Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2277 2290 Research India Publications http://www.ripublication.com/gjpam.htm Solution of First Order Initial Value Problem by Sixth Order Predictor Corrector Method R. M. Dhaigude P. G. Department of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati-444 604 (M.S) India. R. K. Devkate Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004 (M.S) India. Abstract In this paper, we developed a new sixth order predictor-corrector method by using Newton s forward interpolation formula to obtain the solution of first order initial value problem. The basic properties of the derived corrector is investigated and found to be zero stable, consistent and convergent. The efficiency of the method is tested through numerical examples. Also, it found that this method gives better solution, comparative to other methods. AMS subject classification: 65L05, 65L06, 65L20. Keywords: Newton s forward interpolation formula, Initial value problem, Error analysis, Sixth order predictor-corrector method. 1. Introduction Initial value problems plays a very important role in the study of science, engineering and technology. Many problems in day today life forms as an initial value problem (IVP). This attracts many researchers have been involved in the field of theory, methods and applications of IVP s. It demands to study reliable and efficient technique to obtain the exact or approximate solution of IVP s. It is easy to solve linear IVP s and
2278 R. M. Dhaigude and R. K. Devkate for which analytical methods are available in literature too. The main difficulties lies between in nonlinear IVP s in such cases numerical methods will help to obtain the solution. The literature is existed on the solution of IVP s by using numerical techniques. James and Adesanya [1] discussed constant order predictor corrector algorithm for the solution of IVP. James et al. [2] developed starting order seven method accurately for the solution of first IVP s of first order ordinary differential equations. Adesanya et al. [3] obtained starting the five steps Stomer-Cowell method by Adams-Bashforth method for the solution of the first order ordinary differential equations. Sunday and Odekunle [5] employed new numerical integrator for the solution of IVP s in ordinary differential equations. Odekunle et al. [8] gave the direct integration of first order ordinary differential equations by using 4-Point block method. Furthermore, Abbas [10] employed fourth-order block method for the solution of first order IVP s. Though these method existed but accuracy, stability and convergence of the method matters. At the same time error analysis plays crucial role in numerical methods. In this paper, we made an attempt to obtain the solution of first order IVP by sixth order predictor corrector method (SOPCM). The plan of the paper is as follows. In section 2 we define some basic definitions and concepts. The section 3 is devoted to develop predictor and corrector by using Newton s forward interpolation formula. The consistency, zero stability and convergence of the sixth order corrector is proved in section 4. In last section test problems are solved. It shows that efficiency of the new method is good in comparison of the existing method. Also, conclusions is given at the end of the paper. 2. Preliminary In this section, we define some basic definitions and interpolation formulae [4, 6, 7, 9, 11] which are useful in this paper. Definition 2.1. equation with initial condition Initial value problem Consider the first order ordinary differential dy dx = f(x,y), (2.1) y(x 0 ) = y 0, (2.2) the equation (2.1) with (2.2) is called initial value problem of first order. The existence and uniqueness theorem for IVP (2.1) (2.2) is stated as follows. Theorem 2.2. Let f(x,y) be defined and continuous for all points (x, y) in the region D defined by {(x, y) : a x b, <y< }, where a and b finite, and let there exist a constant L such that for every x,y,y such that
Solution of First Order Initial Value Problem 2279 (x, y) and (x, y ) are both in D: f(x,y) f(x,y ) L y y. (2.3) Then, if η is any number, there exist a unique solution y(x) of the IVP (2.1)-(2.2), where y(x) is continuous and differentiable for all (x, y) in D. The inequality (2.3) is known as a Lipschitz condition where L called Lipschitz constant. Definition 2.3. Newton s forward interpolation formula n(n 1) f(x,y) = f 0 + n f 0 + 2 f 0 + 2! n(n 1)(n 2)(n 3) + 4 f 0 4! + n(n 1)(n 2) 3 f 0 3! n(n 1)(n 2)(n 3)(n 4) 5 f 0 +, 5! (2.4) this is the Newton s forward interpolation formula. 3. Methodology In this section, we are developed predictor and corrector formula by using Newton s forward interpolation formula. 3.1. Development of the Predictor Integrating IVP (2.1) (2.2) from x 0 to x 0 + 5h and using initial condition, we have y 5 = y 0 + x0 +5h x 0 f(x,y)dx. By using Newton s forward interpolation formula (2.4), we get x0 +5h [ ( n 2 n y 5 = y 0 + f 0 + n f 0 + x 0 2 ( n 4 6n 3 + 11n 2 ) 6n + 4 f 0 24 ( n 5 10n 4 + 35n 3 50n 2 + 24n + 120 ) 2 f 0 + ( n 3 3n 2 + 2n 6 ) ] 5 f 0 + dx, ) 3 f 0
2280 R. M. Dhaigude and R. K. Devkate put x = x 0 + nh dx = hdn, 5 [ ( n 2 n y 5 = y 0 + h f 0 + n f 0 + 0 2 ( n 4 6n 3 + 11n 2 ) 6n + 4 f 0 24 ( n 5 10n 4 + 35n 3 50n 2 + 24n + 120 ( = y 0 + h [nf 0 + n2 n 3 2 f 0 + 1 2 + 1 ( n 5 24 5 3n4 2 + 11n3 3 ( n 6 + 1 120 After simplifying, we have 6 2n5 + 35n4 4 3 n2 2 3n 2 ) 4 f 0 50n3 3 ) 2 f 0 + ( n 3 3n 2 + 2n 6 ) ] 5 f 0 + dn, ) 3 f 0 ) 2 f 0 + 1 ( n 4 ) 6 4 n3 + n 2 3 f 0 ] 5 + 12n ) 2 5 f 0 +. 0 [ y 5 = y 0 +h 5f 0 + 25 2 f 0 + 175 12 2 f 0 + 75 8 3 f 0 + 425 144 4 f 0 + 95 ] 288 5 f 0 +.... (3.5) Neglecting fifth and higher order differences and expressing f 0, 2 f 0, 3 f 0 and 4 f 0 in terms of the function, we get. [ y 5 = y 0 + h 5f 0 + 25 ( ) 175 f1 f 0 + 2 12 + 425 ( ) ] f4 4f 3 + 6f 2 4f 1 + f 0, 144 [( = y 0 + h 5 25 2 + 175 12 75 8 + 425 ) ( 25 f 0 + 144 2 175 6 + 225 8 425 36 ( 175 + 12 225 8 + 425 ) ( 75 f 2 + 24 8 425 ) f 3 + 425 ] 36 144 f 4, [ 95 = y 0 + h 144 f 0 25 72 f 1 + 25 6 f 2 175 72 f 3 + 425 ] 144 f 4, y 5 = y 0 + 5h [ ] 19f0 10f 1 + 120f 2 70f 3 + 85f 4, 144 which is called a predictor formula. ( ) 75( ) f2 2f 1 + f 0 + f3 3f 2 + 3f 1 f 0 8 ) f 1
Solution of First Order Initial Value Problem 2281 3.2. Development of the Corrector In the equation (3.5) neglect sixth and higher order differences and expressing f 0, 2 f 0, 3 f 0, 4 f 0 and 5 f 0 in terms of the function, we get. [ y 5 = y 0 + h 5f 0 + 25 ( ) 175( ) 75( ) f1 f 0 + f2 2f 1 + f 0 + f3 3f 2 + 3f 1 f 0 2 12 8 + 425 ( ) 95 ( ) ] f4 4f 3 + 6f 2 4f 1 + f 0 + f5 5f 4 + 10f 3 10f 2 + 5f 1 f 0, 144 288 [( = y 0 + h 5 25 2 + 175 12 75 8 + 425 144 95 ) ( 25 f 0 + 288 2 175 6 + 225 8 425 36 + 475 ) f 1 288 ( 175 + 12 225 8 + 425 24 950 ) ( 75 f 2 + 288 8 425 36 + 950 ) ( 425 f 3 + 288 144 475 ) f 4 + 95 ] 288 288 f 5, [ 95 = y 0 + h 288 f 0 + 125 96 f 1 + 125 144 f 2 + 125 144 f 3 + 125 96 f 4 + 95 ] 288 f 5, y 5 = y 0 + 5h [ ] 19f0 + 75f 1 + 50f 2 + 50f 3 + 75f 4 + 19f 5, 288 which is called a corrector formula. Hence, the following theorem is obvious. Theorem 3.1. If the first order initial value problem with initial condition dy dx = f(x,y), y(x 0 ) = y 0, and f(x,y) is continuous and satisfies a Lipschits condition in the variable y, and f 0 = f(x 0,y 0 ), f 1 = f(x 0 + h, y 1 ), f 2 = f(x 0 + 2h, y 2 ), f 3 = f(x 0 + 3h, y 3 ) and f 4 = f(x 0 + 4h, y 4 ) then predictor y n+1 = y n 4 + 5h [ ] 19fn 4 10f n 3 + 120f n 2 70f n 1 + 85f n, (3.6) 144 and corrector y n+1 = y n 4 + 5h [ ] 19fn 4 + 75f n 3 + 50f n 2 + 50f n 1 + 75f n + 19f n+1, (3.7) 288 for n = 4, 5, 6, 7, 8,... Remark 3.2. Sixth order predictor-corrector method is not a self-starting method. Four additional starting values y 1,y 2,y 3 and y 4 must be given. They are usually computed using the Picard s method or Taylor⣠s series method or Runge Kutta method of fourth order.
2282 R. M. Dhaigude and R. K. Devkate 4. Analysis of the Basic Properties of the Corrector In this section, we will discuss the basic properties of the corrector. 4.1. Order and Error Constant of the Corrector Consider the linear operator L{y(x n ); h} associated with the linear multistep method (3.7) be defined as L{y(x n ); h} =y(x n+1 ) ay(x n 4 ) h 5 b i f n i+1. (4.8) Expanding (4.8) using Taylor s series expansion and comparing the coefficients of h gives where c 0 = 1 a L{y(x n ; h} =c 0 y(x n ) + c 1 hy (x n ) + c 2 h 2 y (x n ) + + c p h p y p (x n ) + c p+1 h p+1 y p+1 (x n ) +, (4.9) c q = 1 q! [1 a( 4)q ] 1 (q 1)! i=0 5 b i (1 i) q 1,q = 1, 2, 3,...,p. i=0 Definition 4.1. The linear operator L and the associated continuous linear multistep method (3.7) is said to be of order p if, in (4.9) c 0 = c 1 = c 2 = = c p = 0 and c p+1 = 0. c p+1 /σ (1) is called the error constat and implies that the local truncation is given by In this method: T n+1 = c p+1 h p+1 y p+1 (x) + O(h p+2 ). (4.10) c 0 = c 1 = c 2 = c 3 = c 4 = c 5 = c 6 = 0 and c 7 = 0 Therefore, the new predictor corrector method is of order six with error constant 247, where σ(1) = 5. 60480 4.2. Zero Stability of the Corrector Definition 4.2. A linear multistep method (3.7) is said to be zero stable, if all the roots of the first characteristic polynomial ρ(ξ) satisfies ξ 1. In this method all the roots of the first characteristic polynomial ρ(ξ) = ξ 5 1 = 0 satisfies ξ 1. Hence SOPC method is zero stable.
Solution of First Order Initial Value Problem 2283 4.3. Consistency of the Corrector Definition 4.3. A linear multistep method (3.7) is said to be consistent, if it has order p 1 and if ρ(1) = 0, ρ (1) = σ(1). Where ρ(ξ) is the first characteristic polynomial and σ(ξ) is the second characteristic polynomial for this method defines as follows respectively. ρ(ξ) = ξ 5 1, σ(ξ) = 95 288 ξ 5 + 125 96 ξ 4 + 125 144 ξ 3 + 125 144 ξ 2 + 125 96 ξ + 95 288. Therefore ρ(1) = 0, ρ (1) = σ(1). Hence SOPC method is consistent. 4.4. Convergence The necessary and sufficient condition for a linear multistep method to be convergent is that it must be consistent and zero stable. Hence new sixth order predictor corrector method is convergent. 5. Numerical Test Problems The following notations are used in the following tables. RKFO Runge-Kutta Method of Fourth Order. MPCM Milne s Predictor-Corrector method. SOPCM Sixth Order Predictor-Corrector method. NNI A new Numerical Integrator (2012). Problem 5.1. Consider a first order initial value problem dy dx The exact solution of problem (5.1) is = x + y, y(0) = 1, 0 x 1, h = 0.1. y(x) = (1 + x) + 2e x. In this Problem result in table 1 shows that solution by RKFO, MPCM and SOPCM and the comparison with the existing method is shown in table 2. Problem 5.2. Consider a first order initial value problem dy dx = xy, y(0) = 1, 0 x 1, h = 0.1.
2284 R. M. Dhaigude and R. K. Devkate Table 1: Numerical results for problem 5.1 x Exact solution RKFO MPCM SOPCM 0.1 1.110341836 1.110341667 1.110341667 1.110341667 0.2 1.242805516 1.242805142 1.242805142 1.242805142 0.3 1.399717616 1.399716994 1.399716995 1.399716995 0.4 1.583649396 1.583648480 1.583649220 1.583648482 0.5 1.797442542 1.797441277 1.797442197 1.797442312 0.6 2.044237600 2.044235924 2.044237745 2.044237208 0.7 2.327505414 2.327503253 2.327505486 2.327504783 0.8 2.651081856 2.651079126 2.651082484 2.651080939 0.9 3.019206222 3.019202827 3.019206897 3.019205009 1 3.436563656 3.436559488 3.436564986 3.436563067 Figure 1: Comparison of solutions of problem 5.1 The exact solution of problem (5.2) is y(x) = e x2 2 In this Problem result in table 3 shows that solution by MPCM and SOPCM and the comparison with the existing method is shown in table 4. Problem 5.3. (Logistic Model) The logistic model finds applications in various fields, among which are; neural networks, Statistics, Medicine, Physics and so on. In Neural
Solution of First Order Initial Value Problem 2285 Table 2: Absolute error in problem 5.1 x RKFO MPCM SOPCM 0.1 0.000000169 0.000000169 0.000000169 0.2 0.000000374 0.000000374 0.000000374 0.3 0.000000622 0.000000621 0.000000621 0.4 0.000000916 0.000000176 0.000000914 0.5 0.000001265 0.000000345 0.000000230 0.6 0.000001676 0.000000145 0.000000392 0.7 0.000002161 0.000000072 0.000000631 0.8 0.000002730 0.000000628 0.000000917 0.9 0.000003395 0.000000675 0.000001213 1 0.000004168 0.000001330 0.000000589 Table 3: Numerical results for problem 5.2 x Exact solution MPCM SOPCM 0.1 1.005012521 1.005012521 1.005012521 0.2 1.020201340 1.020201340 1.020201340 0.3 1.046027860 1.046027860 1.046027860 0.4 1.083287068 1.083287632 1.083287066 0.5 1.133148453 1.133149301 1.133148521 0.6 1.197217363 1.197219137 1.197217472 0.7 1.277621313 1.277623878 1.277621476 0.8 1.377127764 1.377131933 1.377128001 0.9 1.499302500 1.499308427 1.499302840 1 1.648721271 1.648730134 1.648721829 networks for example, the logistic model is used to introduce nonlinearity in the model and/or to clamp signals to within a specific range. In statistics, they are used to model how the probability p of an event may be affected by one or more explanatory variables. In medicine, they are being used to model the growth of tumors. In chemistry, the concentration of reactants and products in autocatalytic reactions follows the logistic function. If the special initial conditions are applied to the logistic model then it forms logistic nonlinear first order IVP dy dx = y(1 y), y(0) = 0.5, 0 x 1, h = 0.1.
2286 R. M. Dhaigude and R. K. Devkate Table 4: Absolute error in problem 5.2 x MPCM SOPCM 0.1 0.000000000 0.000000000 0.2 0.000000000 0.000000000 0.3 0.000000000 0.000000000 0.4 0.000000564 0.000000002 0.5 0.000000848 0.000000068 0.6 0.000001774 0.000000109 0.7 0.000002565 0.000000163 0.8 0.000004169 0.000000237 0.9 0.000005927 0.000000340 1 0.000008863 0.000000558 Figure 2: Comparison of solutions of problem 5.2 The exact solution of problem (5.3) is y(x) = 1 1 + e x This problem was solved by NNI [5]. The numerical result it shown in table 5 and the comparison with the existing method is shown in table 6.
Solution of First Order Initial Value Problem 2287 Table 5: Numerical results for problem 5.3 x Exact solution NNI SOPCM 0.1 0.52497919 0.52497977 0.52497919 0.2 0.54983398 0.54983515 0.54983401 0.3 0.57444252 0.57444417 0.57444253 0.4 0.59868768 0.59868979 0.59868768 0.5 0.62245932 0.62246192 0.62245933 0.6 0.64565632 0.64565927 0.64565631 0.7 0.66818777 0.66819102 0.66818778 0.8 0.68997446 0.68997794 0.68997449 0.9 0.71094948 0.71095312 0.71094952 1 0.73105860 0.73106223 0.73105857 Table 6: Absolute error in problem 5.3 x NNI SOPCM 0.1 0.00000058 0.00000000 0.2 0.00000117 0.00000003 0.3 0.00000165 0.00000001 0.4 0.00000211 0.00000000 0.5 0.00000260 0.00000001 0.6 0.00000295 0.00000001 0.7 0.00000325 0.00000001 0.8 0.00000348 0.00000003 0.9 0.00000364 0.00000004 1 0.00000363 0.00000003 Problem 5.4. Consider a nonlinear first order initial value problem dy dx = xy2, y(0) = 2, 0 x 0.5, h = 0.05. The exact solution of problem (5.4) is y(x) = 2 x 2 1 6. Conclusion In this paper, we have developed a new sixth order predictor-corrector method for the solution of first order initial value problem. It has been shown that the method is zero
2288 R. M. Dhaigude and R. K. Devkate Figure 3: Comparison of solutions of problem 5.3 Table 7: Comparison of numerical results for problem 5.4 x Exact solution Computed Solution Absolute error Relative error % error 0.05 2.005012532 2.005012534 0.000000002 0.000000001 0.0000001 0.10 2.020202020 2.020202030 0.000000010 0.000000005 0.0000005 0.15 2.046035806 2.046035830 0.000000024 0.000000012 0.0000012 0.20 2.083333334 2.083333380 0.000000046 0.000000022 0.0000022 0.25 2.133333334 2.133333581 0.000000247 0.000000116 0.0000116 0.30 2.197802198 2.197802650 0.000000452 0.000000206 0.0000206 0.35 2.279202280 2.279203086 0.000000806 0.000000354 0.0000354 0.40 2.380952380 2.380953839 0.000001459 0.000000613 0.0000613 0.45 2.507836990 2.507839701 0.000002711 0.000001081 0.0001081 0.50 2.666666666 2.666672078 0.000005412 0.000002030 0.0002030 stable and consistent hence convergent. The numerical problems have shown that sixth order predictor-corrector method gives better and consistent approximate solution than existing methods.
Solution of First Order Initial Value Problem 2289 Figure 4: Comparison of solutions of problem 5.4 References [1] A. A. James and A. O. Adesanya (2014). A Note on the construction of constant order predictor corrector algorithm for the solution of dy/dx=f(x,y), J. British Math. and Comp. Sci., 4(6), 886 895. [2] A. A. James, A. O. Adesanya and M. K. Fasasi (2013). Starting order seven method accurately for the solution of first initial value problems of first order ordinary differential equations, Prog. in Appl. Math., 6(1), 30 39. [3] A. O. Adesanya, M. R. Odekunle, M. A. Alkali and A. B. Abubakar (2013). Starting the five steps Stomer-Cowell method by Adams-Bashforth method for the solution of the first order ordinary differential equations, J. African Math. and Comp. Sci. Res., 6(5), 89 93. [4] J. D. Lambert (1973). Computational Methods in Ordinary Differential Equations, John Willey and Sons, New York. [5] J. Sunday and M.R. Odekunle (2012). A New Numerical integrator for the solution of initial value problems in ordinary differential equations, J. The Pacific Sci. and Tech., 13(1), 221 227. [6] K. S. Rao (2004). Numerical Methods for Scientist and Engineers, Second Edition, Prentice-Hall of India, New Delhi.
2290 R. M. Dhaigude and R. K. Devkate [7] M. K. Jain, S. R. K. Iyengar and R. K. Jain (2008). Numerical Methods for Scientific and Engineering Computation, Fifth Edition, New age International Private Ltd Publishers, New Delhi. [8] M. R. Odekunle, A. O. Adesanya and J. Sunday (2012). 4-Point block method for direct integration of first-order ordinary differential equations, J. Int. Engg. Research and Appl., 2(5), 1182 1187. [9] P. B. Patil and U. P. Verma (2009). Numerical Computational Methods, Revised Edition, Narosa Publishing House Private Ltd, New Delhi. [10] S. H. Abbas (2006). Fourth-order block methods for the numerical solution of first order initial value problems, Sarajevo J. of Math., 2(15), 247 258. [11] S. S. Sastry (1997). Introductry Methods of Numerical Analysis, Second Edition, Prentice-Hall of India Private Ltd, New Delhi.