TAYLOR-NEWTON HOMOTOPY METHOD FOR COMPUTING THE DEPTH OF FLOW RATE FOR A CHANNEL

TAYLOR-NEWTON HOMOTOPY METHOD FOR COMPUTING THE DEPTH OF FLOW RATE FOR A CHANNEL Talib Hahim Haan 1 and MD Sazzad Hoien Chowdhury 2 1 Department of Science in Engineering, Faculty of Engineering, International Ilamic Univerity Malayia, P.O Box 10 Kuala Lumpur 50728, Malayia. talib 2 Department of Science in Engineering, Faculty of Engineering, International Ilamic Univerity Malayia, P.O. Box 10 Kuala Lumpur 50728, Malayia talib@iiu.edu.my, chowdhury@iiu.edu.my Abtract Homotopy approximation method (HAM) can be conidered a one of the new method belong to the general claification of the computational method which can be ued to find the numerical olution of many type of the problem in cience and engineering. The general problem relate to the flow and the depth of water in open channel uch a river and canal i a nonlinear algebraic equation which i nown a continuity equation. The olution of thi equation i the depth of the water. Thi paper repreent attempt to olve the equation of depth and flow uing Newton homotopy baed on Taylor erie. Numerical example i given to how the effectivene of the purpoed method uing MATLAB language. Key Word: Newton homotopy, Taylor-Newton homotopy method, open channel, MATLAB. 1. Introduction It wa nown that finding the depth of the water and it velocity in open channel flow i one of the mot important problem in civil engineering. To determine it, we need to olve a nonlinear algebraic equation called continuity equation. There are many claical numerical method ued to olve thi equation, uch a Newton-Raphon method. Thi method ha been proven globally convergent only under unrealitically retrictive condition. They ometime fail becaue it i difficult to provide a tarting point ufficiently cloe to an unnown olution [1]. To overcome thi convergence problem, globally convergent homotopy method have been tudied by many reearcher from variou view point. By thee tudiehe application of the homotopy method in practical engineering imulation ha been remarably developed homotopy problem with the theoretical guarantee of global convergence. Howeverhe programming of ophiticated homotopy method i often difficult for non-expert or beginner.[1-23, 25-29] There are everal type of homotopy method, a one of the efficient method for olving Van der Waal problem [11]; the Newton homotopy (NH) method i well nown [6]. For thi method, many tudie have been formed from variou viewpoint. Since the idea of Newton homotopy i introducedhe path following often become mooth. However, in thi methodhe initial point i ometime far from the olution [3]. In thi paper, we dicu the ue of Newton homotopy method (NHM) baed on Taylor erie of the multiple verion to olve the nonlinear algebraic equation called continuity equation for finding the depth of the water in open channel flow. Newton homotopy connect a trivial olution of thi nonlinear algebraic equation to the deired unnown olution. The propoed method i almot globally convergent algorithm [10]. By numerical example, it i hown that the propoed method efficient than the conventional method. It i alo hown that the propoed method can be eaily implemented in MATLAB oftware. 2. The Equation of Open-Channel Flow The mot fundamental relationhip between flow q (m 3 /) and depth h (m) i the continuity equation q = ua c (1) where a c = the cro-ectional area of the channel (m 2 ) and u = a pecific velocity (m/). Depending on the channel hapehe area can be related to the depth by ome functional relationhip. For the rectangular channel depicted in figure 1, where b = the width (m) a c = bh (2)

example, for a rectangular channel, it i defined a p = b + 2h (6) Although the ytem of nonlinear equation (3) and (4) can be olved imultaneouly uing Newton-Raphon approach, an eaier approach would be to combine the equation. Equation (a) can be ubtituted into equation (3) to give bh q = r 1/ 3 (7) n Then the hydraulic radiu, Equation (5), along with the variou relationhip for the rectangular channel can be ubtituted, Figure 1 Subtituting thi relation into equation (1) give q = ubh (3) Now, although equation (3) certainly relate the channel parameter, it i not ufficient to find the depth h and the velocity u becaue auming that b i pecified, we have one equation and two unnown (u and h). We therefore require an additional equation. For uniform flow (meaning that the flow doe not vary patially and tempo7rallyhe Irih engineer Robert Manning propoed the following emi-empirical formula (appropriately called the Manning equation) 1 u = r 2/3 2/ 3 n (4) where n = the Manning roughne coefficient (a dimenionle number ued to parameterize the channel friction), = the channel lope (dimenionle, meter drop per meter length), and r = the hydraulic radiu (m), which i related to more fundamental parameter by a r = c (5) p where p = the wetted perimeter (m). A the name impliehe wetted perimeter i the length of the channel ide and bottom that i under water. For 1/ 2 ( bh) q = (8) n ( b + 2h) Thuhe equation now contain a ingle unnown h along with the given value for q and the channel parameter (n,, and b). Although we have one equation with an unnown, it i impoible to olve explicitly for h. Howeverhe depth can be determined numerically by reformulating the equation a a root problem, 1/ 2 ( bh) f ( h) = q = 0 (9) n( b + 2h) Equation (9) can be olved readily with any of the root-location method either braceting method uch a biection and fale poition or open method lie the Newton-Raphon and ecant method. The complexity of braceting method i that to determine whether we can etimate lower and upper guee that alway bracet a ingle root. For the cae of initial gueehe problem of initial guee i complicated by the iue of convergence. For thi reaon, we will apply the method of Taylor Newton Homotopy decribed in ection 3 to olve equation (9) [44]. 3. Taylor-Newton Homotopy Method Firt, we decribe roughly the theoretical bai of the method. Suppoe we need to olve the following equation: f(x) =0 (10)

where f i a continuouly differentiable function from R n into itelf. To find the olution, we can contruct a homotopy. The term homotopy mean a continuou mapping H i defined on the product R n+1 : R n I to R n, H: R n I R n where I i the unit interval [0, 1] uch that H(x) = tf(x) + (1 t)g(x) (11) from g(x) = 0 to f(x) = 0, where the olution of g(x) = 0 can be found trivially. For example, chooe g(x) = f(x) f(x 0 ) (12) Subtitute equation (5) in equation (4) and uppoe that x i a function of t, we obtained a new verion of homotopy H(x) = f(x) + (t 1)f(x 0 ) (13) Thi form of the homotopy i termed the Newton homotopy becaue ome of the idea behind it come from the wor of Sir Iaac Newton (1642-1727) himelf [8]. It i een from equation (13) that, at t = 0he olution of the equation (13) i already nown. For different value of the equation will reult in different olution. At t =1he olution of equation (13) i identical to the deired unnown olution. To follow the above predetermined trajectory, define H ={(x) = 0/ H(x) = 0} (14) a the et of all olution (x) R n+1 to the ytem H(x) = 0. If H i continuouly differentiable, we can mae a linear approximation of it near (x ) uing multivariable verion of the Taylor erie: or H(x +1 +1 ) = H(x ) + (x +1 x )H x (x ) + (t +1 t )H t (x ) + (15) H(x +1 +1 ) H(x ) + (x +1 x )H x (x ) + (t +1 t )H t (x ) (16) The ymbol mean approximately equal and H x = H x H, and H t = t. Study point (x +1 +1 ) in H that are near the point (x ), a we want to how that they are on a path through. By definition, all uch point mut atify H(x +1 +1 ) = 0. (17) Moreover, if we aume that H i actually linear near (x )hen by (16) and (17), we get 0 = H(x ) + (x +1 x )H x (x ) + (t +1 t )H t (x ) (18) In other word, if H where actually linear near (x )hen any (x +1 +1 ) H point near (x ) would have to atify thi equation [8]. Auming that H x i invertible at (x ), 1 H ( x ) th ( x ) x + + = x t ; H ( x x ) t=t +1 t and = 0, 1, 2, (19) or x + 1 = x [ ( x )] [ H( x ) ( x Hx + tht )]; = 0, 1, 2, (20) Here t [0, 1]. We named (19) & (20) a Taylor-Newton homotopy (T-NH) formula. 4. The Propoed Algorithm (T-NH Algorithm) Baed on the above theory, we propoe a new application of Newton homotopy baed on Taylor erie (T-NH) for olving the continuity equation a follow: 1/ 2 ( bh) f ( h) = q n( b + 2h) H(h) = f(h) + (1 t)f(h 0 ) (21) where h i the depth of the water. It i oberved that the firt term of the above equation i equal to te continuity equation and h 0 i the initial value for the depth of the water h. The value of h can be obtained by olving (21) uing (22); h (, ) (, + 1 H h t + th t h t ) = h ; H ( x h ) = 0, 1, 2, (22)

Thuhe propoed method i ummarized a follow: 1. Chooe the initial value h 0. 2. Chooe t. 3. Follow the trajectory from t = 0 to t = 1 to find the deired olution uing (22). 4. Stop when the norm le than or equal the allowance tolerance ε (epilon: Small poitive real number) ; f (h) ε.or f (h) 0 5. Numerical Example Compute the depth for the water in open vhannel uing the following provided data: q = 5 m 3 /, n = 0.03, b = 20m, = 0.0002, Solution 0.471405(20h) f ( h) = 5 (20 + 2h) 15.7135(20h) 12.5708(20h) f ( h) =. (20 + 2h) (20 + 2h) We chooe the minimum value of h a the initial value of, h 0 = 0.001 f(0.001) = 4.999 f (0.001) = 0.1571 H t = f(h 0 ) & H v = f (h) By chooing t = 0.25he calculation to determine h are hown by the table in below, Table(1) No. t h h +1 N 1 0 0.001 3.1833 49.0168 2 0.25 3.1833 2.3094 28.1215 3 0.50 2.3094 1.0200 04.1334 4 0.75 1.0200 0.7023 4.4e-05 From the table above, it appear that when the parameter t = 0.75, = 4 then t +1 = 1 and the depth h +1 = 0.7023 with norm N = 4.4288e-005. Since thi norm i very cloe to zero then the obtained value of h; h = 0.7023 can be the deired value for the depth of the water. 6. Concluion The Taylor-Newton homotopy method i conceptually very imple in finding the depth of the water in open channel. We conclude that the norm become maller when the parameter t increae from t = 0 to t = 1. The value of norm howed that thi method i globally convergent. To gain more effective reult, we can chooe t maller and repeat the ame calculation. Another advantage i with certain number of iterationhe deired olution will be obtained. Baed thi advantage, in thi method, we didn t need to control the value of the error a in the claical method a Biection, fixed-point and Newton-Raphon method. In particularhe propoed method i epecially attractive compared with the exiting method. Naturally, the propoed method tae more computing time, thi i the common diadvantage of homotopy method. At the expene of computing peed we can achieve a much wider convergent region. 7. Reference [1] A. Kevorian, "A Decompoition Methodology and a Cla of Algorithm for the Solution of Nonlinear Equation" Mathematic of operation reearch, Vol. 6, No. 2, 1981, pp. 277-292. [2] A. Uhida and L. Chua, "Tracing Solution Curve of Nonlinear Equation with Sharp Turning Point" Circuit Theory and Application, Vol. 12, 1984, pp. 1-21. [3] Andrew J. Sommee and Charle W. Wampler, The numerical olution of ytem of polynomial. World Scientific Publihing Co. Pte. Ltd., Hacenac, NJ, 2005. [4] Andrew Sommee, Solving Polynomial Sytem by Homotopy Continuation, Algebraic Geometry and Application Seminar, IMA, 2006. [5] Anton Leyin and Jan Verchelde. Interfacing with the numerical homotopy algorithm in PHCpac, In Nobui Taayama and Andre Igleia, editor, Proceeding of ICMS 2006, pp. 354 360, 2006. [6] C. B. Garcia and W. B Zangwill, Pathway to Solution, Fixed Point, and Equilibrium Prentice Hall, 1981. [7] C. Dea, K. M. Irani, C. J. Ribben, L. T. Waton, and H. F. Waler,, "Preconditioned Iterative Method for Homotopy Curve Tracing" SIAM J. SCI. STAT. COMPUT., Vol. 13, No. 1, 1992, pp. 33-46. [8] C. Garcia, and W. Zangwill, Pathway to Solution, Fixed Point, and Equilibrium Handboo, Prentice Hall. U.S.A, 1981. [9] D. Wolf and S. Sander, "Multiparameter

Homotopy Method for Solving DC Operating Point of Nonlinear Circuit" IEEE Tranaction on Circuit and Sytem- I: Fundamental Theory and Application, Vol. 43, No. 10, 1996, pp. 824-838. [10] E Allgower and K. Georg, Numerical Path Following Internet Document, Colorado, U.S.A., 1994. [11] E. Allgower and K. Georg, Simplicial and Continuation Method for Approximating Fixed Poit and Solution to Sytem of Equation SIAM REVIEW, Vol. 22, No. 1, 1980, pp. 28-85. [12] E. Allgower, and K. Georg, 1994, "Numerical Path Following" handboo of numerical analyi, volume 5, pp. 3-207, North-Holland. [13] E. Allgower, and K. Georg, "Piecewie Linear Method for Nonlinear Equation and Optimization",http://www.math.colotate.ed u/~georg/, 1999, pp. 1-18. [14] J. M. Todd, Exploiting Structur in Piecewie-Linear Homotopy Algoritm for Solving Equation School of Operation Reearch and Indutrial Engineering, College of Engineering, Cornell Univerity, Ithaca, NY, U.S.A., 1978. [15] J. Verchelde and A. Haegeman, "Homotopie for olving polynomial ytem within a bounded domain" Theoretical Computer Science 133, 1994, pp. 165-185. [16] Kapil Ahuja, Probability-One Homotopy Map for Mixed Complementarity Problem, Thei ubmitted to the faculty of the Virginia Polytechnic, Intitute and State Univerityin partial fulfillment of the requirement for the degree of Mater of Science in Computer Science & Application, 2007. [17] K. M. Irani, M. P. Kamat, C. J. Ribben, H. F. Waler and L. T. Waton, Experiment with Conjugate Gradient Algorithm for Homotopy Curve Tracing SIAM J. Optimization, Vol. 1, N0. 2, 1991, pp. 222-251. [18] K. Yamamura and K. Horiuchi, Quadratic Convergence of the Homotopy Method Uing A Rectangular ubdiviion Tran. IEICE, Vol. E 72, No. 3, 1989, pp. 188-193. [19] K. Yamamura and K. Horiuchi, Solving Nonlinear Reitive Networ by a Homotopy Method Uing a Rectangular Subdiviion The Tranaction of the IEICE, Vol. E 72, No. 5, 1989, pp. 584-549. [20] K. Yamamura and K. Horiuchi, A Globally and Quadratically Convergent Algorithm for Solving Nonlinear Reitive Networ IEEE tranaction on computer-aided deign, Vol. 9, No. 5, 1990, pp. 487-499. [21] K. Yamamura and M. K. Kiyo, A Piecewie-Linear Homotopy Method with the Ue of the Newton Homotopy and a Polyhedral Subdiviion the tranaction of the IEICE, Vol. E 73, No. 1, 1990, pp 140-148. [22] K. Yamamura K., K. Horiuchi. and S. Oihi,, "A Decompoition Method Baed on Simplicial Approximation for the Numerical Analyi of Nonlinear Sytem" proceeding of ISCAS, 1985, pp. 635-638. [23] K. Yamamura, H. Kubo and K. Horiuchi, A Globally Convergent Algorithm Baed on Fixed-Point Homotopy for the Solution of Nonlinear Reitive Circuit, Tran. IEICE, J70-A, 10, 1987, pp. 1430-1438. [24] S. C. Chapra and R. P. Canale, Numerical Method for Engineer, Fifth Edition, 2008. [25] S. Lui, B. Keller and T. Kwo, "Homotopy Method for the Large, Spare, Real Nonymmetric Eigenvalue Problem" SIAM J. Matrix Anal. Appl. Vol.18, No. 2, 1997. pp 312-333. [26] Talib. Hahim and F. Soeianto, "A Newton Homotopy Method for Solving Sytem of Nonlinear Equation" to appear. 2000. [27] W. Kuroi, K. Yamamura, and S. Furui, An Efficient Variable Gain Homotopy Method Uing the SPICE-Oriented Approach, IEEE Tranaction on Circuit and Sytem-II: Expre Brief, VOL. 54, No. 7., 2007. [28] W. Lin, and G. Lutzer, "An application of the homotopy method to the ceneralized ymmetric eigenvalue problem" J. Autral. Math. Soc. Ser. B 30, 1988. pp. 230-249. [29] Xudong Zha and Qiuming Xiao, Homotopy Method for Bac calculation of Pavement Layer Moduli, International Sympoium on Non-Detructive Teting in Civil Engineering, 2003. Brief Biography Dr. Talib Hahim Haan received the B.Sc. degree in mathematic and phyic from the Baghdad Univerity, Baghdad in 1983, and the M.Sc degree in Mathematic from Gadjah Mada Univerity, Yogyaarta-Indoneia in 1997. He received the Ph.D. degree in mathematic and natural cience from Gajah Mada Univerity, Yogyaarta-Indoneia in 2001. He wored a a

lecturer in the department of Mathematic, Faculty of Education-Baghdad Univerity from 1984 until 1994. He alo wor a a lecturer and aitant profeor in 16 Univerity, Intitute, College and Academy ince 1996 until 2007. He i preently an Aitant Profeor with the Department of Science in Engineering at the Faculty of Engineering, International Ilamic Univerity Malayia, Kuala Lumpur. Hi reearch interet in Homotopy Approximation Method (HAM), Numerical computation, Modeling, Dynamical Sytem, Nonlinear ytem, and Fluid Dynamic. Dr. Md. Sazzad Hoien Chowdhury received hi Bachelor degree in Mathematic from Univerity Univerity of Chittagong, Chittagong, Bangladeh in 1995. He obtained hi Mater degree in Applied Mathematic with a poition from Univerity of Chittagong, Chittagong, Bangladeh in 1999. He obtained hi Ph.D. in Applied Mathematic from Univerity Kebangaan Malayia (UKM), Malayia in 2007. Currently, he i aitant profeor at the department of Science in Engineering, Faculty of Engineering, International Ilamic Univerity Malayia (IIUM), Kuala Lumpur, Malayia.