Research Article Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation

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1 Mathematical Problems in Engineering Article ID 856 pages Research Article Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation Shazalina Mat Zin Ahmad Abd Maid Ahmad Izani Md Ismail and Muhammad Abbas School of Mathematical Sciences Universiti Sains Malaysia 8 Pulau Pinang Malaysia Institute of Engineering Mathematics Universiti Malaysia Perlis 6 Pauh Perlis Malaysia Department of Mathematics University of Sargodha Sargodha Pakistan Correspondence should be addressed to Shazalina Mat Zin; shazalina@unimapedumy Received June ; Accepted December ; Published December Academic Editor: Anuar Ishak Copyright Shazalina Mat Zin et al This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited The generalized nonlinear Klien-Gordon equation is important in quantum mechanics and related fields In this paper a semiimplicit approach based on hybrid cubic B-spline is presented for the approximate solution of the nonlinear Klien-Gordon equation The usual finite difference approach is used to discretize the time derivative while hybrid cubic B-spline is applied as an interpolating function in the space dimension The results of applications to several test problems indicate good agreement with known solutions Introduction Consider a generalized nonlinear Klien-Gordon (KG) equation in the form of [] u tt +αu xx +βu+g(u) =f(x t) () a x b t T subect to the initial conditions u (x ) =ω (x) a x b (a) u t (x ) =ω (x) a x b (b) and the Dirichlet boundary conditions u (a t) =φ (t) t T (c) u (b t) =φ (t) t T (d) where u(x t) denotes the wave displacement at position x and time t G(u) is a nonlinear function in u α and β are constants andf(x t) ω (x) ω (x) φ (x) andφ (x) are known functions In the last decade or so spline functions have been utilized to solve differential equations For example Caglar et al [] haveintroducedacubicb-splineinterpolationmethod to solve two-point boundary value problems The results obtained were compared to finite difference finite element and finite volume method Caglar et al [] concludedthat the B-spline interpolation is a better method to interpolate any smooth functions than others Hamid et al [] have developed an alternative cubic trigonometric B-spline interpolation method for the same problem They have found that the trigonometric B-spline gives better approximation compared to technique used by Caglar et al Goh et al [] have presented a comparison between cubic B-spline and extended cubic B-spline collocation method for solving heat equation It was concluded that the extended cubic B-spline gives better results By using the same method Abbas et al [5]havesolvedcoupledreactiondiffusionsystemTheyfound that the B-spline function approximates the system very well and the results are in good agreement with known solutions Great deals of research on solving KG equation have been carried out and the results can be found in [6 ] Dehghan and Shokri [] have approximated the numerical solution of the nonlinear KG equation using thin plate splines (TPS)

2 Mathematical Problems in Engineering radial basis functions The implementation of the method has been claimed to be simple as the finite difference method and the numerical results obtained were more accurate than others in literature Khuri and Sayfy [] have solved the generalized nonlinear KG equation using a finite element collocation approach based on third degree B-spline polynomials Six examples of nonlinear KG equation including Sine- Gordon equation have been analyzed The proposed method gives compatible results and better approximation compared to Dehghan and Shokri s method in [] Rashidinia et al [5] have presented a cubic B-spline collocation method for solving linear KG equation The results show the proposed scheme is effective and accurate In this paper a new approach by combining hybrid cubic B-spline function and central finite difference is proposed to solve the KG equation The finite difference approach is used for the time derivative and the hybrid cubic collocation method is applied to interpolate the solutions at space dimensiontheschemeobtainedisanalyzedbyvonneumann stability analysis To show the feasibility and accuracy of the method three problems are considered Numerical solutions absolute errors maximum error and order of convergence are calculated Temporal Discretization Consider a uniform mesh Ω with grid points (x t k ) to discretize the grid region Δ = [a b] [ T] with x = a+hand t k = kδtwhere = n and k = NThevaluesofh and Δt denote mesh space size and time step size respectively The Klien-Gordon equation is approximated at t k+ th time level as follows [6]: u k+ u k +uk (Δt) where g k =α(u xx) k + β(u)k gk+ + ( θ) g k +θgk+ +G(u k ) =fk () =α(u xx ) k+ + β(u) k+ θ and the subscripts k and k+ are successive time levels Central difference approach has been used to discretize the time derivative In order to produce Crank-Nicolson scheme θ is chosen to be 5 Hence the scheme becomes u k+ + 5 (Δt) g k+ =u k 5 (Δt) g k + (Δt) [f k G(uk )] uk At k = thereistermu that is outside of domain Therefore initial condition (b) is approximated by the following central difference approach Thus () u =u Δtω (x) (5) Hybrid Cubic B-Spline Collocation Method Through this section hybrid cubic B-spline (HCuBS) is used to solve nonlinear KG equation The approximate solution u(x t) to the analyticalsolutionu(x t) is considered as u (x t) = n = C (t) H (x) (6) where C (t) are time-dependent unknowns to be determined and H (x) is hybrid cubic B-spline basis function of order as H (x) =γb (x) +( γ)t (x) (7) where B (x) is cubic B-spline basis function given as B (x) (x x ) [x x + ] h +h (x x + ) + h (x x + ) = { (x x + ) [x + x + ] 6h h +h (x + x) + h (x + x) (x + x) [x + x + ] { {(x + x) [x + x + ] and T (x) is cubic trigonometric B-spline basis function [8 9]givenas T (x) p (x ) x [x x + ] p(x )(p(x )q(x + )+q(x + )p(x + )) = { +q (x + )p (x + ) x [x + x + ] κ q(x + )(p(x + )q(x + )+q(x + )p(x + )) +p (x )q (x + ) x [x + x + ] { { q (x + ) x [x + x + ] (9) with p(x )=sin((x x )/) q(x )=sin((x x)/)and κ=sin(h/) sin(h) sin(h/)thevalueofγ plays an important role in the hybrid cubic basis function If γ=thebasis function is equal to cubic trigonometric B-spline basis function and if γ= the basis function is equal to cubic B-spline basis function Hence this work ust considers the value of <γ< Due to local support properties of B-spline basis function there are only three nonzero basis functions; namely H (x ) H (x )andh (x ) are included over subinterval [x x + ] Thus the approximate solution and its derivatives with respect to x at (x t k ) are u k =A C k +A C k +A C k (8) The whole scheme is solved numerically by substituting hybrid cubic B-spline function discussed in next section for =nat each time level k (u x ) k =A C k A C k (u xx ) k =A C k (t k)+a 5 C k (t k)+a C k (t k) ()

3 Mathematical Problems in Engineering where with where A i =γσ i + ( γ) η i for i=5 () σ = 6 σ = 6 σ = h σ = h σ 5 = h η = κ κ κ η = κ κ η = η κ = (κ κ +κ ) 8κ κ κ η 5 = (κ +κ κ ) κ κ κ κ = sin ( h ) κ = sin (h) κ = sin ( h ) κ = sin (h) () () These approximations are substituted into () to produce the matrix system of order (n + ) with (n + ) unknown In order to generate a unique solution two additional equations are needed in the system Hence boundary conditions (c) and (d) are approximated as follows: u(at k+ )=A C k+ +A C k+ +A C k+ =φ (t k+ ) u(bt k+ )=A C k+ n +A C k+ n +A C k+ n =φ (t k+ ) Thus the resulting system can be written as where () MC k+ = NC k PC k + Q (5) A A A m m m m m m M = d ( m m m ) m m m ( A A A ) n n n n n n N = ( d ) n n n n n n ( ) A A A A A A P = ( d ) A A A A A A ( ) φ (t k+ ) (Δt) [f k G(uk )] Q = ( ) (Δt) [f k n G(uk n )] ( φ (t k+ ) ) (6) with m =A + 5(Δt) [α(a )+β(a )] m =A + 5(Δt) [α(a 5 )+β(a )] n =A 5(Δt) [α(a )+β(a )] and n =A 5(Δt) [α(a 5 )+β(a )] This tridiagonal matrix system can be solved using Thomas Algorithm repeatedly for k=n Initial State C The initial vector C is obtained from initial condition (a) and boundary values of the derivatives of the initial condition as follows [ 6]: (i) (u x ) =ω (x ) for = (ii) u =ω (x ) for =n (iii) (u x ) =ω (x ) for =n This operation yields (n + ) (n + ) matrix system: AC = B A A A A A A A A ( d ) A A A A A A ( A A )

4 Mathematical Problems in Engineering ( ( C C C C n C n C n ω (x ) ω (x ) = ( ) ) ω (x n ) ( ω (x n) ) ) (7) The solution of the tridiagonal system is obtained by using the Thomas Algorithm [] 5 Von Neumann Stability Analysis The growth of error in single Fourier mode is considered as C k =δk e iηh (8) where i = and η isthemodenumberitisknown that this method is applicable to linear scheme Hence () is linearized by assuming all nonlinear terms equal zero [5] The following equation is obtained after substituting () into the linear scheme: p C k+ +p C k+ +p C k+ where =p C k +p C k +p C k A C k A C k A C k p =A +θ(δt) (αa +βa ) p =A +θ(δt) (αa 5 +βa ) p =A ( θ)(δt) (αa +βa ) p =A ( θ)(δt) (αa 5 +βa ) (9) () By substituting (8) into (9) the following characteristic equation is generated: Aδ Bδ+C= () where A=p [ cos(ηh)] + p B= p [ cos(ηh)] + p and C=A [ cos(ηh)] + A Based on Routh-Hurwitz criterion the transformation δ = ( + V)/( V) isappliedtothe characteristic equation [5 ] Then the equation becomes (A+B+C) V +(A C) V + (A B+C) = () The necessary and sufficient conditions for δ are A+B+ C A C and A B+C Thusthefollowingterms have been proved: A cos (ηh) + A () [A β+a α] cos (ηh) + [A β+a 5 α] Hence this scheme is concluded to be unconditionally stable u(x t) t 6 x Figure : Space-time graph for analytical solution of Problem 6 Numerical Results and Discussions In this section two problems involving KG equation with initial conditions and Dirichlet boundary conditions are tested In order to measure the accuracy of the method absolute errors and maximum error are calculated using [] Absolute error = u i u i Maximum error =L = max i u i u i () where u i and u i are analytical solution and approximate solution respectively The numerical order of convergence p is obtained by using [] p= Log (L (n)) Log (L (n)) (5) Log (T/n) Log (T/n) where L (n) and L (n) are the L at number of partitions n and nrespectively Problem Consider the following nonlinear Klien-Gordon equation as [ ] u tt u xx +u = xcos t+x cos t x subect to the initial conditions and boundary conditions t 5 8 (6) u (x ) =x u t (x ) = (7) u ( t) = u( t) = cos t (8) The analytical solution is given by u(x t) = x cos t Figure showsthespace-timeplotforthisanalyticalsolution Initially this problem is tested by h=and Δt = 5 Numerical solutions of this problem at t = 5 are listed in Table The absolute errors of this problem at the same point are tabulated in Table The absolute errors at t = 5

5 Mathematical Problems in Engineering 5 Table : Numerical solution of Problem at t=5with h = and Δt = 5 x Analytical solution HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) Table : Absolute error of Problem at t =5withh = and Δt = 5 x Mat Zin et al [7] HCuBS(γ = ) HCuBS (γ = 5) HCuBS (γ = 9) Table : Maximum error of Problem compared to Khuri and Sayfy [] Dehghan and Shokri [] and Mat Zin et al [7] t 5 For (h = Δt = ) Dehghan and Shokri [] For (h = Δt = 5) Khuri and Sayfy [] Mat Zin et al [7] HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) Table : Numerical solution of Problem at t =5withh = and Δt = x Analytical solution HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) with γ = γ = 5 andγ = 9 arealsodepicted in Figure It can be seen numerically and graphically that HCuBS with γ = 9 is capable of giving better results This observation is established by comparing the maximum error obtained with the maximum error obtained by Khuri and Sayfy [] Dehghan and Shokri [] and Mat Zin et al [7] Table tabulates the comparison value of maximum errors at different time levels Then this problem is tested by h = and Δt = Tables and 5 illustrate the numerical solutions and absolute errors of this problem at t=5 respectively Graphically the absolute error of this problem at t = 5 with γ = γ = 5 andγ = 9 is plotted in Figure Thecomparisonof maximum error obtained from present method with Khuri and Sayfy [] Dehghan and Shokri [] and Mat Zin et al [7] methodsis shown in Table 6 Clearly from the tables and figures HCuBS withγ = 9 approximates this problem very well compared to Khuri and Sayfy [] Dehghan and Shokri [] and Mat Zin et al [7] The order of convergence of the present problem is tabulated in Table 7 An examination of this table indicates that the method has a nearly second order of convergence Problem The nonlinear Klien-Gordon equation is considered as [] u tt 5 u xx +u+ u = x t (9)

6 6 Mathematical Problems in Engineering Table 5: Absolute error of Problem at t =5withh = and Δt = x Mat Zin et al [7] HCuBS(γ = ) HCuBS (γ = 5) HCuBS (γ = 9) Table 6: Maximum error of Problem compared to Khuri and Sayfy [] Dehghan and Shokri [] and Mat Zin et al [7] t 5 For (h = Δt = ) Dehghan and Shokri [] For (h = Δt = ) Khuri and Sayfy [] Mat Zin et al [7] HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) γ = γ = 5 γ = 9 Figure : Absolute error of Problem with h = and Δt = γ = γ = 5 γ = 9 Figure : Absolute error of Problem with h = and Δt = Table 7: The maximum error and order of convergence of Problem at t =5 Δt = 5 Δt = n L p n L p with the initial and boundary conditions u (x ) =Btan (Kx) u t (x ) =cbksec (Kx) u ( t) =Btan (Kct) u( t) =Btan [K (+ct)] ()

7 Mathematical Problems in Engineering 7 Table 8: Numerical solution of Problem at t =withh = and Δt = x Analytical solution HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) Table 9: Absolute error of Problem at t =withh = and Δt = x HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) u(x t) t 6 x 8 Figure : Space-time graph for analytical solution of Problem where B = / K = /( 5 + c )andc = 5 The analytical solution is known to be u(x t) = B tan[k(x + ct)] Space-time plot for this analytical solution is shown in Figure This problem is solved numerically using h = and Δt = Tables 8 and 9 list approximate solutions and absolute errors at t = respectively The graphical plot of absolute errors at t = with three different values of γ is shown in Figure 5 Themaximumerrorofthisproblemis 6 8 γ = γ = 5 γ = 9 Figure 5: Absolute error of Problem with h = and Δt = compared with Dehghan and Shokri [] work which is listed in Table ThetablesindicatethatHCuBSwithγ = gives better results to this problem

8 8 Mathematical Problems in Engineering Table : Maximum error of Problem compared to Dehghan and Shokri []withh = and Δt = t Dehghan and Shokri [] HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) Table : Numerical solution of Problem at t =withh = (ln )/5 and Δt = x Analytical solution HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) (ln )/5 9 (ln )/ (ln )/ (ln )/ ln u(x t) t x 6 Figure 6: Space-time graph for analytical solution of Problem Problem Let the nonlinear Klien-Gordon equation be considered as [] u tt u xx + sin u= x ln t () This equation is also known as Sine-Gordon equation The initial and boundary conditions are given by u (x ) = u ( t) =tan t u t (x ) =sech x u (ln t) =tan ( t 5 ) () The analytical solution of this problem is given as u(x t) = tan (t sech x) Figure 6 presents the space-time graph of this analytical solution HCuBS collocation method is used to solve this problem numerically with parameters h = (ln )/5 and Δt = Table tabulates the approximate solutions of this problem at t=and Table tabulates the absolute errors of this problem also at t= The graphical comparison of the absolute errors with different value of γ at t=is depicted in Figure γ = γ = 5 γ = 9 Figure 7: Absolute error of Problem with h=(ln )/5 and Δt = The maximum errors of this problem are compared with Khuri and Sayfy [] and Mat Zin et al[7] work(table ) The table shows that Mat Zin et al produce better results which are the errors ust slightly better than HCuBS with γ= 7 Conclusion In this work the generalized Klien-Gordon equation was successfully solved using HCuBS collocation method Specifically the central difference approach has been applied to discretize the time derivative and hybrid cubic B-spline function hadbeenusedtointerpolatethesolutioninspacedimension Three problems have been tested using the proposed method and the solutions obtained were in good agreement with

9 Mathematical Problems in Engineering 9 Table : Absolute error of Problem at t =withh = (ln )/5 and Δt = x Mat Zin et al [7] HCuBS(γ = ) HCuBS (γ = 5) HCuBS (γ = 9) (ln )/ (ln )/ (ln )/ (ln )/ Table : Maximum error of Problem compared to Khuri and Sayfy []andmatzinetal[7]withh = (ln )/5 and Δt = t 5 Khuri and Sayfy [] Mat Zin et al [7] HCuBS (γ = ) HCuBS (γ = 5) HCuBS (γ = 9) the analytical solution Through this method an accurate solution at an intermediate point can be easily calculated Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper Acknowledgment The authors acknowledge financial support by Fundamental Research Grant Scheme (FRGS) with no /PMATHS/ 67 from School of Mathematical Sciences Universiti Sains Malaysia (USM) Penang Malaysia References [] S A Khuri and A Sayfy A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation Applied Mathematics and Computation vol6no pp 7 56 []HCaglarNCaglarandKElfaituri B-splineinterpolation compared with finite difference finite element and finite volume methods which applied to two-point boundary value problems Applied Mathematics and Computationvol75nopp [] NNHamidAAMaidandAIMIsmail Cubictrigonometric B-spline applied to linear two-point boundary value problems of order two World Academic of Science Engineering and Technologyno7pp78 8 []JGohAAMaidandAIIsmail Acomparisonofsome splines-based methods for the one-dimensional heat equation Proceedings of World Academy of Science Engineering and Technologyvol7no7pp [5] MAbbasAAMaidAIMdIsmailandARashid Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system PLoS ONE vol 9 no Article ID e865 [6] W M Cao and B Y Guo Fourier collocation method for solving nonlinear Klein-Gordon equation Computational Physicsvol8nopp [7]EYDeebaandSAKhuri Adecompositionmethodfor solving the nonlinear Klein-Gordon equation Computational Physicsvolnopp 8996 [8] G Ben-Yu L Xun and L Vázquez A Legendre spectral method for solving the nonlinear Klein-Gordon equation Computational and Applied Mathematicsvol5nopp [9] YSWongQChangandLGong Aninitial-boundaryvalue problem of a nonlinear Klein-Gordon equation Applied Mathematics and Computationvol8nopp [] A-M Wazwaz The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein- Gordon equation Applied Mathematics and Computationvol 67 no pp [] Sirendaorei Auxiliary equation method and new solutions of Klein-Gordon equations Chaos Solitons & Fractalsvolno pp [] U Yücel Homotopy analysis method for the sine-gordon equation with initial conditions Applied Mathematics and Computationvolnopp [] M S Chowdhury and I Hashim Application of homotopyperturbation method to Klein-GORdon and sine-gordon equations Chaos Solitons and Fractalsvol9nopp [] M Dehghan and A Shokri Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions Computational and Applied Mathematicsvolno pp 9 [5] J Rashidinia F Asfahani and S Jamalzadeh B-spline collocation approach for solution of Klien-Gordon equation International Mathematical Modelling and Computations volnopp5 [6] I DaǧDIrkandBSaka AnumericalsolutionoftheBurgers equation using cubic B-splines Applied Mathematics and Computationvol6nopp99 5 [7] S Mat Zin M Abbas A Abd Maid and A I Md Ismail A new trigonometric spline approach to numerical solution of generalized nonlinear klien-gordon equation PLoS ONEvol9no 5 Article ID e9577 [8] M Abbas A A Maid A I Ismail and A Rashid The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems Applied Mathematics and Computationvol9pp7 88

10 Mathematical Problems in Engineering [9] M Abbas A A Maid A I Ismail and A Rashid Numerical method using cubic trigonometric B-spline technique for nonclassical diffusion problems Abstract and Applied Analysis vol ArticleID8968pages [] D U V Rosenberg Methods for Solution of Partial Differential EquationsvolElsevierNewYorkNYUSA969 [] S S Siddiqi and S Arshed Quintic B-spline for the numerical solution of the good Boussinesq equation the Egyptian Mathematical Societyvolnopp9 [] M Saadian Numerical solutions of Korteweg de Vries and Korteweg de Vries-Burger s equations using computer programming International Nonlinear Science vol5 no pp 69 79

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