Khyber Pakhtunkhwa Agricultural University, Peshawar, Pakistan

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1 ttp:// Septic B-Spline Collocation metod for numerical solution of te Equal Widt Wave (EW) equation 1 Fazal-i-Haq, 2 Inayat Ali sa and 3 Sakeel Amad 1 Department of Matematics, Statistics and Computer Science, Kyber Paktunkwa Agricultural University, Pesawar, Pakistan 2 Department of Matematics, Islamia College Pesawar (CU), Kyber Paktunkwa, Pakistan 3 Department of Matematics, Islamia College Pesawar (CU), Kyber Paktunkwa, Pakistan Abstract: Numerical solutions of te Equal Widt Wave (EW) equation are obtained by Septic B-Spline collocation metod using Rubin and Graves linearization tecnique [16]. Te motion of a single solitary wave and interaction of two solitary waves are studied to validate te accuracy and efficiency of te proposed metod. Accuracy of te metod is discussed by computing te errors norms, and conservative quantities. Te analytic values of invariants and along wit oter results sow tat te present metod is a successful numerical tecnique for solving EW equation. Tis numerical sceme is based on forward difference sceme in time and teta-weigted sceme in space and is unconditionally stable. [ 1 Fazal-i-Haq, Inayat Ali sa and Sakeel Amad. Septic B-Spline Collocation metod for numerical solution of te Equal Widt Wave (EW) equation. Life Sci J 2013;10(1s): ] (ISSN: ). ttp:// 41 Key words: Collocation, Septic B-Spline, Linearization, Solitary waves. 1. INTRODUCTION Tis paper is concerned wit te numerical solution of te Equal Widt Wave (EW) equation based on collocation metod using septic B-Spline. Tis equation was first introduced by Morrison et. al. [12] as model equation to describe te non-linear dispersive waves. Many metods ave been proposed to solve te EW and Modified Equal Widt Wave (MEW) equations. Garacia-Arcilla [5] used te spectral metod for te solution of EW equation. Dag and Saka [1] solved te EW equation by using cubic B-Spline collocation metod. Saka [17] used a finite elemet metod for numerical solution of EW equation. Ramos [14] investigated solitary waves of EW and Regularized Long Wave (RLW) equations. Zaki [23] worked on solitary waves introduced by te boundary forced EW equation. Kalifa and Raslan [10] used te finite difference metods for EW equation. Gardner and G.A Gardner [6] solved te EW equation wit te Galerkin metod using Cubic B-Spline as a trial and test function. Zaki [21] obtained te numerical solution of te EW equation by using least-squares metod. Esen applied a Lumped Galerkin metod [4] on quadratic B- splines finite element for solving EW and MEW equations. Raslan [15] studied te Generalized Equal Widt Wave (GEW) equation by using collocation metod based on quadratic B-spine to obtain te numerical solution of a single solitary wave. Dag et. al. [18] obtained te numerical solutions of te EW equation by tree different metods. Kalifa et. al. [9] studied te numerical analysis for te EW Equation. Evans and Raslan [2] obtained te solution of solitary wave for te GEW equation. Wazwaz [20] used te tan and sine-cosine metods to obtained numerical approximation of MEW equation and its invariants. Hamdi et al. [19] obtained te exact solutions of te GEW equation. Zaki [22] worked on solitary wave interactions for te modified equal widt equation. Te Equal Widt Wave Equation as te form + μ = 0, (1) were U is te amplitude, is a positive parameter, te subscripts x and t denote te space and time partial differentiation. EW equation represents an alternative to te RLW equation [8]. We seek numerical solution subject to te following initial and boundary conditions 0 as x ±, U(x,0) = f(x), and (, )= 0, (, )= 0, (, )= 0, (, )= 0, (, )= 0, (, )= 0, were and f(x) is a localized disturbance inside te closed interval [a,b]. Te organization of tis paper is as follows. Septic B-splines are explained in section 2. Numerical stability is given in section 3. Numerical results are provided in section 4. Finally some conclusions are drawn. 2. Septic B-spline Collocation Metod Te interval [a,b] is partitioned into N finite elements of uniformly equal lengt by te knots, m = 0,1,2,3 N suc tat = < <.< = and =. Te septic B-spline function ( ), = 3, 2, 1., + 3 at tese knots is defined as 253

2 ttp:// ( )= 1 ( ), ( ) 8( ), ( ) 8( ) + 28( ), ( ) 8( ) + 28( ) 56( ), ( ) 8( ) + 28( ) 56( ), ( ) 8( ) + 28( ), [, ] ( ) 8( ), ( ), 0,. (2) Te set of B-splines {,,. } forms a basia for te functions over teinterval [a,b]. A global approximation (, ) to te exact solution u(x,t) takes te form = ( ) = 210 ( ), (4) (, )= ( ) ( ), (3) were ( ) are unknown time dependent quantities wic are determined from collocation boundary and initial conditions. Te nodal values,,, at te knots are obtained from Eqs (2) and (3) in te following form = ( )= , = ( ) = ( ) = 7 ( ), = 42 ( ), were dases represent differentiation wit respect to te space variable. Eq. (1) can be written as ( ) + = 0. (5) Te time derivative of Eq. (5) is discretized by a first order accurate forward difference formula and by using te -weigted (0 1), sceme to te space derivative at two time levels to get te equation ( ) + ( ) +(1 )( ) = 0, (6) were, is time step and te superscripts n and n+1 are successive time levels. In tis work we take =. Hence Eq. (6) is written as ( ) ( ) + ( ) ( ) = 0. (7) Te non-linear term in Eq. (7) is approximated by applying Taylor series given in [12] as ( ) ( ) + ( ) ( ). (8) At te nt time step we denote,, at te knots by te following expressions = , 254

3 ttp:// = 7 ( ), = 42 ( ), = ( ). (9) Using te knots, = 0,1,2, as te collocation points, te following recurrence relation at te point is obtained using Eqs. (6)-(9) = 2 ( ), (10) were = 7 84, = , = , = , = , = = , = (2 + ), (11) were m=0,1,2.n. Te Eq. (10) relates parameters at adjacent time levels. From te above general sceme as stated in Eq. (10) and using te values of m=0,1,2,.n, a septa diagonal matrix is produced containing N+1 equations in N+7 unknowns in te form of,i=-3,-2-1 N+3. In order to obtained a unique solution, we eliminate te parameters {,,,,, } from Eq. (10). Te values of tese parameters are obtained from Eq. (9) and te collocation boundary conditions as given below = , = = , = , = , = , (12) By eliminating te parameters from Eq. (10) given in Eq. (12) a linear system of (N+1) equations in (N+1) unknowns parameters, i=0,1,2,3,..n is obtained, wic is solved by a septa diagonal solver for,, = (1,2,3.). Finally te approximate solutions U(x,t) are obtained from Eq. (10). Te initial parameters, are determined by using te initial and boundary conditions wit te elp of te following expressions = 7 ( )= 0, = 42 ( )= 0, = 210 ( )= 0, = = ( ), = 0,1,2,, = 210 ( )= 0, = 42 ( )= 0, (.0)=,0 = ( )= 0. (13) Eq. (13), consists of ( + 1) ( + 1) system of equations wic can also be solved by a septa-diagonal solver. 3. Stability Analysis Te non-linear term in te sceme is linear zed by equating U as a constant k, using te Von- Neumann [11] stability metod.te linear zed from of te proposed sceme is given as = , (14) were = 2 7 8, = , = , = , = , = , 255

4 ttp:// = , = , = , = , = , = , = , = Substitution of = exp ( ) in te general sceme (14), were = 1 leads to (( exp( 3 )+ ( 2 )+ exp( )+ + exp( )+ exp(2 )+ exp(3 ) = exp( 3 )+ ( 2 )+ exp( )+ + exp( )+ exp(2 )+ exp(3 )). (15) Simplifying Eq. (15) we get =,were A= (4 168 )cos(3 )+ ( )cos(2 )+ ( )cos( )+ ( ), B= (14 )sin(3 )+ (784 )sin(2 )+ (3430 )sin( )+ ( ), so tat = = 1, ence = 1,sows tat te sceme for EW equation is unconditionally stable. 4. Te numerical tests and problems Te numerical metod proposed in te previous section is tested for single solitary wave and interaction of two solitary waves. Te accuracy of te metod is measured using te following error norms =, = were u and U denote te exact and approximate solutions respectively. Te analytical solution of EW equation given in te literature [18] is written as (, )= 3 sec ( ), (16) were = measures widt of te wave pulse, v=c is te wave velocity and is an arbitrary constant. Te initial condition is given by (, )= 3 sec ( ), (17) Te conservative properties of te EW equation related to mass, momentum and energy given in [13] are determined by assessing te following tree invariants, te exact solution wereas te following boundary conditions are used (0, )= (0, )= (0, )= 0, (30, )= (30, )= (30, )= 0, Te results are compared wit [3,7,21]., and te tree invariants, are recorded in Table 1 and Table 2. Fig. 1. Solitary wave profile for amplitude 0.3 corresponding to problem 1. Fig2. Error grap for c=0.1 corresponding to problem 1. =, = ( + ( ) ) =. (18) 4.1 Motion of single solitary wave Problem 1. We use te parameters = 1, = 10, = , =0.03,0.15 and time step = 0.05, run up time t=80 so tat te solitary waves ave amplitudes 0.3 and Te initial conditions are extracted from 256

5 ttp:// Fig3. Error grap for c = 0.3 corresponding to problem 1. Table 1. Invariants and norms for te single solitary wave T [3] [7] [21] Fig. 5 Solitary wave profile for amplitude 0.09 corresponding to problem 1. Table 2. Invariants and norms for te single solitary wave T [21] [3] Fig. 4. Solitary wave profile for amplitude 0.3 Te exact values of tese invariants are given in reference [18] as = = 1.2, = + = and = = (19) Wen c = 0.03, ten te analytical values Given in te reference [18] are = 0.36, = = (20) Fig. 6. Motion of single solitary wave at Selected times corresponding to problem 1. It is clear from Table 1 tat te error norms of te present metod are smaller tan tose of [3,7,21]. Te values of te tree invariants and two error norms obtained by te present metod are sown in Table 1,2 for different amplitudes c=0.1, Te analytical 257

6 ttp:// values of te tree invariants are given in Eqs. (19)- (20). 4.2 Interaction of two solitary waves Consider Eq. (1) along wit collocation boundary conditions (0, )= (0, )= (0, )= 0, (80, )= (80, )= (80, )= 0, and te initial condition (,0)= 3 sec ( ) + 3 sec ( ). (19) wic represents interaction of two solitary waves, one wit amplitude 3 and oter wit amplitude 3 placed initially at = and =. Problem 2. For interaction of two solitary waves, using te parameters = 1, = 10, = 25, = = 0.5, = 1.5, = 0.75 = 0.1, =0.1, = 0.1, and te runup time t=30. Te values of te tree invariants obtained by te present metod are recorded in Table 3. Tey are compared wit some earlier metods [15,18] in te literature. Te results of te present metod are in a good agreement. Te analytical values given in [18] are = 12( + )= 27, = 28.8( + )= 81, = 57.6( + )= Table 3. Values of tree invariants at different time. T [15] [18] Fig. 7. Interaction of two solitary waves at t = 0, t = 30 corresponding to problem 2. Fig.8 Interaction of two solitary waves at selected times corresponding to problem 2 Table 4. Values of te tree invariants at various times. T Fig. 9. Interaction of two solitary waves at t = 0, t = 15 and t = 30 corresponding to problem

7 ttp:// 5. Conclusion We studied motion of a single solitary wave, interaction of two solitary waves and interaction of two solitons using septic B-spline collocation metod.tree tests problems are given and te results are compared wit some earlier work from literature. In tis work, te performance and accuracy of te septic B-spline collocation metod was demonstrated by evaluating te two error norms, and tree invariants on te motion of signal solitary wave and interaction of two solitary waves. Te error norms are sufficiently small and te invariants are very close to te analytic values. Fig. 10. Motion of single solitary wave at Selected times corresponding to problem Interaction of solitons For tis simulation we use Gardner [24], wic as simulated te interaction of two waves (a positive and a negative) for te EW equation. Te collision was confirmed given in references [25-28]. Te initial condition is using as (,0)= +,were = 3 sec( ( )), = 1,2. Problem 3. We use te parameters = 0, = 80, = 1, = 23, = 38, = = 0.5, = 1.2, = 1.4, = 0.1, = 0.1, = 0.1 = 10. Te values of te tree invariants obtained by te present metod are recorded in Table 4 for two solitary waves. Te analytical values of te tree invariants given in te reference [29] as listed below = 12( + )= 2.4, = 28.8( + )= 97.9, = 57.6( + )= Fig11. Interaction of two opposite solitons corresponding to problem 3. References [1] I. Dag and B. Saka. A cubic b- spline collocation metod for te ew equation, Matematical Computational Applications 9 (2004), no.3, [2] D.J.E. vans and K.R. Raslan, Solitary waves for te generalized equal widt equation (GEW). Int. J. Comput. Mat 82 (4) (2005), [3] A. Dogan, Application of galerkin s metod to equal widt wave equation, Appl.Mat.Comput 160 (2005), [4] A. Esen, A numerical solution of te equal widt wave equation by a lumped galerkin metod, Appl. Mat.Comput 168 (2005), [5] B.Garcia-Arcilla, A spectrial metod for te equal widt equation, J comput. [6] L.R.T. Gardner and G.A. Gardner, Solitary waves of te equal widt wave equation, J Comput.Pys 101 (1992), [7] L.R.T. Gardner, G.A. Gardner, F.A. Ayoub, and N.K. Amein, Simulation of te ew undular Bore,Commun.Numer.Met.Eng 13 (1997), [8] I. Dag.B.Saka. and D. Irk. Application of cubic b- spline for numerical solution of te rlw eqution, APPL. Mat.comput 159 (2004), [9] AK. Kalifa, A.H Ali, and K.R. Raslan, Numrical study for te equal widt wave equation (e.w.e), J.Memoris of te Faculty of Science, Koci, Japan, Series A 20 (1999), [10] A.K. Kalifa, and K.R. Raslan, Finite difference metods for te equal widt wave equation, J. Egypt. Mat.Sos 7 (1999), [11] A.R. Mitcell and D.F. Griffit, Te finite difference eqution, in partial differential equation, Jon Wiley Jon Wiley and sons, New York (1980). [12] P. J. Morrison, Meiss JD, and Carey JR, Scattering of RLW solitary waves, pysica 11D (1981),

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