Quartic B-spline Differential Quadrature Method

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.11(2011) No.4,pp Quartic B-spline Differential Quadrature Method Alper Korkmaz 1, A. Murat Aksoy 2, İdris Dağ 2 1 Çankırı Karatekin University, Faculty of Science, Department of Mathematics,Çankırı, Turkey. 2 Eskişehir Osmangazi University, Department of Mathematics and Computer Science, Eskişehir, Turkey. (Received 26 March 2011, accepted 29 May 2011) Abstract: A new differential quadrature method based on quartic B-spline functions is introduced. The weighting coefficients are determined via a semi-explicit algorithm containing an algebraic equation system with four-band coefficient matrix. In order to validate the proposed method, the Burgers Equation is selected as test problem. The shock wave and the sinusoidal disturbance solutions of the Burgers equation are obtained numerically.the error between numerical and analytical solutions is measured by discrete maximum error norm and discrete root mean square error norm. Comparisons with some earlier works are also given. Numerical rate of convergence is studied in both cases. The stability of the method is studied by matrix stability analysis. Keywords: differential quadrature method; Burgers equation; B-spline; shock waves 1 Introduction Differential quadrature method (DQM) was originally developed by simple analogy with integral quadrature[1]. Since it was first introduced, many researchers used various basis functions to develop various types of DQMs. Legendre polynomials, Lagrange interpolation polynomials, spline functions, radial basis functions, etc can be counted some of them[1 5]. Shu and Richards obtained the weighting coefficients explicitly by using Lagrange polynomials as test function[6]. Shu and Xue [8] have developed the harmonic differential quadrature method derived by Striz et al.[7]. Eventhough Striz et al. obtained weighting coefficients implicitly by solving a set of algebraic equations, Shu and Xue obtained some explicit formulations to determine the weighting coefficients with the help of trigonometric functions. Recently, many engineering or physics problem have been solved successfully using various differential quadrature methods[9 19] due to its easy applicability, stability, high accuracy and adaptation with other numerical schemes. The form of one-dimensional Burgers equation is given as: with the initial and boundary conditions U t + UU x νu xx = 0, (1) U(x, 0) = f(x), a x b (2) U(a, t) = g 1 (t), U(b, t) = g 2 (t), t [0, T ]. (3) where ν denotes the kinematic viscosity constant. This equation is used as a model in fluid dynamics and in many branches of engineering as a simplified model for turbulence, boundary layer behavior, shock wave formation, traffic congestion, motion of gas, acoustics, gas dynamics, star formation and expansion, and mass transport. The Burgers equation is similar to the one dimensional Navier-Stokes equation without the stress term. The first time in a paper from Burger, is the model for the solution of Navier-Stokes equation and is applied to laminar and turbulence flows as well[20]. With appropriate initial and boundary conditions, Cole and Hopf studied this equation giving a theoretical solution based on Fourier series analysis[21, 22]. A great deal of effort has been expended in the last few years to compute efficiently the numerical solutions of the Burgers equation for small values of the kinematic viscosity. One of the major difficulties is generated by the steepening effect of the nonlinear advection term in Burgers equation. Recently, manyresearchershavestudied on Corresponding author. address: akorkmaz@karatekin.edu.tr Copyright c World Academic Press, World Academic Union IJNS /490

2 404 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp analyticalsolutions of theburgers equation[23-27]. The existence of the analytical solutions increases the importance of the Burgers equation for the researchers studying numerical schemes. So far various numerical algorithms such as B- spline finite element [23], cubic spline collocation [24], Galerkin [25], quartic B-spline collocation[26], b-spline Galerkin with graded mesh[27], quadratic and cubic B-spline collocation[28], polynomial differential quadrature[29], cubic b- spline differential quadrature method (CBCDQ)[30], automatic differentiation method[31] etc. have been developed to solve the Burgers equation. 2 Quartic B-spline differential quadrature method (QRTDQ) Differential quadrature is the approximation of derivatives with the help of weighted sums of functional values. It replaces a partial space derivative by a linear weighted sum of the function values at the discrete points. As a result, a partial differential equation reduces to a set of algebraic equations or a set of ordinary differential equation depending on timedependence or time-independence. The main approximation of the derivative of the function U at the grid x i is given by U (p) x (x i ) = N j=1 ij U(x j), i = 1, 2,..., N (4) where U x (p) denotes the pth order derivative of the function U with respect to the variable x and ij is the weighting coefficient of the pth order derivative. The key of this method is the determination of the weighting coefficients. Let Q i (x) be the quartic B-splines with knots at the points x i where the uniformly distributed N grid points are chosen as a = x 1 < x 2 <... < x N = b on the real axis. Then, the splines {Q 1, Q 0,..., Q N+1 } form a basis for functions defined over [a, b]. The quartic B-splines Q i (x) are defined by the relationships: Q i (x) = 1 h 4 (x x i 2 ) 4, [x i 2, x i 1 ] (x x i 2 ) 4 5(x x i 1 ) 4, [x i 1, x i ] (x x i 2 ) 4 5(x x i 1 ) (x x i ) 4, [x i, x i+1 ] (x i+3 x) 4 5(x i 2 x) 4, [x i+1, x i+2 ] (x i+3 x) 4, [x i+2, x i+3 ] 0, otherwise where h = x i x i 1 for all i. Using the quartic B-splines as test functions in the fundamental DQM equation (4) leads to the equation (p) Q m (x i ) x (p) = m+2 j=m 1 ij Q m(x j ), i = 1, 2,..., N, m = 1, 0,..., N + 1, (6) (5) An arbitrary choice of i leads to an algebraic equation system Q 1, 2 Q 1, 1 Q 1,0 Q 1,1 Q 0, 1 Q 0,0 Q 0,1 Q 0, Q N+1,N Q N+1,N+1 Q N+1,N+2 Q N+1,N+3 i, 2 i, 1. i,n+3 = Ψ (7) [ (p) ] T Q 1 (x i ) where Q i,j denotes Q i (x j ) and Ψ =, (p) Q 0 (x i ),..., (p) Q N+1 (x i ). The weighting coefficients x (p) x (p) x (p) ij related to the i th grid are determined by solving Eq. (7). The system (7) consists of N +3 equations and N +6 unknowns. In order to be able to solve Eq.(7), the equations (p+1) Q 1 (x i ) x (p+1) = 1 j= 2 i,j Q 1(x j ) IJNS for contribution: editor@nonlinearscience.org.uk

3 A. Korkmaz, A. Murat Aksoy, İ. Dağ: Quartic B-spline Differential Quadrature Method 405 (p+1) Q N (x i ) = N+2 x (p+1) (p+1) Q N+1 (x i ) x (p+1) j=n 1 = N+3 j=n are added to the system (7) and the equation system (7) becomes where M = [ w = Mw = Ψ Q 1, 2 Q 1, 1 Q 1,0 Q 1,1 Q 1, 2 Q 1, 1 Q 1,0 Q 1,1 Q 0, 1 Q 0,0 Q 0,1 Q 0, i, 2, w(p) i, 1 i,j Q N (x j) i,j Q N+1 (x j) Q N,N 1 Q N,N Q N,N+1 Q N,N+2 Q N,N 1 Q N,N Q N,N+1 Q N,N+2 Q N+1,N Q N+1,N+1 Q N+1,N+2 Q N+1,N+3 Q N+1,N Q N+1,N+1 Q N+1,N+2 Q N+1,N+3,..., w(p) i,n+3] T and [ (p+1) Q 1 (x i ) Ψ =, (p) Q 1 (x i ), (p) Q 0 (x i ),..., (p+1) Q N (x i ), (p) Q N+1 (x i ), (p+1) Q N+1 (x i ) x (p+1) x (p) x (p) x (p+1) x (p) x (p+1) Using functional values of the quartic B-splines at knots and eliminating i, 2, w(p) i,n+2, w(p) i,n+3 obtain an algebraic equation system having 4-banded coefficient matrix of the form where M = The nonzero entries of the load vector Ψ are given as, ] T from the system, we Mw = Ψ (8) and w = Ψ 1 = Q (p) 1 (x i) h 4 Q(p+1) 1 (x i ), Ψ i 2 = Q (p) i 2 (x i), Ψ i 1 = Q (p) i 1 (x i), Ψ i = Q (p) i (x i ), i, 1 i,0. i,i 1 i,i i,i+1 i,i+2. i,n 1 i,n i,n+1 IJNS homepage:

4 406 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp Ψ i+1 = Q (p) i+1 (x i), Ψ N = Q (p) N (x i) + h 4 Q(p+1) Ψ N+1 = Q (p) N+1 (x i) + h 4 Q(p+1) N+1 N (x i ) + hq(p+1) N (x i ) 4Q (p) N. The equation system (8) is solved using 4-banded Thomas algorithm derived from 5-banded Thomas algorithm. 3 Numerical discretization and stability of the scheme Substituting the differential quadrature derivative approximations given in Eq.(4) instead of the terms U x and U xx in the Burgers equation and applying the boundary conditions, we obtain the ordinary differential equation U(x i, t) t N 1 = U(x i, t) j=2 N 1 w (1) ij U(x j, t) + ν j=2 w (2) ij U(x j, t) + C i, i = 2, 3,..., N 1 (9) [ ] [ ] where C i = g 1 (t) w (1) i,1 g 1(t) + w (1) i,n g 2(t) + v w (2) i,1 g 1(t) + w (2) i,n g 2(t). Then, the ordinary differential equation (9) can be integrated in time using any method. We have preferred four stage Runge-Kutta algorithm due to its accuracy, stability and memory allocation properties. The stability of a time-dependent problem U = l(u) (10) t is determined with proper initial and boundary conditions, where l is a spatial differential operator. After discretization via DQM, Eq.(10) is reduced into a set of ordinary differential equations in time d {u} dt = [A] {u} + {s} (11) where {u} is an unknown vector of the functional values at the interior grid points, {s} is a known vector containing the non-homogenous part and the boundary conditions and A is the linearized form of the coefficient matrix the entries of which are α i w (1) ij + vw (2) ij where α i = U(x i, 0). It is well-known that the stability of a numerical scheme for numerical integration of Eq.(11) is dependent on the stability of the ordinary differential equation (11). If the ordinary differential equation (11) is unstable, then the stable numerical scheme for temporal discretization may not produce the converged solution. The stability of Eq.(11) is related to the eigenvalues of the matrix A, since its exact solution is directly determined by the eigenvalues of A. Let λ i be the eigenvalues of the coefficient matrix A. The stable solution of {u} as t requires; i) if all eigenvalues are real, 2.78 < Δtλ i < 0 ii) if eigenvalues have only complex components, 2 2 < Δtλ i < 2 2 iii) if eigenvalues are complex, Δtλ i should be in the region, Fig 1. The determination of the eigenvalues should be performed for each test problems owing to the coarse linearization done such as assuming α i = U(x i ), α i constant, during the determination of the entries the coefficient matrix A. Since the eigenvalues of a full matrix is not easy when the matrix is large. So, numerical determination of the eigenvalues is accomplished by the following: (1) The coefficient matrix A is reduced to a Hessenberg matrix H the eigenvalues of which are the same with A. (2) The eigenvalues of H are determined. In fact, the eigenvalues of H are called as Ritz values and they converge to the eigenvalues of A. The QR algorithm that reduces A to H is designed to accept the H is a Hessenberg matrix when H 3,3 < Numerical tests Two well known solutions of the Burgers equation are studied as test problems. The error between the proposed method and the analytical solutions is measured numerically with the discrete root mean square error norm L 2 and maximum error norm L given by: L 2 = N h j=0 j (U N ) j 2, L = U exact U N = max U exact j U exact j (U N ) j IJNS for contribution: editor@nonlinearscience.org.uk

5 A. Korkmaz, A. Murat Aksoy, İ. Dağ: Quartic B-spline Differential Quadrature Method 407 Figure 1: Stability region. The numerical rates of convergence (ROC) is calculated using the following formula ROC ln(e(n 2)/E(N 1 )) ln(n 1 /N 2 ) where E(N j ) is either the L -error or the L 2 -error when using N j grid points. Thus, we perform some further numerical run for various numbers of space steps. Furthermore, we make the computations about the ROC under the assumption that the methods are an algebraically convergent in space. In the otherwords, we suppose that E(N) N p for some p < 0 where E(N) denotes the L 2 or L -error when using N subintervals. 4.1 Shock wave Shock-like solution of the Burgers equation has analytic solution of the form: U(x, t) = x t, t 1, 0 x 1.2 (12) t 1 + exp( x2 t 0 4νt ) where t 0 = exp( 1 ). This solution simulates a deflating shock wave as time goes. The initial condition 8ν and the forced boundary conditions U(x, t) = 1 + x 1 exp( x2 t 0 4ν ) U(0, t) = U(1.2, t) = 0 are chosen. The designed routine is run up to the terminating time t = 3.6 with the kinematic viscosity v = The simulation is performed with a fixed time step Δt = for various space step lengths. The fade out of an initial shock is demonstrated in Fig 2. The error norms and a numerical rate of convergency analysis for various numbers of grids are documented in Table 1. The method is shows a superlinear convergence in space when Table 1 is examined. A comparison with some earlier studies is also given in Table 2. It should be told that the QRTDQ generates little bit more accurate numerical solutions than almost all previous methods. At the terminating time t = 3.6, the L 10 3 = as the L = IJNS homepage:

6 408 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp Figure 2: Fade out of initial shock. Since the problem domain was taken as [0, 1] in the previous studies, the terminating time was selected as t = 3.1 in order to reduce the boundary effect. Table 1. Error norms and rate of convergence for various numbers of grids at t = 3.6 N L ROC(L 2 ) L ROC(L ) Table 2. Comparison with some earlier studies for ν= t = 1.7 t = 2.4 t = 3.1 Method N Δt L L 10 3 L L 10 3 L L 10 3 QRTDQ(present) BS.FEM[23] C.S.C.[24] Galerkin[25] QBCM1[26] QBCM2[26] PDQ[29] CBCDQ[30] t = 1.7 t = 2.5 t = 3.25 GQBG(σ = 1)[27] GQBG(σ = 1.025)[27] GCBG(σ = 1)[27] GCBG(σ = 1.016)[27] QBCM[28] CBCM[28] QRKM[28] A stability analysis is performed for various numbers of grids. When N = 21 is chosen, only two of the eigenvalues are complex as all of the other are negative and real. The maximum real eigenvalue in absolute value is , as the complex eigenvalues are i. The routine is run again and again for various values of N. All the eigenvalues are non-positive for N = 31, 41, 61 and 81. The maximum eigenvalues in absolute value are determined IJNS for contribution: editor@nonlinearscience.org.uk

7 A. Korkmaz, A. Murat Aksoy, İ. Dağ: Quartic B-spline Differential Quadrature Method 409 as for N = 31, for N = 41, for N = 61 and for N = 81. The information about the eigenvalues gives ideas about selection of Δt to provide the stability of the method. 4.2 Sinuzoidal disturbance The function u 0 (x) = sin(2πx) is selected as the initial condition over the interval [0, 1] with the boundary conditions U(0, t) = U(1, t) = 0. This solution of the Burgers equation models sinusoidal disturbance of a shock wave, Fig 3. This solution was selected as a test problem in the study by wavelet Galerkin Method(WGM)[31]. In order to measure the error between the analytical and the numerical solutions obtained by the QRTDQ, the exact solution is calculated using U(x, t) = x ς t exp [ (x ς) 2 4vt exp [ (x ς) 2 4vt ] exp [ 1 ς 2v 0 u 0(η)dη ] dς ] exp [ 1 ς 2v 0 u 0(η)dη ], 0 x 1, t 0 (13) dς where u 0 denotes the initial condition. The almost exact solutions of the analytical solution Eq. (13) are computed Gauss- Hermit quadrature rule. For the sake of comparison with earlier study, the designed procedure is run with the parameters v = π/100, Δt = for various numbers of grid points. The simulation is seen in Fig 3. The maximum error norms are calculated at various times in each case. The measured errors for various numbers of grids, rate of convergence and comprasion with WGM and CBCDQ methods is seen in Table 3. The QRTDQ generates more accurate results than the both methods even less grids and larger time steps are used. For example, the QRTDQ reaches the maximum error with the parameters N = 32 and Δt = as the WGM reaches almost the same accuracy with N = 128 and Δt = at the time t = When N = 64 is selected in the QRTDQ the maximum error norm is computed as at t = Moreover, the rate of convergence analysis shows that the QRTDQ is linearly convergent when the number of grids are increased from 16 to 32 as the convergence is almost cubic when the number of grids are increased from 32 to 64. A stability analysis for this test problem is also performed. The largest real eigenvalue in absolute value is determined as as the complex eigenvalue whose imaginary part is maximum is i for N = 16. When N = 32, the largest real eigenvalue in absolute value is as the complex eigenvalue with largest imaginary part is i1. Unfortunately, when the number of grid is chosen as N = 64, the routine exceed the iteration number during the determination of the Hessenberg matrix H. So, no eigenvalue is determined. Figure 3: Sinusoidal Disturbance simulation. IJNS homepage:

8 410 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp Table 3. Maximum error at various times, rate of convergence and comparison t = 0.14 t = 0.26 t = 0.38 t = 0.50 Method N Δt L L L L ROC QRTDQ WGM[31] CBCDQ[30] Conclusion The quartic b-spline differential quadrature method is constructed to obtain numerical solution of the nonlinear Burgers Equation. The weighting coefficients of the derivative approximations are determined by solving some linear algebraic equation systems with 4-banded coefficient matrix. Once the weighting coefficients are determined, the Burgers equation is discretized in space by using the differential quadrature derivative approximations. Resultant ordinary differential equation system is integrated in time by using the Runge-Kutta method of order four. In order to show the validity of the method, the shock wave and the sinusoidal disturbance solutions of the Burgers equation are selected as test problems. For the sake of the comparison with some earlier works, the viscosity coefficient is selected as the same with them. Error norms prove that the quartic b-spline differential quadrature method generates acceptable results among the other methods. The rate of convergence analysis shows that the quartic b-spline differential quadrature method converges super linearly. The eigenvalues are determined for various numbers of grids. They are the indicators of the stability. Except some eigenvalues are complex for N = 21, all eigenvalues are determined as non-positive and real for various numbers of grids. The second test problem is the sinusoidal disturbance solution of the Burgers equation. The quartic b-spline differential quadrature method simulated this solution successfully, too. The comparison with wavelet Galerkin method shows that the quartic b-spline differential quadrature method generates more accurate results than it even less grids are used during the numerical computation. The quartic b-spline differential quadrature method converges the solution almost cubic rate for this test problem. Eventhough the eigenvalues are determined as complex, all the eigenvalues agree with the stability of the method. It should be mentioned that, the determination of the weighting coefficients of the first order derivative approximation is not always possible owing to the values of N j=1,j =i x x j x i x j when the Lagrange polynomials are used as basis. When the problem domain is large, the mentioned term may exceed the computation limits of the computers. So the usage of the uniform grids is not practical for all problems with differential quadrature method based on the Lagrange polynomials. This difficulty force the researchers to develop new grid distribution. On the other hand, the differential quadrature method based on quartic b-spline is suitable for uniform grid distribution. References [1] R. Bellman, B. G. Kashef, J. Casti. Differential Quadrature: A Tecnique for the Rapid Solution of Nonlinear Differential Equations. Journal of Computational Physics, 10(1972): [2] R. Bellman, Kashef Bayesteh, Lee E. S., Vasudevan R.. Differential quadrature and splines. Computers and mathematics with applications, (1976): [3] C. Shu, Y.L. Wu. Integrated radial basis functions-based differential quadrature method and its performance. Int. J. Numer. Meth. Fluids, 53(2007) : [4] Quan JR, Chang CT. New sightings in involving distributed system equations by the quadrature methods-i. Comput. Chem. Engrg., 13(1989): IJNS for contribution: editor@nonlinearscience.org.uk

9 A. Korkmaz, A. Murat Aksoy, İ. Dağ: Quartic B-spline Differential Quadrature Method 411 [5] Quan JR, Chang CT. New sightings in involving distributed system equations by the quadrature methods-ii. Comput Chem Engrg,13(1989): [6] C. Shu, Richards BE.. Application of generalized differential quadrature to solve two dimensional incompressible Navier Stokes equations. Inter.Jour. for Num. Meth. in Fluids, 15(1992): [7] A. G. Striz, X. Wang, C. W. Bert. Harmonic differential quadrature method and applications to analysis of structural components Acta Mechanica, 111: [8] C. Shu, H. Xue. Explicit Computaion of Weighting Coefficients in the Harmonic Differential Quadrature. Journal of Sound and Vibration, 204(3)(1997): [9] Korkmaz A., Dağ İ.. A Differential Quadrature Algorithm for Simulations of Nonlinear Schrödinger Equation. Computers & Mathematics with Application, 56(9)(2008): [10] B.Saka, İ.Dağ, Y.Dereli, A.Korkmaz. Three different methods for numerical solutions of the EW equation. Engineering Analysis with Boundary Elements, 32(2008): [11] Korkmaz A. and Dağ İ.. A differential quadrature algorithm for nonlinear Schrödinger equation. Nonlinear Dynamics, 56(1-2)(2009). [12] Korkmaz A. and Dağ İ.. Crank-Nicolson Differential Quadrature Algoritms for the Kawahara equation.chaos & Solitons & Fractals, 42(1)(2009): [13] Korkmaz A., Dağ İ.. Solitary Wave Simulations of Complex Modified Korteweg-de Vries Equation using Differential Quadrature Method. Computers Physics Communications, 180(9)(2009): [14] I. Dag, A. Korkmaz, B. Saka. Cosine Expansion-Based Differential Quadrature Algorithm for Numerical Solution of the RLW Equation. Numerical Methods for Partial Differential Equations, 26(3)(2010): [15] Korkmaz A.. Numerical Algorithms for solutions of Korteweg-de Vries Equation. Numer. Meth. Part. D. E., 26(6)(2010): [16] Korkmaz A., Dağ İ.. Numerical Simulations of Complex Modified KdV Equation using Polynomial Differential Quadrature Method. J. of Mathematics and Statistics, 10(11)(2011):1-13. [17] R.C. Mittal, R. Jiwari. Differential Quadrature Method for Two Dimensional Burgers Equations. Int. J. for Comput. Methods in Eng. Science and Mech., 10(2009): [18] R.C. Mittal, R. Jiwari. Numerical Study of Two-Dimensional Reaction-Diffusion Brusselator System by Differential Quadrature Method. Int. J. for Comput. Methods in Eng. Science and Mech., 12(1)(2011): [19] R.C. Mittal, R. Jiwari. A Higher Order Numerical Scheme for Some Nonlinear Differential Equations: Models in Biology. Int. J. for Comput. Methods in Eng. Science and Mech., 12(3) (2011): [20] J. M. Burger. A Mathematical Model Illustrating the Theory of Turbulence. Adv. in App. Mech. I,(1948): [21] Julian D. Cole. On a Quasi-linear Parabolic Equation in Aerodynamics. Quarterly of Applied Math., 9(1951): [22] E. Hopf. The Partial Differential Equation U t + UU x = μu xx,. Comm. Pure App. Math., 3(1950): [23] J. Chung, E. Kim, Y.Kim. Asymptotic agreement of moments and higher order contraction in the Burgers equation.journal of Differential Equations, 248(10)(2010): [24] A. H. Salas. Symbolic computation of solutions for a forced Burgers equation. Applied Mathematics and Computation, 216(1)(2010): [25] L. Zhang, J. Ouyang, X. Wang, X. Zhang. Variational multiscale element-free Galerkin method for 2D Burgers equation. Journal of Computational Physics, 229(19)(2010): [26] S. Lin, C. Wang, Z. Dai. New exact traveling and non-traveling wave solutions for (2 + 1)-dimensional Burgers equation. Applied Mathematics and Computation,216(10)(2010): [27] N H Ibragimov, M Torrisi, R Tracina. Self-adjointness and conservation laws of a generalized Burgers equation. J. Phys. A: Math. Theor., doi: / /44/14/145201( 2011). [28] İ. Dağ, D. Irk, A.Şahin. B-Spline Collocation Methods for Numerical Solutions of the Burgers Equation. Mathematical Problems in Engineering, 5 (2005) : [29] A. Korkmaz. Numerical Solutions of some Nonlinear Partial Differential Equations using Differential Quadrature Method. Thesis of Master Degree, Eskişehir Osmangazi University,(2006). [30] A.Korkmaz. Numerical Solutions of Some One Dimensional Partial Differential Equations using B-spline Differential Quadrature Methods.Doctoral Dissertation, Eskişehir Osmangazi University, (2010). [31] A. Asaithambi. Numerical solution of the Burgers equation by automatic differentiation. Applied Mathematics and Computation, 216(9) (2010): IJNS homepage: