Convergence of Differential Transform Method for Ordinary Differential Equations

Transcription

1 Journal of Advances in Mathematics and Computer Science 246: 1-17, 2017; Article no.jamcs Previously nown as British Journal of Mathematics & Computer Science ISSN: Convergence of Differential Transform Method for Ordinary Differential Equations A. S. Oe 1 1 Department of Mathematical Sciences, Faculty of Science, Adeunle Ajasin University, Aungba Aoo, Ondo State, Nigeria. Author s contribution The sole author designed, analyzed and interpreted and prepared the manuscript. Article Information DOI: /JAMCS/2017/36489 Editors: 1 Lei Zhang, Winston-Salem State University, North Carolina, USA. 2 Kai-Long Hsiao, Taiwan Shoufu University, Taiwan. Reviewers: 1 Sherin Zafar, Manav Rachna International University Faridabad, India. 2 Jagdish Praash, University of Botswana, Botswana. 3 Alo Kumar Pandey, Rooree Institute of Technology, India. Complete Peer review History: Received: 30 th August 2017 Accepted: 17 th September 2017 Original Research Article Published: 10 th October 2017 Abstract Differential transform method DTM as a method for approximating solutions to differential equations have many theorems that are often used without recourse to their proofs. In this paper, attempts are made to compile these proofs. This paper also proceeds to establish the convergence of the DTM for ordinary differential equations. This paper establishes that if the solution of an ordinary differential equation can be written as Taylors expansion, then the solution can be obtained using the DTM. This is also demonstrated with some numerical examples. Keywords: Differential transform; inverse differential transform; homogeneous ODE; nonhomogeneous ODE Mathematics Subject Classification: 34L99. *Corresponding author: abayomi.oe@aaua.edu.ng, oeabayomisamuel@gmail.com

2 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Introduction One of the main focus of research in recent times is on the methods for solving nonlinear ordinary differential equations since most partial differential equations can be transformed to some nonlinear ordinary differential equations by means of some rescaling terms. Perturbation Methods, Adomian Decomposition Method ADM [1, 2, 3, 4, 5], Variational Iterative Method VIM [6, 7, 8], Differential Transform Method DTM [9, 10, 11, 12, 13], Homotopy Perturbation Method [14] and Homotopy Analysis Method HAM are some of the widely used methods for solving nonlinear problems. The Perturbation Methods, ADM, DTM and VIM are very effective in handling wealy nonlinear problems while HAM can handle wealy as well as strongly nonlinear problems. Despite the wide applicability of the Homotopy Analysis Method, it requires the solution of some differential equations and some quadratures. When the problem involved is large, the quadratures become too cumbersome and uneasy to handle but the DTM has an advantage over this setbac. The DTM reduces the problem to a set of recursive equations that can easily be handled recursively. DTM has been applied to solve linear and nonlinear systems of ordinary differential equations [15, 11, 16, 17, 18, 12, 19, 20, 7, 3, 21, 13, 22, 23] and it is applied to solve some biological equations in [24, 25]. Based on the assumption that the reader is familiar with DTM, most authors omit the proofs of some theorems. This assumption inspires the first part of this paper. Proofs of some important theorems which are often omitted are presented. These theorems are then extended with proofs. Moreso, we further show that the DTM converges to the exact solution when the problem involved is linear whether homogeneous or nonhomogeneous. Finally, we establish these proofs with some examples. 2 Differential Transform The Differential Transform Method is an extension of the Taylor Expansion Method. The Taylor expansion for a function yx about the point x x 0 is defined as x a d yx! dx yx. 2.1 xa Define the differential transform [9, 10, 26] of the function yx about x a as DT {y x} Y a [] 1 d! dx yx 2.2 xa and the inverse transform as DT 1 {Y a []} y x Y [] x x We also define the differential transform of yx about x 0 as DT {y x} Y [] 1 d! dx yx 2.4 and the inverse transform as DT 1 {Y []} y x Y [] x

3 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs We define the Mth order approximation of yx as y x 3 Theorems in Differential Transform M Y [] x. 2.6 The following theorems and some of the proofs can be found in [9, 13, 10, 26, 27, 28, 29]. In the following theorems, we shall suppose α, β, γ, c are constants and that DT {yx} Y [], DT {a j x} A j [], DT {fx} F [], DT {gx} G [], DT {a x} A [] and define the delta function as δ,n { 0 n 1 n. 3.1 Theorem 3.1. Linear combination. Linear combination is closed under differential transform i.e. if yx αfx ± βgx, then Y [] αf [] + βg[]. Proof. DT {yx} 1 d αf x ± βg x! dx αf [] ± βg []. Corollary 3.2. Scalar Multiplication. If yx αfx, then Y [] αf []. Proof. This follows from theorem 3.1 by setting β 0 Theorem 3.3. Polynomial function. If y x cx n, then Y [] cδ n, Proof. By definition Y [] 1! { d dx cxn 1 n c 0 n cδ n,. Corollary 3.4. Constant function. If y x c, then Y [] cδ 0, Proof. By setting n 0 in theorem 3.3, the proof is complete. Theorem 3.5. Exponential function. If y x e ax+b, then Y [] e b a! Proof. By definition Y [] 1! d dx eax+b e b a!. Theorem 3.6. Trigonometric functions. If y x sin ax + b and y x cos ax + b then Y [] a π! sin 2 + b and Y [] a π! cos 2 + b 3.2 respectively. 3

4 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Proof. By definition Y [] 1 d sin ax + b! dx 1! a sin ax + b + π a π 2! sin 2 + b. and similarly, Y [] a π! cos 2 + b. Theorem 3.7. nth derivative. If y x dn f x, then dx n n Y [] + r F [ + n]. Proof. By definition Y [] 1 d x d n! dx dx f 1 d +n f x n! dx+n n 1 d +n n + r f x + n! dx+n + r F [ + n]. Corollary 3.8. The differential transform of f x, f x, f x are + 1 F [ + 1], F [ + 2], F [ + 3] respectively. Proof. These results are obtained by simply setting n 1, 2, 3 respectively in theorem 3.7. Theorem 3.9. Convolution theorem. If the convolution of F [] and G [] is defined as then Proof. Let then r0 F [] G [] F [r] G [ r] r0 DT {f x g x} F [] G [] G [] F [] y x f x g x Y [] 1 d f x g x 1 d r d r! dx f x g x r! r! dxr dx r r0 1 d r r! dx f x 1 d r r g x r! dx r F [r] G [ r] 3.3 by interchanging the roles of f and g, we also have Hence, Y [] G [r] F [ r] r0 r0 F [ r] G [r]. 3.4 r0 DT {f x g x} F [] G [] G [] F []

5 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Theorem Suppose then Y [] 1 r 1 0 r 2 0 n 2 r n 1 0 y x n f i x, i1 F 1 [r 1] F 2 [r 2] F n 1 [r n 1] F n [ n 1] where p p r i, 1 p < n i1 Proof. By definition so, r 1 0 Y [] 1! d n dx i1 f i x Y [] 1! d r 1! r 1 0 1!r 1! dx f r 1 x d 1 n 1 f dx i x 1 i2 1 d 1 n F 1 [r 1] f 1! dx i x 1 repeating this process again, we have Y [] F 1 [r 1 ] r r 2 0 F 2 [r 2 ] i2 1 2! and continuing in this manner for n 1 times, we have d 2 n dx 2 i3 f i x Y [] F 1 [r 1] r 1 0 F 1 [r 1] r r 1 0 r r r 2 0 n 1 r n 1 0 F 2 [r 2] F 2 [r 2] n 1 r n 1 0 n 1 r n 1 0 F n 1 [r n 1] 1 n 1! F n 1 [r n 1] F [ n 1] d n 1 dx n 1 F 1 [r 1] F 2 [r 2] F n 1 [r n 1] F [ n 1]. fn x Theorem Suppose then Y [] y x dn dm f x dxn dx g x m n m r + i r + j F [r + n] G [ r + m]. r0 i1 j1 5

6 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Proof. By definition Y [] 1 d d n dm f x! dx dxn dx g x m 1 d r d n d r d m C r! dx r dx f x n dx r dx g x m r0 1 d n+r 1 d m+ r f x g x r! dxn+r r! dxm+ r thus, Y [] r0 n m 1 d n+r 1 d r+m r + i r + j f x g x r + n! dxn+r r + m! dx r+m i1 j1 n m r + i r + j F [r + n] G [ r + m] r0 r0 i1 j1 Theorem If yx 1 + x m then Y [] m C Proof. By definition Y [] 1! d 1 + xm dx m C 1 + x m m C. Theorem If then Y [] yx x 0 ftdt { 0 0 F [ 1] > 0. Proof. Let f x be the anti-derivative of f x, then and When 0, Y [] 1! x 0 ftdt f x f 0 d x ftdt dx 0 1! d f x f 0 dx and when > 0, we have Y [] 1! Y [0] f 0 f 0 0. d f x f 0 dx 1! f x 1 d 1 dx F [ 1]. 6

7 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Theorem If yx f ax then Y [] a F []. Proof. By using the inverse differential transform, we have f x F [] x and clearly, f ax and the differential transform is F [] ax a F [] x Y [] DT {f ax} a F []. Theorem Y [] and Y a [] are related by the relations Y [] r C Y a [r] and Y a [] r C Y [r] r r Proof. If Y a [] 1 d! dx yx, 3.6 xa then the inverse differential transform is yx Y a [] x a. By substituting the Binomial expansion, we have yx and therefore Similarly, Y a [] C r x r a r C r Y a [] x r a r r0 r C Y a [r] x r0 r r Y [] r C Y a [r]. r r C Y a [r] x yx Y [] x Y [] x a + a By substituting the Binomial expansion, we have yx Y [] C r x a r C r Y [] x a r r0 r C Y [r] x a r0 r r r C Y [r] x a 7

8 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs and therefore Y a [] r C Y [r]. r 4 Main Results Theorem 4.1. If then 1 yx x 0 f t g t dt { 0 0 Y [] 1. r0 F [r] G[ 1 r] > 0 Proof. Let ȳ x be the anti-derivative of f x gx, then and x 0 Y [] 1! f t g t dt ȳ x ȳ 0 1! d dx x 0 f t g t dt d ȳ x ȳ 0 dx When 0, Y [0] ȳ 0 ȳ 0 0. and when > 0, we have Y [] 1! 1 d ȳ x ȳ 0 dx 1 1! d 1 dx f x g x 1 By applying the convolution theorem 3.9, we have Y [] 1 1 F [r] G [ 1 r] r0 Theorem 4.2. Initial conditions. If d n dx yx n α then Y [n] α n!. Proof. By definition Y [] 1! d dx yx 8

9 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs and so, when n, we have Y [n] 1 n! and inserting the initial condition, we have d n dx yx n Y [n] α n!. Corollary 4.3. If y 0 α 0, y 0 α 1, y 0 α 2, y 0 α 3, then Y [0] α 0, Y [1] α 1, Y [2] α 2 2! and Y [3] α 3 3!. Proof. This is a direct consequence of theorem 4.2 by simply setting n 0, 1, 2, 3 respectively. Theorem 4.4. Boundary conditions. If y b β then β Proof. By definition of the inverse differential transform and so, by substituting x b, we have y x β Y [] b. Y [] x Y [] b. Remar 4.1. In application, we tae the Mth order approximation and we therefore set β M Y [] b. Theorem 4.5. If a, b are constants in the first order ordinary differential equation, a dy + by 0, y 0 α, 4.1 dx then the differential transform method converges to the exact solution. Proof. The differential equation has a general solution yx αe bx/a Taing the differential transform of equation 4.1, we have a + 1 Y [ + 1] + by [] 0, Y [0] α and on rearranging, we have b Y [ + 1] a + 1 Y []. 9

10 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Solving this recurrence relation, we have Y [] 1! b α a and thus, the solution obtained from the differential transform method is 1 b yx αx! a 1 bx α! a αe bx/a. Theorem 4.6. If a, b, c are constants in the second order ordinary differential equation, a d2 y dx 2 + b dy dx + cy 0, y 0 α 0, y 0 α then the differential transform method converges to the exact solution. Proof. Taing the differential transform of equation 4.2, we have a Y [ + 2] + b + 1 Y [ + 1] + cy [] 0, Y [0] α 0, Y [1] α and the differential transform solution is y x Y [] x. 4.4 Substituting the series 4.4 in the original problem 4.2, we have a 1 Y [] x 2 + b Y [] x 1 + c Y [] x 0 2 and on rearranging, we have 2 a Y [ + 2] + b + 1 Y [ + 1] + cy [] x 0, and comparing coefficients gives a Y [ + 2] + b + 1 Y [ + 1] + cy [] Hence, the recurrence relation from the DTM is satisfied. Theorem 4.7. If a i s i 1, 2,, n are constants in the nth order homogeneous ordinary differential equation, a 0 y + d r y dx r 0, y0 α 0, then the differential transform method converges to the exact solution. 2 d i dx y x i α i, i 1, 2,, n

11 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Proof. Taing the differential transform of equation 4.6, we have r a 0Y [] + + i Y [ + r] 0, Y [0] α 0, Y [i] αi, i 1, 2,, n i! i1 and the differential transform solution is y x Substituting the series 4.8 in the original problem 4.6, we have 0 a 0 Y [] x d r + Y [] x dx r r 1 a 0 Y [] x + Y [] i Y [] x. 4.8 r r a 0 Y [] x + Y [] i0 r 1 a 0 Y [] x + Y [ + r] a 0Y [] + and comparing coefficients gives a 0Y [] + r i1 r i1 + i + i Hence, the recurrence relation from the DTM is satisfied. i i0 r 1 x r x r + r i i0 Y [ + r] x x Y [ + r] Theorem 4.8. For a variable coefficient nth order homogeneous ordinary differential equation a 0 x y + x dr y dx 0, y0 α0, d i r dx y x i α i, i 1, 2,, n the differential transform method converges to the exact solution. Proof. Taing the differential transform of equation 4.10, we have r A 0 [m] Y [ m] + A r [m] m + j Y [ m + r] 0, 4.11 m0 m0 subject to Y [0] α 0, Y [i] αi, i 1, 2,, n 1. i! Rearranging, we have r A 0 [m] Y [ m] + A r [m]y [ m + r] m + j 0. m0 j1 j1 11

12 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Substituting the differential transforms y x Y [] x, x A r [] x into the original problem 4.10, we have 0 A 0 [] x m0 Y [] x + A 0 [m] Y [ m] x + A r [] x d r Y [] x dx r r 1 A r [] x Y [m + r] m0 j0 m + r j x m A 0 [m] Y [ m] x + A r [m] Y [ m + r] m0 m0 m0 A 0 [m] Y [ m] + A r [m] Y [ m + r] r 1 r 1 m + r j j0 m + r j r A 0 [m] Y [ m] + A r [m] Y [ m + r] m + j j0 m0 j1 x x x and comparing coefficients, we have r A 0 [m] Y [ m] + A r [m] Y [ m + r] m + j m0 j1 Hence, the recurrence relation from the DTM is satisfied. Theorem 4.9. If a i s i 1, 2,, n are constants in the nth order nonhomogeneous ordinary differential equation, a 0 y + d r y dx r f x, y0 α 0, then the differential transform method converges to the exact solution. d i dx y x i α i, i 1, 2,, n Proof. Taing the differential transform of equation 4.14, we have a 0Y [] + r i1 + i Substituting the differential transforms Y [ + r] F [], Y [0] α 0, Y [i] αi, i 1, 2,, n i! y x Y [] x, f x F [] x

13 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs in the original problem 4.14, we have F [] x a 0 Y [] x + a 0 Y [] x + r d r Y [] x dx r r 1 Y [] i r a 0 Y [] x + Y [] i0 r 1 a 0 Y [] x + Y [ + r] n a 0 Y [] x + i i0 r 1 x r x r + r i i0 r 1 Y [ + r] i0 i0 + r i r 1 a 0 Y [] + Y [ + r] + r i a 0Y [] + and comparing coefficients gives a 0 Y [] + r i1 r i1 + i Hence, the recurrence relation from the DTM is satisfied. + i Y [ + r] x x x x Y [ + r] F [] Theorem For a variable coefficient nth order nonhomogeneous ordinary differential equation a 0 x y + x dr y dx f x, y0 α d i 0, r dx y x i α i, i 1, 2,, n the differential transform method converges to the exact solution. Proof. Taing the differential transform of equation 4.18, we have r A 0 [m] Y [ m] + A r [m] m + j Y [ m + r] F [], 4.19 m0 m0 subject to Y [0] α 0, Y [i] α i, i 1, 2,, n 1. i! Rearranging, we have r A 0 [m] Y [ m] + A r[m]y [ m + r] m + j F []. m0 Substituting the differential transform 4.20 j1 y x Y [] x, x A r [] x, f x F [] x j1 13

14 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs in the original problem 4.18, we have and comparing coefficients, we have r A 0 [m] Y [ m] + A r [m] Y [ m + r] m + j F [] m0 Hence, the recurrence relation from the DTM is satisfied. 5 Numerical Examples Example 5.1. Consider the homogeneous second order linear ordinary differential equation The general solution is and to O7,we have j1 y + 5y + 6y 0; y 0 0, y 0 1. y e 2x e 3x 2 3 x y x 5 2 x x x x x6 + Ox We tae the Differential Transform and we get subject to By rearranging, we have so that the 6th order approximation is yx Y [ + 2] Y [ + 1] + 6Y [] 0, M Y [0] 0, Y [1] 1. 5Y [ + 1] 6Y [] Y [ + 2] , Y [] x x 5 2 x x x x x6 + Clearly, this series converges to the exact solution as shown in equation 5.1.! 14

15 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs Example 5.2. Consider the equation the nonhomogeneous second order linear ordinary differential equation y + 5y + 6y e x ; y 0 0, y 0 1. The general solution is y e 2x 9e 3x + e x x 2x x x x x6 + We now proceed to tae the differential transform of the problem Y [ + 2] Y [ + 1] + 6Y [] 1, Y [0] 0, Y [1] 1! on rearranging, we have Y [ + 2] 1 5Y [ + 1] 6Y [] + 2! So that Y [2] 2, Y [3] , Y [4], Y [5] , Y [6] 7 10, and thus we have the solution as which converges to the exact solution. y x 2x x x x x6 + Example 5.3. Consider the nonhomogeneous first order linear ordinary differential equation The exact solution is 1 + x dy dx + y 1 + x ex, y 0 0. y x 1 + x 1 e x x x 6 Conclusion n0 m0 1 m n m! x n n0 x m m! m0 x n x x3 1 3 x x5 + The first part of this wor is dedicated to proving some theorems whose proofs have been long ignored. Most authors assume the nowledge of these theorems, so they do not bother to prove the theorems. The theorems are therefore proved to serve as eference for any wor that would want to use the theorems without proofs. The later part of this wor establishes the convergence of the solution obtained from the DTM to the exact solution. The DTM solution converges to the exact solution for any ordinary differential equation, whether homogeneous or nonhomogeneous. These theorems are illustrated with some examples. The differential transform method reduces the difficulty of solving an ordinary differential equation to a simple recursive equation that are relatively easy to solve. This wor establishes that without 15

16 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs solving a differential equation to get a closed form solution, we can obtain an approximation up to any term of interest to the solution by solving using the DTM. It is important to mention here that since DTM is a method derived from the Taylors expansion method and Taylors expansion is only valid for functions that are continuous in the region of concern, we note that this method will as well be useful only for solutions that are continuous in the region under consideration. In addition to this, we recognise that the solution is assumed to admit a Taylors expansion. Competing Interests Author has declared that no competing interests exist. References [1] Biazar J, Babolian E, Islam R. Solution of a system of volterra integral equations of the first ind by adomian method. Appl. Math. Comput. 2003;139: [2] El-Sayed S. The decomposition method for studying the lein-gordon equation. Chaos, Solitons & Fractals. 2003;18: [3] Hassan OA-H. Comparison differential transform technique with adomian decomposition method for linear and nonlinear initial value problems. Chaos, Solitons & Fractals. 2008;36: [4] Jafari H, Gejji VD. Revised adomian decomposition method for solving system of ordinary and fractional differential equations. Appl. Math. Comput. 2006;181: [5] Biazar J, Babolian E, Islam R. Solution of the system of ordinary differential equations by adomian decomposition method. Appl. Math. Comput. 2004;1473: [6] Shawagfeh NT, Kaya D. Comparing numerical methods for solutions of ordinary differential equations. Appl. Math. Lett. 2004;17: [7] Biazar J, Hes HG. Variational iteration method for solving linear and nonlinear and system of ordinary differential equations. Appl. Math. Comput. 2007;191: [8] Yusufoglu OE. The variational iteration method for studying the lein-gordon equation. Appl. Math. Lett. 2008;21: [9] Zhou J. Differential transformation and application for electrical circuits. Huazhong University Press, Wuhan, China; [10] Chen CL, Lin SH, Chen C. Application of taylor transformation to nonlinear predictive control problem. Appl. Math. Model. 1996;20: [11] Chen CK, Ho SH. Solving partial differential equations by two-dimensional differential transform method. Appl. Math. Comput. 1999;106: [12] Ayaz F. Solution of the system of differential equations by differential transform method. Appl. Math. Comput. 2004;147: [13] Odibat Z. Differential transform method for solving volterra integral equation with separable ernels. Math. Comput. Model. 2008;48:

17 Oe; JAMCS, 246: 1-17, 2017; Article no.jamcs [14] Javidi M, Golbabai A. Modified homotopy perturbation method for solving non-linear fredholm integral equations. Chaos, Solutions and Fractals; [15] Kangalgil OF, Ayaz OF. Solitary wave solutions for the dv and mdv equations by differential transform method. Chaos, Solitons & Fractals; [16] Jang MJ, Chen CL, Liu Y. Two-dimensional differential transform for partial differential equations. Appl. Math. Comput. 2001;121: [17] Hassan OA-H. Different applications for the differential transformation in the differential equations. Appl. Math. Comput. 2002;129: [18] Ayaz F. On two-dimensional differential transform method. Appl. Math. Comput. 2003;143: [19] Arioglu A, Ozol I. Solution of boundary value problems for integro-differential equations by using transform method. Appl. Math. Comput. 2005;168: [20] Kurnaz A, Oturnaz G, Kiris M. n-dimensional differential transformation method for solving linear and nonlinear pdes. Int. J. Comput. Math. 2005;82: [21] Kanth ASVR, Aruna K. Differential transform method for solving linear and non-linear systems of partial differential equations. Phys. Lett. 2008;A 372: [22] Kanth ASVR, Aruna K. Differential transform method for solving the linear and nonlinear lein-gordon equation. Computer Physics Communications. 2009;180: [23] Mirzaee F. Differential transform method for solving linear and nonlinear systems of ordinary differential equations. Applied Mathematical Sciences. 2011;570: [24] Tari A. The differential transform method for solving the model describing biological species living together. Iranian Journal of Mathematical Sciences and Informatics. 2012;7:2: [25] Moghadam MM, Saeedi H. Application of differential transforms for solving the volterra integro-partial differential equations. Iranian Journal of Science & Technology, Transaction. 2010;A 34:A1. [26] Jang MJ, Chen CL, Liu YC. On solving the initial value problems using the differential transformation method. Appl. Math. Comput. 2000;115: [27] Abbasov A, Bahadir AR. The investigation of the transient regimes in the nonlinear systems by the generalized classical method. Math. Prob. Eng. 2005;5: [28] Hassan IHA-H. Differential transformation technique for solving higher-order initial value problems. Appl. Math. Comput. 2004;154: [29] Mali M, Dang HH. Vibration analysis of continuous system by differential transformation. Appl. Math. Comput. 1998;96: c 2017 Oe; This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Peer-review history: The peer review history for this paper can be accessed here Please copy paste the total lin in your browser address bar 17