Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43
COMPARATIVE STUDY OF COLLOCATION METHOD-SPLINE, FINITE DIFFERENCE AND FINITE ELEMENT METHOD FOR SOLVING FLOW OF ELECTRICITY INA CABLE OF TRANSMISSION LINES
2 Dr A.K.Pathak1 Dr. H.D.Doctor Lecturer, Prof & Head Of H&S.S. Department Mathematics Department S.T.B.S.College Of Diploma Engg. Veer Narmad South Gujarat Uni. Surat, India Surat, India [email protected] [email protected]
Abstract The present work describes spline collocation ,finite difference and finite element approximation for solving flow of electricity in a cable of transmission lines problem. The mathematical problem gives rise to solve a linear partial differential equation which is of parabolic type. The method involves the solution of algebraic linear equation which can be written in the matrix form is main advantage. The solution are obtained by these two methods and compared with analytic solution to demonstrate the justification and simplicity of the spline approximation converge to finite difference approximation as well as analytic solution.
Keywords: Spline collocation, Finite difference, finite element method, Partial differential equation.
1. INTRODUCTION: The present work describes spline collocation, finite difference and finite element method for solving flow of electricity in a cable of transmission lines problem. The mathematical problem gives rise to solve a linear partial differential equation which is of parabolic type. The method involves the solution of algebraic linear equation which can be written in the matrix form is main advantage. The solution are obtained by these three methods and compared with analytic solution to demonstrate the justification and simplicity of the spline approximation converge to finite difference, finite element approximation as well as analytic solution.
2. FINITE DIFFERENCE METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATION:
2.1 Explicit Method: Consider parabolic partial differential equation having one space variable. 2 ut = c u xx ; 0
ui, j+1 = ui, j + r[ui-1, j - 2ui, j + ui+1, j ] . (2.1.2) where r= c 2 Dt/ h2 It is called the EXPLICIT formula it can be shown that this formula is valid only for 0< r £ 1/2.
2.2 Implicit Method: Equation (1.2.1.1) can be replaced by finite difference approximation which can written as
2(1+r)ui, j+1 - r(ui-1, j+1 + ui+1, j+1 ) = 2(1- r)ui, j + r(ui-1, j + ui+1, j ) (2.2.1) where r= c 2 Dt / h2 It is called the IMPLICIT formula it can be shown that this formula is valid for all finite value of r.
36 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 3 FINITE ELEMENT METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATION: Basic concept of finite element method is that to develop a finite element formulation of equation (2.1.1) First we have to know how to formulate the problem as a variational problem and then derive an algebraic system of equation associated with the variational problem and we obtain the equation . [ M 1 ]{u} +[K] {u}= {F} ( 3.1) and usingq family of approximation we can write equation (1.3.1) as set of linear algebraic equation in matrix form L1 L1 [K]{U}n+1 = {F} ( 3.2) L1 where [K]=[ M 1 ] +qDt[K] L1 1 {F}= ([M]- (1-q)Dt[K]){U}n + Dt(q{F}n+1 + (1-q ){F}n ) (3.3) L1 L Here D t is time step and [K]and{F} requires the knowledge of initial and boundary condition
and by solving equation (1.3.2) we get {U}n+1 .
4 SPLINE COLLOCATION METHOD: For solving linear and nonlinear differential equation with the help of the numerical methods required much computational work and time. Bickley [1968] Suggested the method of spline function containing truncated power polynomials to solve a linear boundary value problem Ahlberg et al [1967] used cardinal splines for solving differential equations. Doctor et al [1983,1984] have shown that the method of spline collocation is quite useful for the solution of physical phenomena which give rise to linear parabolic one dimensional partial differential equation. The method demonstrates the use of spline function. Spline functions are piecewise polynomial and their successive derivatives are continuous. They were used for data interpolation initially. In 1967, Blue [1969] suggested the use of spline function for the solution of B.V.P.
y” = f(x,y,y’) (4.1)
with boundary conditions
G1 [Y(0),Y’(0)]
G2 [Y(1),y’(1)] (4.2) The following recurrence relations were used. 2 S”( xi-1 )+4s”( xi )+s”( xi+1 ) = 6/ h (f( xi-1 )-2f( xi )+f( xi+1 ) (4.3)
4.1 Spline Formula To Solve Parabolic Liner Partial Differential Equation Having One Space Variable: Consider parabolic partial differential equation having one space variable. 2 ut = c u xx 0 the second derivative s”( xi ) at j th level explicit scheme in finite difference . We get 2 (ui, j+1 - ui, j )/ c Dt = S"i, j (4.1.2) 37 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 whereS"i, j denotes s”( xi )at j th level Substitute values of S"i, j from (4.1.2) in to equation (4.3) and get ui-1, j+1 + 4 ui, j+1 + ui+1, j+1 = (1+6r) ui-1, j +(4-12r) ui, j +(1+6r) (4.1.3) Where r = c 2 Dt / h 2 .This equation known as cubic Spline Explicit formula to solve equation (4.1.1) The finite difference replacement of (4..1.1) corresponding to implicit scheme is ui, j+1 - ui, j =½(S" + S" ) (4.1.4) c 2 Dt i, j i, j+1 whereS"i, j , S"i, j+1 denote second derivatives at x= xi at the time level j and j+1 respectively. We use the relationship (4.3) and rewrite it in following forms. 2 s"i-1, j +s"i, j +s"i+1, j = (6 / h )(ui-1, j - 2ui, j - ui+1, j ) (4.1.5) 2 s"i-1, j+1 +s"i, j+1 +s"i+1, j+1 = (6 / h )(ui-1, j+1 - 2ui, j+1 - ui+1, j+1 ) (4.1.6) i=1, 2, 3... n-1 putting the values of S"i, j+1 etc from (4.1.4) in to (4.1.5) We get which gives (1- 3r)ui-1, j+1 + (4 + 6r)ui, j+1 + (1- 3r)ui+1, j+1 = (1+ 3r)ui-1, j + (4 - 6r)ui, j + (1+ 3r)ui+1, j (4.1.7) Where r= c 2 Dt / h2 , i=1, 2, 3... (n-1) Above equation (4.1.7) is known as cubic spline implicit formula to solve equation (4.1.1) The convergence and stability of these methods totally depend upon the value of r. For explicit scheme we have to choose the value of r such that 0< r £1/2. Implicit scheme has advantage that there no limitation on the value of r. 5 FLOW OF ELECTRICCITY IN A CABLE OF TRANSMISSION LINES: This is the study of the derivation of partial differential equations from physical principals, we assume the flow of electricity in a cable of transmission line. We assume the cable to be imperfectly insulated so that there is both capacitance and current leakage to ground. Figure (3.1 a) shows such a cable with the electromotive source x=0 and load x=1. Figure (5.1a) Figure (5.1b) Here the current returns through the earth. Consider an element PQ of the cable with P at a distance x from the source .e(x, t) the potential, i (x, t) the current at P at any time t. Let R denote the resistance. L the inductance, G the conductance (leak ness) and C the capacitance per unit length of the cable. Let Dx be the length of the element PQ and i+ D i , e+ D e be the current and the electromotive force, respectively, at Q. Figure (3.1b) gives the electrical circuit for the 38 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 element PQ. Since the drop in voltage from P to Q is due to the resistance RDx and inductance L Dx , We have - De = (R Dx )i + (L Dx ) ¶i / ¶t (5.1) Again, since the drop in the current - Di is due to the conductance G Dx and capacitance C Dx ,we have - Di = (G Dx ) e + (C Dx ) ¶e/ ¶t (5.2) Dividing equation (5.1) and (5.2) by Dx and taking the limit as Dx ® 0, We get ¶e / ¶x + L ( ¶i / ¶t ) + R i = 0 (5.3) and ¶i / ¶x + c ¶e/ ¶t + Ge = 0 (5.4) Differentiating equation (5.3) with respect to x and equation (5.4) with respect to t, we get ¶ 2 e / ¶ x 2 + L ( ¶ 2 i / ¶ x ¶ t ) + R ¶ i / ¶ x = 0 and C ( ¶ 2 i / ¶ x ¶ t ) + C ( ¶ 2 e / ¶ x 2 ) + ¶ e / ¶ x = 0 If we eliminate the term ¶ 2i / ¶x¶t between these two equation and then substitute for ¶i / ¶t from equation (5.3) we obtained ¶ 2e / ¶x 2 = Lc(¶ 2e / ¶t 2 + (Rc + LG)¶e / ¶t + RGe (5.5) ¶ 2i / ¶x 2 = Lc¶ 2i / ¶t 2 + (Rc + LG)¶i / ¶t + RGi (5.6) Equation (5.5) and (5.6) are known as the telephone equations. Two special case of the telephone equation are ¶ 2e/ ¶x 2 = Rc¶e / ¶t (5.7) ¶ 2i / ¶x 2 = Rc¶i / ¶t (5.8) Equation (5.7) and (5.8) are obtained from equations (5.5) and (5.6) by taking G=L=0 i.e. leakage and inductance are negligible. For example, in telegraphic transmission through submarine cables. These equations are known as telegraphic equation. Mathematically these equations are similar to parabolic type linear partial differential having one space variable .Equation (5.7) and (5.8) can be rewrite as U t = (1/Rc) U xx (5.9) where U(x , t) stands for either i(x , t) or e( x , t) Let the initial current in the cable be f(x) ,then the initial condition is U(x,0) = f(x)= sin Õ x ; 0 U(0,t) = 0 and U(1,t) = 0 "t (5.11) 39 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 We shall determine the solution of the equation (5.9) satisfying equations (5.10) and ( 5.11 ) by spline explicit –implicit, finite difference explicit-implicit and finite element method and we can compare the result at t=0.0125 6 RESULT TABLE 6.1 Comparison Of Current In Cable U At T=0.0125 x UFE USE UEXT UFI USI UFEM 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.05 0.156239 0.156241 0.156242 0.156239 0.156241 0.156239 0.10 0.308635 0.308635 0.308636 0.308634 0.308636 0.308636 0.15 0.453429 0.453429 0.453431 0.453431 0.453430 0.453431 0.20 0.587059 0.587059 0.587060 0.587060 0.587060 0.587063 0.25 0.706237 0.706233 0.706235 0.706237 0.706233 0.706232 0.30 0.808021 0.808017 0.808019 0.808020 0.808018 0.808017 0.35 0.889909 0.889907 0.889908 0.889907 0.889907 0.889907 0.40 0.949886 0.949882 0.949884 0.949886 0.949882 0.949882 0.45 0.986474 0.986468 0.986471 0.986471 0.986468 0.986468 0.50 0.998770 0.998764 0.998767 0.998769 0.998765 0.998765 TABLE 6.2 Comparison Of Error Analysis At T=0.0125 x UEXT-UFE UEXT-USE UEXT-UFI UEXT-USI UEXT-UFEM 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.05 0.000003 0.000001 0.000003 0.000001 0.000003 0.10 0.000001 0.000001 0.000002 0.000000 0.000000 0.15 0.000002 0.000002 0.000000 0.000001 0.000000 0.20 0.000001 0.000001 0.000000 0.000000 0.000003 0.25 0.000002 0.000002 0.000002 0.000002 0.000003 0.30 0.000002 0.000002 0.000002 0.000001 0.000002 0.35 0.000001 0.000001 0.000001 0.000001 0.000001 0.40 0.000002 0.000002 0.000002 0.000002 0.000002 0.45 0.000003 0.000003 0.000003 0.000003 0.000003 0.50 0.000003 0.000003 0.000004 0.000002 0.000002 Figure:6.1 Error Analysis of spline explicit and finite difference explicit result 40 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 Figure 6.2 Error Analysis of spline implicit and finite difference implicit result Figure 6.3 Error Analysis of spline explicit- implicit and finite element method result Figure 6.4 Error Analysis of spline explicit- implicit result 41 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 Figure 6.5 Comparison Of Current In The Cable By Spline Explicit ,Implicit And Exact Solution 7 CONCLUTION: It is noticed from figure (6.1),(6.2)and (6.3) that , spline explicit and implicit solution are more accurate than finite difference explicit and implicit and finite element solution. From figure (6.5) it is also clear that analytic solution and spline solution represents a single curve which indicates the accuracy and reliability of spline collocation technique and spline solution are accurate up to five decimal places. Also both the methods required compact calculations and besides these both methods, namely explicit and implicit method, implicit methods have more accurate results than the explicit method. This justifies that spline approximation converge to finite difference, finite element approximation as well as analytic solution. 8 REFERENCES: Ahlberg J H, Nilson E N And Wals J N, The Theory of Spline And Their Application. Academic Press, N.Y,1967. Berdyshev V I, Subbopim J N,Numerical Methods of Approximation of functions (Russian). Sverdlovsk. Ural.Publ.House, 1979 ,118pp Bhatt R V: “An Extended Study Of The Spline Collocation Approach To Partial Differential Equations”, Thesis, South Gujarat University, Surat, 2005. Bickley W G -Piecewise Cubic Interpolation and Two point Boundary Value Problem. Comp. J, 11, 1968,206. Blue J L -„Spline Function Methods for linear Boundary Value problem.” Communications of the ACM. 12, No.6, 1969, 327. Bruch J C And G Zyvoloski: “Transient Two-Dimensional Heat Conduction Problems Solved by the Finite Element Method,” International Journal for Numerical Methods in Engineering, 8, 1974, PP 481-494. Clough R W: The Finite Element Method in Plane Stress Analysis J.Struct. Div., ASCE, Proc.2d conf. Electronic Computation, 1960,pp.345-378. Cook R D: Concepts and Applications of Finite Element Techniques for Fluid Flow, Butterworths, London, 1976. Courant R: Variational Methods for the Solution of Problems of Equilibrium and Vibration Bull. Am. Math. Soc., vol. 49, 1943,pp1-43. Doctor H D And Kalathia,N L - Spline Collocation in the Flow of Non-Newtonian Fluids. Presented at the Annual conf. of National Academic of science. 1984. Doctor H D And Kalthia N L - Spline Approximation of Boundary Value Problems. Presented at Annual Conf.of Indian math.Soc, 49, 1983. Doctor H D, And Kalathia N L -Spline Approximations of Boundary Value Problems Presented at Annual conf. of Indian Mathematical Society, 49, 1983. 42 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future” Dr.A.K.Pathak et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 3, 2014, pp.36-43 Doctor H D, Bulsari A B And Kalathia N L -Spline Collocation Approach to Boundary Value Problems. Int. J. Numer.Fluids, 4,6, 1984 ,511. Doctor H D, Bulsari A B And Kalthia N L - Spline Collocation Approach To Boundary Value Problems. Int. J. Number, Floids., 4, 6, 1984,511. Dr. Grewal B S:- Numerical Methods In Engineering And Science, Fifth ed, Khanna Publishers, 2005. Fyfe D J -The Use Of Cubic Splines In Solutions Of Two Point Boundary Value Problems, Com.P.J., 12, 1969,188-192. Pathak A K and Doctor H D, Comparative Study of Collocation Method- Spline, Finite Difference and Finite Element Method for Solving Partial Differential Equation” Presented at Conf. of Gujarat Science Congress, 2010. Reddy J N : An Introduction To The Finite Element Method. Mc Graw-Hill International Editions, New York, 1984. Sastry S S - Introductory Methods of Numerical Analysis. Prentice-Hall of India Private Limited. New Delhi,2005. AUTHOR’S BIOGRAPHY: Dr.Anant K Pathak: He is working as H.O.D. of Humanities and Social Science Department in S.T.B.S. College of Diploma Engineering. He has published 2 papers in international journal and 1 in national journal .He has presented 3 papers in national conference and 1 in International conference. He has 9 years of teaching experience. 43 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “ASRF promotes research nature, Research nature enriches the world’s future”