Numerical Solution of the System of Six Coupled Nonlinear ODEs by Runge-Kutta Fourth Order Method

Transcription

1 Applied Mahemaical Sciences, Vol. 7,, no. 6, 87-5 Numerical Soluion of he Sysem of Six Coupled Nonlinear ODEs by Runge-Kua Fourh Order Mehod B. S. Desale Deparmen of Mahemaics School of Mahemaical Sciences Norh Maharashra Universiy Jalgaon-45, India Corresponding auhor N. R. Dasre Deparmen of Engineering Sciences Ramrao Adik Insiue of Technology Nerul, Navi Mumbai-476, India Absrac In his paper, we have proposed he numerical soluions of he sysem of six coupled nonlinear Ordinary Differenial Equaions(ODEs), which are aroused in he reducion of sraified Boussinesq Equaion. We have obained he numerical soluions on unsable and sable manifolds by Runge-Kua fourh order mehod. The minimum error in he soluion is of he order and i increases up o 6 for φ, while he minimum error in θ is of he order 4 and i increases up o. These errors can be reduced by reducing sep size. We have coded his programme in C-language. Mahemaics Subjec Classificaion: 4A9, 65L5, 65L99. Keywords: Sraified Boussinesq equaion, Runge-Kua Mehod, Coupled Differenial Equaions, Inegrable sysems Inroducion The sraified Boussinesq equaions form a sysem of Parial Differenial Equaions (PDEs) modelling he movemens of planeary amospheres. The lier-

2 88 B. S. Desale and N. R. Dasre aure also refers Boussinesq approximaion as Oberbeck-Boussinesq approximaion []. In his connecion of Desale [] has given he complee analysis of an ideal roaing uniformly sraified sysem of ODEs. Desale and Sharma [] have given special soluions for roaing sraified Boussinesq equaions. Desale and Dasre [4] have also given soluions o he sysem. In his paper, we have implemened Runge-Kua fourh order mehod o find he numerical soluion of sysem () passing hrough he iniial values on he sable and unsable manifolds. We have discussed he implemenaion of his mehod in he secion. We have given codes for he soluion on sable and unsable manifolds of invarian surface which is obained by four firs inegrals. Preliminary Noes Shrinivasan e al [5] have esed he sysem () as given below for complee inegrabiliy. Also Desale and Shrinivasan [6] have obained singular soluions of he same sysem. The sysem of six coupled nonlinear ODEs, which is aroused in he reducion of sraified Boussinesq equaions is as below. ẇ = g ρ b ê b, ḃ = w b, () where w = (w,w,w ) T, b = (b,b,b ) T and g ρ b is a non-dimensional consan as menioned by Desale [7] in his hesis. The above sysem can be wrien as componenwise as below w = g ρ b b, w = g ρ b b, w =, b = (w b w b ), b = (w b w b ), b = (w b w b ). () More deail mahemaical analysis of sysem () can be obained from Desale[7]. A criical poin (ê, ê ) lies on invarian surface and (b,b,b ) are on he

3 Numerical soluions by Runge-Kua fourh order mehod 89 surface b =. Hence, we have w = b k + b, b +b w = b k + b, b +b w =. () In above equaions k is a funcion of b, and i can be expressed as; k = ( b ) ( + b ) [4g( + b ) ρ b ρ b ]. (4) One may concern Shrinivasan e al [5] for more deails of his analysis. Since b =, we consider spherical co-ordinae sysem Hence, b = cos θ sin φ, b = sin θ sin φ, b = cos φ. (5) k = an 4 ( φ )[8g cos ( φ ) ]. (6) ρ b For k o be real, Shrinivasan e al [5] have pu up he resricion o φ as φ cos ( ρ b ). Wih his limiaion k akes he values negaive, 8g posiive and zero. Wih hese possible choices of k, he invarian surface will be he union of disjoin manifolds corresponding o k>, is unsable manifold, k<, is sable manifold and k =, is a cener manifold. Regarding hese manifolds, readers are advised o refer o Shrinivasan e al [5]. Now for k> he unsable manifold is given by w = an( φ )[cos θ sin θ 8g ρ b cos ( φ ) ], w = an( φ )[cos θ + sin θ 8g ρ b cos ( φ ) ], w =, b = cos θ sin φ, b = sin θ sin φ, b = cos φ, (7) k = an ( φ )[8g ρ b cos ( φ ) ]. On his surface sysem () reduces o

4 9 B. S. Desale and N. R. Dasre dφ d dθ d = an( φ ) 8g ρ b cos ( φ ), = 4 sec ( φ ). Whereas for k<, he sable manifold is given by w = an( φ )[cos θ + sin θ 8g ρ b cos ( φ ) ], w = an( φ )[cos θ sin θ 8g ρ b cos ( φ ) ], w =, b = cos θ sin φ, (8) (9) b = sin θ sin φ, b = cos φ, k = an ( φ 8g )[ ρ b cos ( φ ) ]. On his surface sysem () reduces o dφ d dθ d = an( φ ) 8g ρ b cos ( φ ), = 4 sec ( φ ). () Numerical Soluion In his paper, we have obained he numerical soluions of a sysem () wih he iniial values on sable and unsable manifolds by Runge-Kua fourh order mehod. The deails of his mehod can be obained from [8, 9, ].. Implemenaion of Runge-Kua Fourh Order Mehod For Numerical Soluion We sar wih he iniial condiion = and he iniial value b =(b,b,b ). We calculae he iniial values of (φ,θ )as θ = an ( b ), b b φ = an + b () ( ). b

5 Numerical soluions by Runge-Kua fourh order mehod 9 Now we calculae φ and θ by Runge-Kua fourh order mehod as, φ = φ + k, θ = θ + k. The values of k and k in above equaion are given by k = 6 [k +(k + k )+k 4 ], k = 6 [k +(k + k )+k 4 ]. Wih k = hf (,φ,θ ), k = hf ( + h,φ + k,θ + k ), k = hf ( + h,φ + k,θ + k ), k 4 = hf ( + h, φ + k,θ + k ), k = hf (,θ,φ ), k = hf ( + h,θ + k,φ + k ), k = hf ( + h,θ + k,φ + k ), k 4 = hf ( + h, θ + k,φ + k ). This gives us he nex approximae values of θ and φ. Then is se o + h and he values of θ and φ are ieraed wih he above formulae. The exac soluion on unsable manifold is given by φ() = sin [ k θ() = 4 + an [ an [ G G e 4 G G ], +ke G e 4 k G k e 4 G G ] G + G ] + k. Whereas on sable manifold he exac soluion is given by G e 4 G G G ], φ() = sin [ k +ke θ() = G 4 an [ e 4 G G ] k G + an [ e 4 G + G ] + k, k () ()

6 9 B. S. Desale and N. R. Dasre where k,k are consans and G = 8g ρ b. Now for our calculaions, we ook G =9. wih g =9.8 and ρ b =. We have compared he calculaed soluions and exac soluions. We have found ha he minimum error in φ 5 and minimum error in θ is 5. While in ieraion process hese errors increase up o 6 and for φ and θ respecively wih he consideraion of he sep size h =.. Please noe ha hese errors will be minimized wih he sep size. Afer calculaing φ and θ, we use he mehod of back-subsiuion o obain he values of b(b,b,b ) and w(w,w,w ).. Algorihm For Numerical soluion The following algorihm is used o find he soluion by Runge-Kua fourh order mehod. The deails of algorihm are given as below. Sep : Ener he iniial values of, φ, θ,, g, ρ b and h (sep size). Sep : Calculae he values of b, b, b, w, w, w, k, k and k. Here we have obained he iniial values. Sep : Calculae he values of φ and θ by using Runge-Kua fourh order mehod. Sep 4: Calculae he values of b, b, b, w, w and w by using equaion(7) (or (9)). Sep 5: Calculae he exac values of φ and θ by using equaion() (or ()) hen calculae he exac values of b, b, b, w, w and w by using equaion(7) (or (9)). Sep 6: Prin he required exac and calculaed numerical values. Sep 7: Replace φ by φ, θ by θ and by +h and go o Sep, unil he value of φ is reached o is maximum ( or minimum ) for he given unsable (or sable ) fold. Sep : Plo he graphs o see he difference. Sep : End. Numerical Soluion For Unsable Manifold On his manifold, we have k> and he sysem () reduces o (8). Now we use he following programme o find he soluion on unsable manifold.

7 Numerical soluions by Runge-Kua fourh order mehod 9 #include<sdio.h> #include<sdlib.h> #include<conio.h> #include<mah.h> #include<sys\sa.h> void main() {double f(double p); double f(double a); FILE *fp; double phi,phi,phi,hea,hea,hea,er_hea,er_phi; double h,,,,b,b,b,w,w,w,b,b,b,w,w,w; double eb,eb,eb,ew,ew,ew,be,be,be,we,we,we; double x,y,y,z,z,diff,diff,k,k,k,k4,k; double k,k,k,k4,k,ehea,ephi,g=9.,u,u,u; double k,k,k; /* g=9.8, rho_b=*/ in i=,n; clrscr(); prinf("\n\n\ PROGRAMME FOR RUNGE-KUTTA FOURTH ORDER METHOD ON UNSTABLE MANIFOLD"); fp=fopen("rk4c.xls","w+"); fprinf(fp,"\n\n\ PROGRAMME FOR RUNGE-KUTTA FOURTH ORDER METHOD ON UNSTABLE MANIFOLD"); =.; =5.58; h=.; prinf("\n\ Ener he value of phi="); scanf("%lf",&phi); prinf("\n\ Ener he value of hea="); scanf("%lf",&hea); fprinf(fp,"\n The value of phi=%lf ",phi); fprinf(fp,"\n The value of hea=%lf ",hea); b=cos(hea)*sin(phi); b=sin(hea)*sin(phi); b=cos(phi); y=sqr(9.*cos(phi/.)*cos(phi/.)-); w=an(phi/.)*(cos(hea)-sin(hea)*y); w=an(phi/.)*(sin(hea)+cos(hea)*y);w=.; fprinf(fp,"\n The value of b=%lf ",b); fprinf(fp,"\n The value of b=%lf ",b); fprinf(fp,"\n The value of b=%lf ",b); fprinf(fp,"\n The value of w=%lf ",w); fprinf(fp,"\n The value of w=%lf ",w); fprinf(fp,"\n The value of w=%lf ",w); /*calculaing k and k for exac soluion and k for iniial soluion*/ u=sin(phi/.); k=(sqr((g-.)/g)+sqr(((g-)/g)-u*u))/u; k=hea-aan((sqr(g)/k)-(sqr(g-.))) +aan((sqr(g)/k)+(sqr(g-.))); k=(an(phi/)*an(phi/))*sqr(g*cos(phi/)*cos(phi/)-); prinf("\n\n\the value of k=%f \n\n\the value of k=%f",k,k); prinf("\n\n\the value of k=%.8f ",k); fprinf(fp,"\nthe value of k=%.8f ",k); fprinf(fp,"\nthe value of k=%.8f ",k);

8 94 B. S. Desale and N. R. Dasre fprinf(fp,"\nthe value of k=%.8f ",k); prinf("\n\n\press ENTER o ge sep by sep"); fprinf(fp,"\n\b\b\b\w\w\w\hea\phi\k\exac phi \Exac Thea\Exac b\exac b\exac b\exac w\exac w \Exac w"); prinf("\n\n\error in Thea\\Error in Phi\Value of K"); while(<) { i++; =+h; /* Calculaion of phi */ k=h*f(phi); k=h*f(phi+.5*k); k=h*f(phi+.5*k); k4=h*f(phi+k); k=(k+.*k+.*k+k4)/6.; phi=phi+k; /*Calculaion of hea */ k=h*f(phi); k=h*f(phi+.5*k); k=h*f(phi+.5*k); k4=h*f(phi+k); k=(k+.*k+.*k+k4)/6.; hea=hea+k; /* Calculaion of approximae soluion B and W */ b=cos(hea)*sin(phi); b=sin(hea)*sin(phi); b=cos(phi); y=sqr(9.*cos(phi/.)*cos(phi/.)-); w=an(phi/.)*(cos(hea)-sin(hea)*y); w=an(phi/.)*(sin(hea)+cos(hea)*y); w=.; /* calculaion of exac soluion */ ephi=.*asin((.*k*sqr((g-.)/g)*exp(-(/4.)*sqr(g-.)))/ (.+k*k*exp(-(/.)*sqr(g-.)))); u=aan((sqr(g)*exp((/4.)*sqr(g-.)))/k-sqr(g-.)); u=aan((sqr(g)*exp((/4.)*sqr(g-.)))/k+sqr(g-.)); ehea=(/4.)+u-u+k; k=(an(ehea/.)*an(ehea/.))*sqr(g*cos(ehea/.) *cos(ehea/.)-.); /* calculaion of error in hea and phi*/ er_hea=fabs(hea-ehea); er_phi=fabs(phi-ephi); /* calculaion of B and W */ be=cos(ehea)*sin(ephi); be=sin(ehea)*sin(ephi); be=cos(ephi); y=sqr(9.*cos(ephi/.)*cos(ephi/.)-); we=an(ephi/.)*(cos(ehea)-sin(ehea)*y); we=an(ephi/.)*(sin(ehea)+cos(ehea)*y); we=.; fprinf(fp,"\n%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf \%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf",,b,b,b,w,w, w,hea,phi,k,ehea,ephi,be,be,be,we,we,we); prinf("\n\n\%.lf\%.lf\%lf",er_hea,er_phi,k); phi=phi;hea=hea; =; } gech(); }

9 Numerical soluions by Runge-Kua fourh order mehod 95 double f(double p) { double p_dash; p_dash=.5*(./(cos(p/.)*cos(p/.))); reurn(p_dash); } double f(double a) { double a_dash; a_dash=.5*an(.5*a)*sqr(9.*(cos(a/.)*cos(a/.))-.); reurn(a_dash); }.4 Numerical Soluion For Sable Manifold On his manifold, we have k< and he sysem () reduces o (). Now we use he following programme o find he soluion on sable manifold. #include<sdio.h> #include<sdlib.h> #include<conio.h> #include<mah.h> #include<sys\sa.h> void main() { double f(double p); double f(double a); FILE *fp; double phi,phi,phi,hea,hea,hea,er_hea; double er_phi,h,,,,b,b,b,w,w; double w,b,b,b,w,w,w,eb,eb,eb,ew,ew,ew; double be,be,be,we,we,we,x,y,y,z,z,diff,diff; double k,k,k,k4,k,k,k,k,k4,k; double ehea,ephi,g=9.,u,u,u,k,k,k; /* g=9.8, rho_b=*/ in i=,n; clrscr(); prinf("\n\n\ PROGRAMME FOR RUNGE-KUTTA FOURTH ORDER METHOD ON STABLE MANIFOLD"); fp=fopen("rk4s.xls","w+"); fprinf(fp,"\n\n\ PROGRAMME FOR RUNGE-KUTTA FOURTH ORDER METHOD ON STABLE MANIFOLD"); =.; =.; h=.; prinf("\n\ Ener he value of phi="); scanf("%lf",&phi); prinf("\n\ Ener he value of hea="); scanf("%lf",&hea); fprinf(fp,"\n The value of phi=%lf ",phi); fprinf(fp,"\n The value of hea=%lf ",hea); b=cos(hea)*sin(phi);b=sin(hea)*sin(phi);b=cos(phi); y=sqr(9.*cos(phi/.)*cos(phi/.)-); w=an(phi/.)*(cos(hea)-sin(hea)*y); w=an(phi/.)*(sin(hea)+cos(hea)*y); w=.;

10 96 B. S. Desale and N. R. Dasre fprinf(fp,"\n The value of b=%lf ",b); fprinf(fp,"\n The value of b=%lf ",b); fprinf(fp,"\n The value of b=%lf ",b); fprinf(fp,"\n The value of w=%lf ",w); fprinf(fp,"\n The value of w=%lf ",w); fprinf(fp,"\n The value of w=%lf ",w); /*calculaing k and k for exac soluion and k for iniial soluion*/ u=sin(phi/.);k=(sqr((g-.)/g)+sqr(((g-)/g)-u*u))/u; k=hea+aan((sqr(g)/k)-sqr(g-.))-aan((sqr(g)/k) +sqr(g-.)); k=-(an(phi/.)*an(phi/.))*sqr(g*cos(phi/.) *cos(phi/.)-.); prinf("\n\n\the value of k=%f\n\n\the value of k=%f",k,k); prinf("\n\n\the value of k=%.8f ",k); fprinf(fp,"\nthe value of k=%.8f ",k); fprinf(fp,"\nthe value of k=%.8f ",k); fprinf(fp,"\nthe value of k=%.8f ",k); prinf("\n\n\press ENTER o ge sep by sep"); fprinf(fp,"\n\b\b\b\w\w\w\thea\phi\k\exac Thea\ExacPhi\Exacb\Exacb\Exacb\Exacw\Exac w\exac w"); prinf("\n\n\error in Thea\\Error in Phi\Value of K"); while(<) { i++; =+h; /* Calculaion of phi */ k=h*f(phi); k=h*f(phi+.5*k); k=h*f(phi+.5*k); k4=h*f(phi+k); k=(k+.*k+.*k+k4)/6.; phi=phi+k; /*Calculaion of hea */ k=h*f(phi); k=h*f(phi+.5*k); k=h*f(phi+.5*k); k4=h*f(phi+k); k=(k+.*k+.*k+k4)/6.; hea=hea+k; /* Calculaion of approximae soluion B and W */ b=cos(hea)*sin(phi); b=sin(hea)*sin(phi); b=cos(phi); y=sqr(9.*cos(phi/.)*cos(phi/.)-.); w=an(phi/.)*(cos(hea)+sin(hea)*y); w=an(phi/.)*(sin(hea)-cos(hea)*y); w=.; /* calculaion of exac soluion */ ephi=.*asin((.*k*sqr((g-.)/g)*exp((/4.)*sqr(g-.)))/ (.+k*k*exp((/.)*sqr(g-.)))); u=aan(((sqr(g)*exp(-(/4.)*sqr(g-.)))/k)-sqr(g-.));

11 Numerical soluions by Runge-Kua fourh order mehod 97 u=aan(((sqr(g)*exp(-(/4.)*sqr(g-.)))/k)+sqr(g-.)); ehea=(/4.)-u+u+k; k=-(an(ehea/.)*an(ehea/.))*sqr(g*cos(ehea/.) *cos(ehea/.)-.); /* calculaion of error in hea and phi*/ er_hea=fabs(hea-ehea); er_phi=fabs(phi-ephi); /* calculaion of B and W */ be=cos(ehea)*sin(ephi); be=sin(ehea)*sin(ephi); be=cos(ephi); y=sqr(9.*cos(ephi/.)*cos(ephi/.)-.); we=an(ephi/.)*(cos(ehea)+sin(ehea)*y); we=an(ephi/.)*(sin(ehea)-cos(ehea)*y); we=.; fprinf(fp,"\n%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf \%lf\%lf\%lf\%lf\%lf\%lf\%lf\%lf",,b,b,b,w,w, w,hea,phi,k,ehea,ephi,be,be,be,we,we,we); prinf("\n\n\%.lf\%.lf\%lf",er_hea,er_phi,k); phi=phi; hea=hea; =; } gech(); } double f(double p) { double p_dash; p_dash=.5*(./(cos(p/.)*cos(p/.))); reurn(p_dash); } double f(double a) { double a_dash; a_dash=-.5*an(.5*a)*sqr(9.*(cos(a/.)*cos(a/.))-.); reurn(a_dash); } 4 Experimenal Resuls We have coded he above algorihm wih he help of C-programming. All he graphs are ploed using MATLAB sofware. We have considered he iniial value as φ =. and θ =. on he unsable manifold wih respec o k>. From he limiaions on φ, given by Desale [7], a φ =.8649 he value of k becomes negaive. Hence we have considered φ =.8649 and θ =. on sable manifold. In each figure, he firs graph shows he numerical value calculaed by us; he second graph shows he exac soluion and he hird graph shows he comparison of he firs and he second graphs as shown in Figure() o Figure(6).

12 98 B. S. Desale and N. R. Dasre 4. Figures for Numerical Soluion on Unsable Manifold On unsable manifold, we consider k>, φ =. and θ =.. The comparison of calculaed values and exac values is as shown below in Fig.() o Fig.(8).4 Calculaed b.4 Exac b.. b b Comparision of Exac and Calculaed b Figure : Graphs for b on unsable manifold Calculaed b Exac b.8.8 b.6.4 b b Comparision of Exac and Calculaed Figure : Graphs for b on unsable manifold

13 Numerical soluions by Runge-Kua fourh order mehod 99 Calculaed b Exac b.5.5 b b Comparision of Exac and Calculaed 4 6 b Figure : Graphs for b on unsable manifold Calculaed hea Exac hea hea hea 4 6 Comparision of Exac and Calculaed 4 6 hea Figure 4: Graphs for θ on unsable manifold Calculaed phi Exac phi phi phi 4 6 Comparision of Exac and Calculaed 4 6 phi Figure 5: Graphs for φ on unsable manifold

14 B. S. Desale and N. R. Dasre Calculaed w Exac w w w Comparision of Exac and Calculaed w Figure 6: Graphs for w on unsable manifold Calculaed w Exac w w w 4 6 Comparision of Exac and Calculaed 4 6 w Figure 7: Graphs for w on unsable manifold Calculaed w Exac w.5.5 w w Comparision of Exac and Calculaed 4 6 w Figure 8: Graphs for w on unsable manifold

15 Numerical soluions by Runge-Kua fourh order mehod 4. Figures for Numerical Soluion on Sable Manifold On sable manifold, we consider k<, φ =.8649 and θ =.. Comparison of calculaed values and exac values is as shown in he following figures Fig.(9) o Fig.(6) Calculaed b Exac b.5.5 b b Comparision of Exac and Calculaed.5 b Figure 9: Graphs for b on sable manifold.4 Calculaed b.4 Exac b.. b. b Comparision of Exac and Calculaed 5 5 b Figure : Graphs for b on sable manifold

16 B. S. Desale and N. R. Dasre Calculaed b Exac b.5.5 b b Comparision of Exac and Calculaed 5 5 b Figure : Graphs for b on sable manifold Calculaed hea Exac hea hea hea 5 5 Comparision of Exac and Calculaed 5 5 hea Figure : Graphs for θ on sable manifold Calculaed phi Exac phi phi phi phi Comparision of Exac and Calculaed Figure : Graphs for φ on sable manifold

17 Numerical soluions by Runge-Kua fourh order mehod Calculaed w Exac w w w w Comparision of Exac and Calculaed Figure 4: Graphs for w on sable manifold Calculaed w Exac w w w Comparision of Exac and Calculaed w Figure 5: Graphs for w on sable manifold Calculaed w Exac w.5.5 w w Comparision of Exac and Calculaed 5 5 w Figure 6: Graphs for w on sable manifold

18 4 B. S. Desale and N. R. Dasre 5 Conclusion Here we have presened he scheme of Runge-Kua fourh order Mehod for he numerical soluion of he sysem of six coupled nonlinear ODEs (). In our calculaion iniially we have he error of, while in ieraion process i increases up o in he soluion. I can be reduced as we reduce he sep size. In fuure we will aemp o minimize he error and sharpen he accuracy of he soluion. References [] K. R. Rajagopal, M. Ruzicka and A. R. Srinivasa, On he Oberbeck- Boussinesq Approximaion, Mahemaical Models and Mehods in Applied Sciences, 6 (996), [] B. S. Desale, Complee Analysis of an Ideal Roaing Uniformly Sraified Sysem of ODEs, Nonlinear Dyanamics and Sysems Theory 9() (9), [] B. S. Desale and V. Sharma, Special Soluions o Roaing Sraified Boussinesq Equaions, Nonlinear Dyanamics and Sysems Theory () (), 9-8. [4] B. S. Desale and N. R. Dasre, Numerical Soluions of Sysem of Nonlinear ODEs by Euler Modified Mehod, Nonlinear Dyanamics and Sysems Theory () (), 5-6. [5] G. K. Srinivasan, V. D. Sharma and B. S. Desale, An inegrable sysem of ODE reducions of he sraified Boussinesq equaions, Compuers and Mahemaics wih Applicaions, 5 (7), [6] B. S. Desale and G. K. Srinivasan, Singular Analysis of he Sysem of ODE reducions of he Sraified Boussinesq Equaions, IAENG Inernaional Journal of Applied Mahemaics, 8:4, IJAM (8), [7] B. S. Desale, An inegrable sysem of reduced ODEs of Sraified Boussinesq Equaions, Ph. D. Thesis, subied o IITB, Powai, Mumbai (7). [8] J. H. Mahews, Numerical Mehods for Mahemaics, Science and Engineering Prenice Hall of India, New Delhi, INDIA, 994. [9] E. Kreyszig, Advanced Engineering Mahemaics Wiley India (P) Ld., New Delhi, INDIA,.

19 Numerical soluions by Runge-Kua fourh order mehod 5 [] M. K. Jain and S. R. K. Iyengar, Numerical Mehods for scienific and engineering compuaion New Age Inernaional (P) Ld., Mumbai, India,. Received: June,