta ourt order (RK metod. Also rom igures ad 8 we coclude tat i te step size teds to zero te te errors also teds to zero.. Coclusios I tis paper, ourt order Ruge-Kutta metod ad Butcer s it order Ruge-Kutta metod are applied to solve secod order iitial value problems (IVP o ordiar dieretial equatio (ODE. To id more accurate results we eed to reduce te step size or bot metods. From te resultig tables ad igures we aalzed tat bot metods coverge to te exact solutios or ver small values o te step size,. From te igure we also observe tat bot metods give almost same results but rom Figures - it is clear tat RK5 metod gives more accurate results ta RK metod. Also we state tat te Butcer s it order Ruge-Kutta metod is more appropriate ad proiciet or idig te umerical solutios o iitial value problems

9 America Joural o Computatioal ad Applied Matematics, (5: 9- (IVP ta ourt order Ruge-Kutta metod. Hece rom tis stud we coclude tat to id more accurate result iger order umerical metod is appropriate ta lower order metods. REFERENCES [] Rabiei, F., & Ismail, F. (. Fit-order Improved Ruge-Kutta metod or solvig ordiar dieretial equatio. Australia Joural o Basic ad Applied Scieces, (, 9-5. [] Butcer, J. C. (995. O it order Ruge-Kutta metods. BIT Numerical Matematics, 5(, -9. [] Islam, M. A. (5. A Comparative Stud o Numerical Solutios o Iitial Value Problems (IVP or Ordiar Dieretial Equatios (ODE wit Euler ad RugeKutta Metods. America Joural o Computatioal Matematics, 5(, 9. [] Butcer, J. C. (9. O Ruge-Kutta processes o ig order. Joural o te Australia Matematical Societ, (, 9-9. [5] Islam, M. A. (5. Accurac Aalsis o Numerical solutios o Iitial Value Problems (IVP or Ordiar Dieretial Equatios (ODE. IOSR Joural o Matematics,, 8-. [] Butcer, J. C. (99. A istor o Ruge-Kutta metods. Applied umerical matematics, (, -. [] Goee, D., & Joso, O. (. Ruge Kutta wit iger order derivative appromatios. Applied umerical matematics, (-, -8. [8] Lambert, J. D. (9. Computatioal metods i ordiar dieretial equatios. Wile, New Yor. [9] Matews, J.H. (5. Numerical Metods or Matematics, Sciece ad Egieerig. Pretice-Hall, Idia. [] Hall, G. ad Watt, J.M. (9. Moder Numerical Metods or Ordiar Dieretial Equatios. Oxord Uiversit Press, Oxord. [] Burde, R.L. ad Faires, J.D. (. Numerical Aalsis. Bagalore, Idia. [] Gerald, C.F. ad Weatle, P.O. (. Applied Numerical Aalsis. Pearso Educatio, Idia. [] Hossai, Md.S., Battacarjee, P.K. ad Hossai, Md.E. (. Numerical Aalsis. Titas Publicatios, Daa. [] Sastr, S.S. (. Itroductor Metods o Numerical Aalsis. Pretice-Hall, Idia. [5] Balagurusam, E. (. Numerical Metods. Tata McGraw-Hill, New Deli.

d y f f dy Numerical Solution of Ordinary Differential Equations Consider the 1 st order ordinary differential equation (ODE) . dx

d y f f dy Numerical Solution of Ordinary Differential Equations Consider the 1 st order ordinary differential equation (ODE) . dx umerical Solutio o Ordiar Dieretial Equatios Cosider te st order ordiar dieretial equatio ODE d. d Te iitial coditio ca be tae as. Te we could use a Talor series about ad obtai te complete solutio or......!!!