Fj25991998

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Fj25991998

  1. 1. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998 Five Steps Block Method For The Solution Of Fourth Order Ordinary Differential Equations. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. Department of Mathematics, ModibboAdama University of Technology, Yola, Adamawa State, NigeriaAbstract We considered method of collocation of highest order. For these reasons, this method isthe differential system and interpolation of the inefficient and not suitable for general purpose.approximate solution to generate a continuous Scholars later developed method to solve (1)linear multistep method, which is solved for the directly without reducing to systems of first orderindependent solution to yield a continuous block ordinary differential equations and concluded thatmethod. The resultant method is evaluated at direct method is better than method of reduction,selected grid points to generate a discrete block among these authors are Twizel and Khaliq (1984),method. The basic properties of the method was Awoyemi and Kayode (2004), Vigo-Aguiar andinvestigated and found to be zero stable, Ramos (2006), Adesanyaet.al (2009).consistent and convergent. The method was Scholars developed linear multistep method for thetested on numerical examples solved by the direct solution of higher order ordinary differentialexisting method, our method was found to give equation using power series approximate solutionbetter approximation. adopting interpolation and collocation method toKeywords: collocation, differential system, derived continous linear multistep method.interpolation, approximate solution, independent According to Awoyemi (2001), continous linearsolution, zero stable, consistent, convergent multistep method have greater advantages over theAMS Subject Classification (2010): 65L05, discrete method is that they give better error65L06, 65D30 estimate, provide a simplified form of coefficient for further analytical work at different points and1.0 Introduction guarantee easy approximation of solution at all This paper considered the approximate interior points of the integration interval. Amongsolution to the general fourth order initial value the authors that proposed the linear multistepproblems of the form method are Kayode (2004), Adesanyaet.al (2009), Awoyemi and Kayode (2004) to mention few. y (iv )  f ( x, y, y, y, y), y k ( x)  y0 k They developed an implicit linear multistep method (1) which was implemented in predictor correctorwhere k  1(1)3 and f iscontinuouswithin the mode and adopted Taylor’s series expansion tointerval of integration. provide the starting value.In most application, (1) is solved by reduction to an In spite of the advantages of the linear multistepequivalent system of first order ordinary method, they are usually applied to initial valuedifferential equation and appropriate numerical problems as a single formula and this has somemethod could be employed to solve the resultant inherent disadvantages, for instance thesystem. This approach had been reported by implementation of the method in predictor-scholars, among them are: Bun andVasil’yer corrector mode is very costly and subroutine are(1992), Awoyemi (2001), Adesanyaet. al (2008). In very complicated to write because they requirespite of the success of this approach, the setback of special technique to supply starting values.the method is in writing computer program which In order to cater for the above mentioned setback,is often complicated especially when subroutine are researchers came up with block methods whichincorporated to supply the starting values required simultaneously generate approximation at differentfor the method. The consequence is in longer grid points within the interval of integration. Blockcomputer time and human effort. Furthermore, method is less expensive in terms of the number ofaccording to Vigor-Aguiar and Ramos (2006), this function evaluations compared to the linearmethod does not utilize additional information multistep method or Ringe-Kuttamethod; above all,associated with specificordinary differential it does not require predictor or starting values.equation, such as the oscillatory nature of the Among these authors are Ismail et. al. (2009),solution. In addition, Bun and Vasil’yer (1992) Adesanyaet. al. (2012), Anakeet. al. (2012),reported that more serious disadvantage of the Awoyemiet. al. (2011).method of reduction is the fact that the given In this paper, we proposed block method for thesystem of equation to be solved cannot be solved solution of general fourth order initial valueexplicitly with respect to the derivatives of the problem of ordinary differential equation. This 991 | P a g e
  2. 2. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998method is an extension of the work of Olabode 2.0 Methodology(2009) and Mohammed (2010) who both worked We consider a power series approximateon special fourth order initial value problems of solution in the formordinary differential equation. s  r 1 y ( x)  ax d 0 j j (2)The fourth derivatives of (2) gives s  r 1 y (iv )   d 0 j ( j  1)( j  2)( j  3)a j x j 4 (3)Substituting (3) into (1) gives s  r 1 f ( x, y, y, y, y)   j 0 j ( j  1)( j  2)( j  3)a j x j 4 (4)where s and r are the number of interpolation and collocation points respectively.Interpolating (2) at xn  j , s  1(1)4 and collocating (4) at xn  r , r  0(1)5 gives a system of nonlinearequation AX  U (5) where X   a0 a1 a2 a3 a4 a5 a6 a7 a8 a9  T U   yn1 yn 2 yn3 yn 4 f n f n1 f n 2 f n3 f n 4 f n5  T 1 xn 1 xn 1 xn 1 xn 1 2 3 4 5 xn 1 6 xn 1 7 xn 1 8 xn 1 xn 1  9  2 3 4 5 6 7 8 9  1 xn  2 xn  2 xn  2 xn  2 xn  2 xn  2 xn  2 xn  2 xn  2  1 xn 3 xn 3 xn 3 xn 3 2 3 4 5 xn 3 6 xn 3 7 xn 3 8 xn 3 xn 3  9  2 3 4 5 6 7 8 9  1 xn  4 xn  4 xn  4 xn  4 xn  4 xn  4 xn  4 xn  4 xn  4  0 0 0 0 24 120 xn 360 xn 2 840 xn 3 1680 xn 4 3024 xn  4 A  0 0 24 120 xn 1 360 xn 1 840 xn 1 1680 xn 1 3024 xn 1  2 3 4 4 0 0 0 0 0 0 24 120 xn  2 360 xn  2 840 xn  2 1680 xn  2 3024 xn  2  2 3 4 4   0 0 0 0 24 120 xn 3 360 xn 3 840 xn 3 1680 xn 3 3024 xn 3  2 3 4 4  2 3 4 4  0 0 0 0 24 120 xn  4 360 xn  4 840 xn  4 1680 xn  4 3024 xn  4  0 0  0 0 24 120 xn 5 360 xn 5 840 xn 5 1680 xn 5 3024 xn 5  2 3 4 4 Solving (5) for the a j s using Gaussian elimination method, given a continuous linear multistep method in theform 4 5 y ( x)    j ( x) yn j h 4   j ( x) f n j j 1 j 0 (6) Where: yn j  y ( xn  jh) f n j  f ( xn  jh, yn  jh, y n  jh, y n  jh, y n  jh, ) yn  i f n i The coefficient and are given as 992 | P a g e
  3. 3. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998 1 3 1   (t  9t 2  26t  24) 6 1 3 2  (t  8t 2  19t  12) 2 1  3   (t 3  7t 2  14t  8) 2 1 1  (t 3  6t 2  11t  e) 6 10   (5t 9  135t 8  1530t 7  9450t 6  34524t 5  75600t 4  95335t 3  60975t 2  12346t  2520) 1814400 11  (5t 9  126t 8  127t 7  6468t 6  15120t 5  84383t 4  192318t 2  180240t  62496) 362880 12   (5t 9  117t 8  1062t 7  4494t 6  7560t 5  13603t 3  127941t 2  229770t  117448) 1814400 13   (5t 9  108t 8  882t 7  3276t 6  5040t 5  12647t 3  36396t 2  57540t  31248) 1814400 14   (5t 9  99t 8  738t 7  2562t 6  3780t 5  5633t 3  4677t 2  1410t  504) 362880 15   (5t 9  90t 8  630t 7  2100t 6  3024t 5  4415t 3  340t 2  340t ) 1814400 x  xnt h Solving (6) for the independent solution yn  j , j  0(1)5 given a continuous block method in the form ( jh)m ( m ) 3 5 yn  j   y n h4   j f n  j m 0 m ! j 0 (7) where: 1 0   (5t 9  135t 8  1530t 7  9450t 6  34524t 5  75600t 4 ) 1814400 1 1  (5t 9  126t 8  1278t 7  6468t 6  15120t 5 ) 362880 1 2   (5t 9  117t 8  1062t 7  4494t 6  7560t 5 ) 1814400 1 3   (5t 9  108t 8  882t 7  3276t 6  5040t 5 ) 1814400 1 4   (5t 9  99t 8  738t 7  2562t 6  3780t 5 ) 362880 1 5   (5t 9  90t 8  630t 7  2100t 6  3024t 5 ) 1814400Evaluating (7), the first, second and third derivatives at t = 1(1)5 gives a discrete block formula in the form 3 i A0Ym(i )   hi ei yn (i ) h4i df ( yn )  h4ibF (Ym ) i 0 (8) where: 993 | P a g e
  4. 4. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998 Ym   yn 1 , yn  2 , yn 3 , yn  4 , yn 5  T F (Ym )   f n 1 , f n  2 , f n 3 , f n  4 , f n 5  T f ( yn )   f n 1 , f n  2 , f n 3 , f n  4 , f n 5  T yn ( i )   yn 1( i ) , yn  2 ( i ) , yn 3( i ) , yn  4 ( i ) , yn ( i )   A0  5  5 identity matrix 0 0 0 0 1 0 0 0 0 1 0  0 0  0 0 0 1  0 0 2 e0   0 0 0 0 1 , e1  0 0 0 0 3 ,     0 0 0 0 1 0 0 0 0 4 0  0 0 0 1 0 0  0 0 5 0 0 0 0 2  1 0 0 0 0 1 6  0 0 0 0 2 0 0 4     0 0 3  e2  0 0 0 0 9  , e3  0 0 0 0 27  ,   2   2 32 0 0 0 0 8 0 0 0 0 3  0  0 0 0 25  2  0 0  0 0 125  6 when i  0, 0 0 0 0 49126 1814400   1814400 49045 40160 1814400 25430 1814400 9310 1814400 1469 1814400  0 0 0 0 4264   7960 4910 3080 1120 176   14175   14175 14175 14175 14175 14175 d  0 0 0 0 25488  b   63315 26460 19170 7020 1107    22400  116480  22400 22400 22400 22400 22400 40448 29440 33280 11360 1792 0 0 0 0 14175   14175 14175 14175 14175 14175  0  0 0 0 418250 72576    1315625  72576 175000 72576 418750 72576 106250 72576 18625 72576  when i  1, 0 0 0 0 3929   40320 4975 3862 40320 2422 40320 883 40320 139 40320 0 40320   734 380 244 89 14   0 0 0 317 630   630 630 630 630 630   b   4480 4374 1539 243 d  0 5181 16119 4230 0 0 0  2336 32  4480 4480 4480 4480   4480 224 704 200  315 315  712 0 0 0 0 315  315 315 315 0 29125   101875  8064 1250 38250 4375 1375  8064   0 0 0 8064  8064 8064 8064when i  2, 0 0 0 0 2462 10080   10080 4315 3044 10080 1882 10080 682 10080 107 10080  0 0 0 0 355   1088 370 272 101 16   630   630 630 630 630 630 d  0 0 0 0 984  b   1120 3501 72 870 288 45     1424  1120 1120 1120 1120 1120 376 176 608 80 16 0 0 0 0 315   315 315 315 315 315  0  0 0 0 3050 2016    11875  2016 2500 2016 6250 2016 1250 2016 275 2016   994 | P a g e
  5. 5. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998When i  3 0 0 0 0 475 1440   1427 1440 798 1440 482 1440 173 1440 27 1440  0 0 0 0 28   129 14 14 6 1   90   90 90 90 90 90  d  0 0 0 0 51  b   160 219 114 114 21 3   14   14 14  160 160 160 160 160 64 24 64 0 0 0 0 45   45 45 45 45 45  0  0 0 0 95  280   375  288 250 288 250 288 375 288 95  288 3.0 Analysis of the basic properties of the method(3.1) Order of the methodLet the linear operator L  y ( x) : h associated with the block formula (8) be defined as: 3 iL  y ( x) : h  A0 ym(i )   hi ei yn (i ) h 4i  df ( yn )  bF ( ym )  (9) i 0Expanding (9) in Taylor series and comparing the coefficients of h givesL  y( x) : h  C0 y( x)  C1hy ( x)  ...  C p h p y p ( x)  C p1h p1 y p1 ( x)  C p 2h p2 y p2 ( x)  ...Definition:-the linear operator L and associated linear multistep method are said to be of order p ifC0  C1  C2  C p  C p1  0 and C p  2  0, C p  2 is called the error constant and implies that the localtruncation error is given by tn  k  C p  2 h p  2 y p  2 ( x)  0(h p 3 ) .For our method, expanding (8) at i 0 in Taylor series gives  (h) j j   j! y n  yn  hy n  2! y n  3! y n  1814400 h y n   1814400 j! y n 49045(1)  40160(2)  25430(3)  9310(4) 1469(5)  h2 h3 49126 4 iv h j 4 j 4 j j j j j    j 0 j 0   (2 h ) j j   j! y n  yn  2hy n  2! y n  3! y n  14175 h y n   14175( j!) 7960(1)  4910(2)  3080(3)  1120(4) 176(5)  2 3 j 4  (2 h ) (2 h ) 4264 4 iv h j j j j j   j 0 j 0  j   (3hj!) y j n  yn  3hy n  (32!) y n  (33!) y n  22400 h 4 y iv n   22400( j!) 63315(1) j  26460(2) j  19170(3) j  7020(4) j  1107(5) j  h j 4  2 3 h h 25488 j 0    j 0 j   (4 h ) y j n  yn  4hy n  (4 h )2 y n  (4 h)3 y n  40448 h 4 y iv n   h j4 116480(1) j  29400 (2) j  33280 (3) j  11360 (4) j  1792 (5) j   14175( j!)   j 0 j! 2! 3! 14175 j 0 14175 14175 14175 14175    (5h ) j y j  y  5hy  (5h)2 y  (5 h)3 y  418250 h4 y iv  1315625(1) j  175000(2) j  418750(3) j  10625(4) j  18625(5) j   j! n n n  72576( j !)  h j 4 n 2! n 3! n 72576  j 0 j 0 comparing the coefficient of h given C0  C1  C2  C3  C4  C5  C6  0  C7  C8  C9  0  2323 137 1737 1408 29375 and the error constant C10    3628800 14175 44800 14175 145152  (3.2) Zero stabilityA block method is said to be zero stable if as h  0 , the root rj j  1(1)k of the first characteristicpolynomial  ( R)  0 that is  ( R)  det  A0 R K 1   0 satisfying R  1 and   for those root with R  1 must have multiplicity equal to unity (see Fatunla (1991) for detail). 995 | P a g e
  6. 6. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998  1 0 0 0 0  0 0 0 0 1      0 1 0 0 0  0   0 0 0 1    ( R)   R 0 0 1 0 0   0 0 0 0 1   0       0 0 0 1 0  0 0 0 0 1   0 0 0 0 1  0 0 0 0 1       ( R)  R ( R  1)  0, R  0, 0, 0, 0,1 4Hence the block is zero stable4.0 Numerical examplesProblem I: We consider a special fourth order initial value problemy iv  x, y(0)  0, y (0)  1, y (0)  y (0)  0, h  0.1 x5 y ( x)  xExact solution: 120This problem was solved by Olabode (2009) using block method developed for special fourth order odes oforder six with h=0.1We compare our result with their result as shown in Table1.Problem II: We consider a linear fourth order initial value problem y iv  y  0 0  x   2y(0)  0, y (0)  7250 , y (0)  1441  , y (0)  1441.2  1.1 100 100 1  x  cos x  1.2sin xExact solution: y ( x)  144  100This problem was solved by Kayode (2008) adopting predictor corrector method, where the corrector of order 1six was proposed with step length h  . We solve this problem using our method for step length h  0.01 320.. We compare our result with Kayode (2008) as shown in table II.Problem III: We consider a nonlinear initial value problemyiv  ( y )2  y( y )  4 x 2  e x (1  4 x  x 2 ) 0  x  1y(0)  1, y (0)  1, y (0)  3, y (0)  1Exact solution: y( x)  x  e 2 xWe compare our result with Kayode (2008) who adopted predictor corrector method where a method of order 1six as proposed with step length h  .We solve this problem for step length h  0.01 . Our result is 320.shown in table III.yiv  ( y )2  yy (3)  4 x2  e x (1  4 x  x 2 ), 0  x  1y(0)  1, y (0)  1, y (0)  3, y (0)  1y ( x)  x 2  e x 996 | P a g e
  7. 7. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998Error = Exact result  Computed result Table 1 Showing result of problem I, h= 0.1 X EXACT RESULT COMPUTED RESULT ERROR Error in Olabode(2009) 0.1 0.100000083333333 0.100000083333333 0.0000+00 7.0000(-10) 0.2 0.200002666666667 0.200002666666667 0.0000+00 8.9999(-10) 0.3 0.300020250000000 0.300020250000000 0.0000+00 2.0999(-09) 0.4 0.400085333333334 0.400085333333334 0.0000+00 5.1000(-09) 0.5 0.500260416666667 0.500260416666667 0.0000+00 7.7999 (-09) 0.6 0.600648000000000 0.600648000000000 0.0000+00 1.1800 (-08) 0.7 0.701400583333333 0.701400583333333 0.0000+00 1.2400 (-08) 0.8 0.802730666666667 0.802730666666667 0.0000+00 1.4100 (-08) 0.9 0.904920750000001 0.904920750000001 0.0000+00 1.8800 (-08) 1.0 1.008333333333333 1.008333333333333 0.0000+00 2.6000 (-08) Table II Showing result of problem II h=0.01 ERROR IN X EXACT RESULT COMPUTED RESULT ERROR Kayode (2008) 0.1 0.0012623718420566 0.0012623718420566 6.5052(-19) 4.8355(-17) 0.2 0.0024592829189318 0.0024592829189318 1.3010(-18) 1.3933(-16) 0.3 0.0035846460422381 0.0035846460422381 4.7704(-18) 6.6893(-16) 0.4 0.0046330889070900 0.0046330889070900 1.7347(-17) 2.0129(-15) 0.5 0.0056000077703975 0.0056000077703976 4.3368(-17) 4.6736 (-15) 0.6 0.0064816134499462 0.0064816134499463 9.5409(-17) 9.1874 (-15) 0.7 0.0072749691846527 0.0072749691846529 1.8127(-16) 1.6069 (-14) 0.8 0.0079780199777112 0.0079780199777115 3.1571(-16) 2.5407 (-14) 0.9 0.0085896131294417 0.0085896131294422 5.1868(-16) 3.8108 (-14) 1.0 0.0091095097546848 0.0091095097546856 8.0491(-16) 5.4051 (-14) Table III Showing result of problem III ERROR IN X EXACT RESULT COMPUTED RESULT ERROR Kayode (2008) 0.1 1.1151709180756477 1.1151709180747431 9.0460(-13) 6.6613(-15) 0.2 1.2614027581601699 1.2614027581580183 2.1516(-12) 9.7477(-14) 0.3 1.4398588075760033 1.4398588075722483 3.7549(-12) 4.6185(-13) 0.4 1.6518246976412703 1.6518246976355817 5.6885(-12) 1.4224(-12) 0.5 1.8987212707001282 1.8987212706922463 7.8819(-12) 3.4254 (-12) 0.6 2.1821188003905090 2.1821188003802967 1.0212(-11) 6.8736 (-12) 0.7 2.5037527074704768 2.5037527074579797 1.2497(-11) 1.2636 (-11) 0.8 2.8655409284924680 2.8655409284779814 1.4486(-11) 2.1388 (-11) 0.9 3.2696031111569508 3.2696031111411017 1.5849(-11) 3.3989 (-11) 1.0 3.7182818284590464 3.7182818284428873 1.6159(-11) 5.1402 (-11)5.0 CONCLUSION noted that the continuous block method has the We have proposed a five steps numerical same advantages as the continuous linear multistepintegrator for the solution of fourth order initial method which is extensively discussed byvalue problems which was implemented in Awoyemi (1991). The results show that our methodcontinuous block method. Continuous block gave better approximation than the existingmethod has advantage of evaluation at all selected methods that we compared our results with.points within the interval of integration. It must be 997 | P a g e
  8. 8. Adesanya, A. Olaide, A. A. Momoh, Alkali, M. Adamu and Tahir, A. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.991-998REFENCES 14. S. O. Fatunla, (1991), Block method for 1. O. Adesanya, T. A. Anake, and G. J. second order IVPs.Intern. J. of Comp. Oghonyon, (2009), Continuous implicit math, 42(9), 53-63 method for the solution of general second 15. S. J. kayode (2008), An efficient zero order ordinary differential equation. J. stable numerical method for fourth –order Nigeria Association of Math. Phys. 15, differential equation. 71-78. 16. Intern. J. Math and Maths sci. Doc: 10: 2. O. Adesanya, T. A. Aneke, and M. O. 1155/2008/364021. Udoh, (2008); improved continuous 17. S. J. kayode (2004), A class of maximal method for direct solution of general order linear multistep collocation methods second order differential equation, J. of for direct solution of ordinary differential Nigeria Association of Math. Phys. 13, equations. Unpublished doctoral 59-62. dissertation, Federal University of 3. O. Adesanya, M. R. Odekunle and A. A. Technology, Akure, Nigeria. James, (2012), order seven continuous 18. T. A. Anake, D. O. Awoyemi, A. O. hybrid method for the solution of first Adesanya, and Famewo, M. M. (2012), order differential equations, Canadian J. of solving general second order ordinary Sci. and Eng. Math. 3(4),154-158. differential equation by a one-step hybrid 4. T. Olabode (2009): A six step sheme for collocation method, intern. J. of Sci. and the solution of fourth order differential Eng. 2(4), 164-168. equation. Pacific J. of Sci and Tech. 10(1), 19. U. Mohammed, (2010), A six step block 143-148. method for the solution of fourth order 5. O. Awoyemi, E. A. Adebile, A. O ordinary differential equations, pacific J. Adesanya, and T. A. Anake (2011): of Sci and Tech. 11(1), 259-265. modified block method for the 6. direct solution of second order ordinary differential equations, Intern. J. of Appl. Math and Comp 3(3), 181-188. 7. O. Awoyemi, (2001), A class of continuous linear multistep method for general second order initial 8. value problems in ordinary differential equation, Intern. J. Math and Comp 372, 29-37. 9. O. Awoyemi, and S. J. Kayode, (2004), Maximal order multi derivative collocation method for the direct solution of fourth order initial value problems ordinary differential equation. J. Nig. Math. Soc.23, 53-64. 10. Ismail, Y. L. Ken and M. Othman (2009), Explicit and Implicit 3-point block methods for solving special second order ordinary differential equation directly, intern. J. Math. Analy, 3(5), 239-254 11. H. Twizel and Khaliq, A. Q. M. (1984), Multi derivative method for periodic IVPs, SIAM J. of Numer Anal, 21, 111-121 12. J. Vigo-Aguilar and H. Ramos (2006): Variable step size implementation of multistep methods for y  f ( x, y, y,), J. of comp. & Appl. Math. 192, 114-131. 13. R. A. Bun and Y. D. Varsolyer (1992), A numerical method for solving differential equation of any orders. Comp. Math. Phys. 32(3), 317-330 998 | P a g e

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