references - shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf ·...

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125 REFERENCES Abazari, R. and Borhanifar, A. 2010. Numerical study of the solution of the Burgers' and coupled Burgers' equations by a differential transformation. Meth. Comp. Math. Applic. 59: 2711-2722. Abbasbandy, S. and Darvish, M.T. 2005. A numerical solution of Burgers’ equation by time discretization of Adomians decomposition method. Appl. Math. Comput. 170: 95–102. Abeynaike, A., Sederman, A.J., Khan Y., Johns, M.L., Davidson, J.F. and Mackley, M.R. 2012. The experimental measurement and modelling of sedimentation and creaming for glycerol/ biodiesel droplet dispersions. Chem. Eng. Sci. 79: 125–137. Adomaitis, R.A. and Lin, Y.H. 2000. A collocation/quadrature-based Sturm-Liouville problem solver. Appl. Math. Comput. 110: 205-223. Ahlberg, J.H. and Ito, T. 1975. A collocation method for two point boundary value problems. Math. Comput. 29 (131): 761-776. Aksan, E.N. 2006. Quadratic B-spline finite element method for numerical solution of the Burger’s equation. Appl. Math. Comput. 174: 884-896. Akyildiz, F.T. and Vajravelu, K. 2007. Orthogonal cubic spline collocation method for the nonlinear parabolic equation arising in non-Newtonian fluid flow. Appl. Math. Comput. 189: 462–471. Ali, A.H.A., Gardner, G.A. and Gardner, L.R.T. 1992. A collocation solution for Burgers' equation using cubic B-spline finite elements. Comp. Meth. Appl. Mech. Eng. 100: 325–337. Al-Jabari, M., Van Heiningen, A.R.P. and Van de Ven, T.G.M. 1994. Modeling the flow and the deposition of fillers in packed bed of pulp fibers. J. Pulp Paper Sci. 20(9): 249-253. Archer, D. 1977. An 4 ( ) Oh cubic spline collocation method for quasilinear parabolic equations. SIAM J. Numer. Ana. 14(4): 620-637. Archer, D. and Diaz, J.C. 1978. A family of modified collocation methods for second order two point boundary value problems. SIAM J. Numer. Ana. 15(2): 242-254. Archer, D. and Diaz, J.C. 1982. A collocation-Galerkin method for a first order hyperbolic equation with Space and time dependent coefficient. Math. Comput. 38 (157): 37-53.

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Page 1: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

125

REFERENCES

Abazari, R. and Borhanifar, A. 2010. Numerical study of the solution of the Burgers' and coupled Burgers' equations by a differential transformation. Meth. Comp. Math. Applic. 59: 2711-2722.

Abbasbandy, S. and Darvish, M.T. 2005. A numerical solution of Burgers’ equation by time discretization of Adomians decomposition method. Appl. Math. Comput. 170: 95–102.

Abeynaike, A., Sederman, A.J., Khan Y., Johns, M.L., Davidson, J.F. and Mackley, M.R. 2012. The experimental measurement and modelling of sedimentation and creaming for glycerol/ biodiesel droplet dispersions. Chem. Eng. Sci. 79: 125–137.

Adomaitis, R.A. and Lin, Y.H. 2000. A collocation/quadrature-based Sturm-Liouville problem solver. Appl. Math. Comput. 110: 205-223.

Ahlberg, J.H. and Ito, T. 1975. A collocation method for two point boundary value problems. Math. Comput. 29 (131): 761-776.

Aksan, E.N. 2006. Quadratic B-spline finite element method for numerical solution of the Burger’s equation. Appl. Math. Comput. 174: 884-896.

Akyildiz, F.T. and Vajravelu, K. 2007. Orthogonal cubic spline collocation method for the nonlinear parabolic equation arising in non-Newtonian fluid flow. Appl. Math. Comput. 189: 462–471.

Ali, A.H.A., Gardner, G.A. and Gardner, L.R.T. 1992. A collocation solution for Burgers' equation using cubic B-spline finite elements. Comp. Meth. Appl. Mech. Eng. 100: 325–337.

Al-Jabari, M., Van Heiningen, A.R.P. and Van de Ven, T.G.M. 1994. Modeling the flow and the deposition of fillers in packed bed of pulp fibers. J. Pulp Paper Sci. 20(9): 249-253.

Archer, D. 1977. An 4( )O h cubic spline collocation method for quasilinear parabolic equations. SIAM J. Numer. Ana. 14(4): 620-637.

Archer, D. and Diaz, J.C. 1978. A family of modified collocation methods for second order two point boundary value problems. SIAM J. Numer. Ana. 15(2): 242-254.

Archer, D. and Diaz, J.C. 1982. A collocation-Galerkin method for a first order hyperbolic equation with Space and time dependent coefficient. Math. Comput. 38 (157): 37-53.

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126

Arnold, D. N. and Wendland, W.L. 1983. On the asymptotic convergence of collocation methods. Math. Comput. 41(164): 349-381.

Arora, S. 2007. Solutions of partial differential equations involving diffusion dispersion phenomenon using weighted residual methods. Punjab Technical University, Jalandhar, INDIA.

Arora, S., Dhaliwal, S.S. and Kukreja, V.K. 2005. Solution of two point boundary value problems using orthogonal collocation on finite elements. Appl. Math. Comput., 171(1): 358-370.

Arora, S., Dhaliwal, S.S. and Kukreja, V.K. 2006a. Simulation of washing of packed bed of porous particles by orthogonal collocation on finite elements. Comp. Chem. Eng. 30(6-7): 1054-1060

Arora, S., Dhaliwal, S.S. and Kukreja, V.K. 2006b. A computationally efficient technique for solving two point boundary value problems in porous media. Appl. Math. Comput. 183: 1170-1180.

Asaithambi, A. 2010. Numerical solution of the Burgers’ equation by automatic differentiation. Appl. Math. Comput. 216: 2700–-708.

Ascher, U. 1986. Collocation for two point boundary value problems revisited. SIAM J. Numer. Ana. 23(3): 596-609.

Ascher, U. and Bader, G. 1986. Stability of collocation at Gaussian points. SIAM J. Numer. Ana. 23(2): 412-422.

Babolian, E. and Saeidian, J. 2009. Analytic approximate solutions to Burgers', Fisher, Huxley equations and two combined forms of these equations. Commun. Nonlinear Sci. Numer. Simulat. 14: 1984-1992.

Bahadir, A.R. and Saglam, M. 2005. A mixed finite difference and boundary element approach to one-dimensional Burgers’ equation. Appl. Math. Comput. 160: 663-673.

Baroth, J. Bode, L., Bressolette, Ph. and Fogli, M. 2006. SFE method using Hermite polynomials: An approach for solving nonlinear mechanical problems with uncertain parameters. Comput. Methods Appl. Mech. Eng. 195: 6479-6501.

Bateman, H. 1915. Some recent researchers on the motion of fluids. Mon. Weather Rev. 43: 163-170.

Bednarik, M., Konicek, P. and Cervenka, M. 2002. Solution of the Burgers' equation in the time domain. Acta Polytechnica. 42(2): 71-75.

Page 3: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

127

Benammou, S. and Omrani, K. 2002. A finite element method for the Sivashinsky equation. J. Comput. Appl. Math. 142: 419–431.

Benton, E. and Platzman, G.W. 1972. A table of solutions of the one-dimensional Burgers equations. Quart. Appl. Math. 30: 195-212.

Besong, D.O. 2010. A new transformation of Burgers' equation for an exact solution in a bounded region necessary for certain boundary conditions. Appl. Math. Comput. 215: 3455-3460.

Bialecki, B. and Fairweather, G. 2001. Orthogonal spline collocation methods for partial differential equations. J. Comput. Appl. Math. 128: 55-82.

Biswas, A. and Swanson, D. 2007. Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in nR . J. Diff. Eqs. 240: 145-163.

Bo, L. and Jiang, Y. 2013. Large deviation for the nonlocal Kuramoto-Sivashinsky SPDE, Nonlinear Ana. 82: 100-114.

Bozhkov, Y. and Dimas, S. 2013. Group classification and conservation laws for a two-dimensional generalized Kuramoto-Sivashinsky equation. Nonlinear Ana. 84: 117-135.

Brenner, H. 1962. The diffusion model of longitudinal mixing in beds of finite length- numerical values. Chem. Eng. Sci. 17: 229-243.

Brill, S.H. 2001. Hermite collocation solution of partial differential equation via preconditioned Krylov methods. Numer. Meth. partial diff. eqs. 17(2): 120-136.

Burger, J.M. 1948. A mathematical model illustrating the theory of turbulence: Advances in applied mechanics I. Academic Press, New York.

Caglar, H., Caglar, N. and Elfaituri, K. 2006. B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems. Appl. Math. Comput. 175: 72-79.

Caldwell, J., Wanless, P. and Cook, A.E. 1987. Solution of Burgers’ equation for large Reynolds number using finite elements with moving nodes. Appl. Math. Model. 11: 211-214.

Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of heat in solids. Oxford University Press.

Cecchi, M.M., Nociforo, R. and Grego, P.P. 1996. Space-time finite elements numerical solution of Burgers problems. LE MATEMATICHE. LI Fasc. I: 43–57.

Page 4: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

128

Cermakova, J., Scargiali, F., Siyakatshana, N., Kudrna, V., Brucato, A. and Machon, V. 2006. Axial dispersion model for solid flow in liquid suspension in system of two mixers in total recycle. Chem. Eng. J. 117: 101-107.

Chawala, M.M. and Subramanian, R. 1988. A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems. J. Comput. Appl. Math. 23: l-10.

Chawla, T.C., Leaf, G., Chen W.L. and Grolmes, M.A. 1975. The application of the collocation method using Hermite cubic splines to nonlinear transient one-dimensional heat conduction problems. Transactions of the ASME : 562-569.

Chen, S., Wu, X., Wang, Y. and Kong, W. 2010. The Laplace transform method for Burgers’ equation. Int. J. Numer. Meth. Fluids. 63(9): 1060-1076.

Chen, X. and Xiang, J. 2011. Solving diffusion equation using wavelet method. Appl. Math. Comput. 217: 6426-6432.

Cheskidov, A. and Foias, C. 2001. On the non-homogeneous stationary Kuramoto–Sivashinsky equation. Physica D. 154: 1-14.

Cocero, M.J. and Garcia, J. 2001. Mathematical model of supercritical extraction applied to oil seed extraction by CO2 + saturated alcohol – I. Desorption model. J. Supercritical Fluids 20: 229-243.

Cole, J.D. 1951. On a quasi linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9: 225-236.

Crotogino, R. H., Poirier, N. A. and Trinh, D. T. 1987. The principles of pulp washing. Tappi J. 70(6): 95-103.

Cruz-Diaz, M., Rivera, F.F., Rivero, E.P. and Gonzalez I. 2012. The FM01-LC reactor modeling using axial dispersion model with a reaction term coupled with a continuous stirred tank. Electrochimica Acta 63: 47-54.

Dag, I., Canivar, A. and Sahin, A. 2011. Taylor–Galerkin and Taylor-collocation methods for the numerical solutions of Burgers’ equation using B-splines. Commun. Nonlinear Sci. Numer. Simulat. 16: 2696-2708.

Dag, I., Irk, D. and Sahin, A. 2005. B-spline collocation methods for numerical solutions of the Burgers’ equation. Math. Prob. Eng. 5: 521-538.

Dag, I., Irk, D. and Saka, B. 2005. A numerical solution of the Burgers' equation using cubic B-splines. Appl. Math. Comput. 163: 199-211.

Page 5: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

129

Danumjaya, P. and Pani, A.K. 2004. Numerical methods for the extended Fisher-Kolmogorov equation. Int. J. Numer. Ana. Model. 3(2): 186-210

Danumjaya, P. and Pani, A.K. 2005. Orthogonal cubic spline collocation method for the extended Fisher–Kolmogorov equation. J. Comput. Appl. Math. 174: 101-117.

Darvishi, M.T. and Javidi, M. 2006. A numerical solution of Burgers’ equation by pseudospectral method and Darvishi’s preconditioning. Appl. Math. Comput. 173: 421-429.

Davis, P. and Rabinowitz, P. 1984. Method of numerical integration. Academic Press, New York, 2nd Edition.

De Boor, C. 2001. A practical guide to splines. Revised Edition, Springer Verlag.

De Boor, C. 1972. On calculating with B-splines. J. Approx. Theory 6: 50-62.

De Boor, C. and Swartz, B. 1973. Collocation at Gaussian Points. SIAM J. Numer. Ana. 10(4): 582-606.

Dhawan, S., Kapoor, S., Kumara, S. and Rawat, S. 2012. Contemporary review of techniques for the solution of nonlinear Burgers' equation. J. Comput. Sci. 3: 405-419.

Díaz, M.C., Rivera, F.F., Rivero, E.P, Gonzalez, I. 2012. The FM01-LC reactor modeling using axial dispersion model with a reaction term coupled with a continuous stirred tank. Electrochimica Acta 63: 47-54.

Dogan, A. 2004. A Galerkin finite element approach to Burgers’ equation. Appl. Math. Comput. 157: 331-346.

Douglas, J. Jr. and Dupont, T. 1974. Collocation methods for parabolic equations in single space variables. Lecture Notes in Mathematics, 385, Springer-Verlag, New York.

Douglas, J. and Dupont, T. 1973. A finite element collocation method for quasilinear parabolic equations. Math. Comput. 121: 17-28.

Duan, J. and Ervin, V.J. 2001. On the stochastic Kuramoto-Sivashinsky equation. Nonlinear Ana. 44: 205-216.

Dubljevic, S. 2010. Boundary model predictive control of Kuramoto–Sivashinsky equation with input and state constraints. Comp. Chem. Eng. 34: 1655-1661.

Dyksen, W.R. and Lynch, R.E. 2000. A new decoupling technique for the Hermite cubic collocation equations arising from boundary value problems. Math. Comput. Simul. 54: 359-372.

Page 6: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

130

Edoh, K.D., Russell, R.D. and Sun, W. 2000. Computation of invariant tori by orthogonal collocation. Appl. Numer. Math. 32: 273-289.

Engelberg, S. 1998. The stability of the shock profiles of the Burgers’ equation. Appl. Math. letter 11(5): 97-101.

Eriksson, G., Rasmuson, A. and Theliander, H. 1996. Displacement washing of lime mud: Tailing effects. Sep. Tech. 6: 201-210.

Fang, Q., Tsuchiya, T. and Yamamoto, T. 2002. Finite difference, finite element and finite volume methods applied to two-point boundary value problems. J. Comput. App. Math. 139: 9-19.

Farooq, S. and Karimi, I.A. 2003. Dispersed plug flow model for steady state laminar flow in a tube with a first order sink at the wall. Chem. Eng. Sci. 58: 71-80.

Feiz, M. 1997. A 1-D multigroup diffusion equation nodal model using the orthogonal collocation method. Annals of Nuclear Energy 24: 187-196.

Felgueroso, L.C. and Peraire, J. 2008. A time-adaptive finite volume method for the Cahn-Hilliard and Kuramoto-Sivashinsky equations. J. Comput. Physics. 227: 9985-10017.

Finlayson, B.A. 1980. Non linear analysis in chemical engineering. McGraw-Hill, New York.

Grahs, L.E. 1974. Washing of cellulose fibres: Analysis of displacement washing operation. Ph.D. Thesis, Chalmers University of Technology, Göteborg, Sweden.

Gupta, B. and Kukreja, V.K. 2012. Numerical approach for solving diffusion problems using cubic B-spline collocation method. Appl. Math. Comput. 219(4): 2087-2099.

Ha, S.N. 2001. A nonlinear shooting method for two point boundary value problems. Comput. Math. Appl. 42: 1411-1420.

Hamid, B.A. 1999. An exact solution to the Kuramoto-Sivashinsky equation. Physics Letters A. 263: 338-340.

Haq, S., Bibi, N., Tirmizi, S.I.A. and Usman, M. 2010. Meshless method of lines for the numerical solution of generalized Kuramoto-Sivashinsky equation. Appl. Math. Comput. 217: 2404-2413.

Haq, S., Islam, S. and Marjan Uddin. 2009. A mesh-free method for the numerical solution of the KdV-Burgers equation. Appl. Math. Model. 33: 3442-3449.

Page 7: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

131

Hassanien, I.A., Salama, A.A. and Hosham, H.A. 2005. Fourth-order finite difference method for solving Burgers equation. Appl. Math. Comput. 170: 781-800.

Higham, D.J. 1992. Monotonic piecewise cubic interpolation with applications to ODE plotting. J. Comput. Appl. Math. 39: 287-294.

Hon, Y.C. and Mao, X. Z. 1998. An efficient numerical scheme for Burgers’ equation. Appl. Math. Comput. 95: 37-50.

Hopf, E. 1950. The partial differential equation t x xxU UU Uµ+ = . Commun. Pure Appl. Math. 3: 201-230.

Huang, P. and Abduwali, A. 2010. The modified local Crank-Nicolson method for one and two-dimensional Burgers' equations. Comp. Math. Applic. 59: 2452-2463.

Hyman J.M., Nicolaenko. B. and Zaleski, S. 1986. Order and complexity in the Kuramoto–Sivashinsky model of weakly turbulent interfaces. Physica D 23: 265-92.

Inc, M. 2005.On numerical solutions of one-dimensional nonlinear Burgers’ equation and convergence of the decomposition method. Appl. Math. Comput. 170: 76-85.

Inc, M. 2008. On numerical solution of Burgers’ equation by homotopy analysis method. Physics Letters A 372: 356-360.

Javidi, M. 2011. A modified Chebyshev pseudospectral DD algorithm for the GBH equation. Comp. Math. Applic. 62: 3366-3377.

Jawad, A.J.M., Petkovic, M.D. and Biswas, A. 2011. Applications of He’s principles to partial differential equations. Appl. Math. Comput. 217: 7039-7047.

Jiwari, R. 2012. A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation, Comp. Physics Commun. 183: 2413-2423.

Jiwari, R., Mittal, R.C. and Sharma, K.K. 2013. A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation. Appl. Math. Comput. 219: 6680-6691.

Jumarhon, B., Amini, S. and Chen, K. 2000. The Hermite collocation method using radial basis functions. Eng. Ana. Boundary Elements 24: 607-611.

Juncu, G., Bildea, S. and Floarea, O. 1994. Steady-state multiplicity analysis of the heterogeneous axial dispersion fixed-bed reactor. Chem. Eng. Sci. 49:123-130.

Kadalbajoo, M.K. and Awasthi, A. 2006. A numerical method based on Crank-Nicolson scheme for Burgers’ equation. Appl. Math. Comput. 182: 1430-1442.

Page 8: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

132

Kadri, T. and Omrani, K. 2011. A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation. Comp. Math. Applic. 61: 451-459.

Kaya, D. and Sayed, S.M. 2004. A numerical simulation and explicit solutions of the generalized Burgers–Fisher equation. Appl. Math. Comput. 152: 403-413.

Kazem, S., Rad, J.A. and Parand, K. 2012. Radial basis functions methods for solving Fokker–Planck equation. Eng. Ana. Boundary Elements 36: 181-189.

Khan, I.A. and Loughlin, K.F. 2003. Kinectics of sorption in deactivated zeolite crystal adsorbents. Comp. Chem. Eng. 27: 689-696.

Khater, A.H. and Temsah, R.S. 2008. Numerical solutions of the generalized Kuramoto-Sivashinsky equation by Chebyshev spectral collocation methods. Comput. Math. Applic. 56: 1465-1472.

Khater, A.H., Temsah, R.S. and Hassan, M.M. 2008. A Chebyshev spectral collocation method for solving Burgers’ type equations. J. Comput. Appl. Math. 222: 333-350.

Khiari, N. and Omrani, K. 2011. Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions. Comp. Math. Applic. 62: 4151-4160.

Korkmaz, A. and Dag, I. 2011. Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation. J. Franklin Inst. 348: 2863-2875.

Krajnc, M. 2009. Geometric Hermite interpolation by cubic G1 splines. Nonlinear Ana. 70: 2614-2626.

Kukreja, V.K. 1996. Modelling of washing of brown stock on rotary vaccum washer. Ph.D. Thesis, University of Roorkee, Roorkee, India.

Kukreja, V.K. and Ray, A.K. 2009. Mathematical modeling of a rotary vacuum washer used for pulp washing: A case study of a lab scale washer. Cellulose Chem. Technol. 43: 25-30.

Kumar, M.A. 2003. Second order finite difference method and its convergence for a class of singular two-point boundary value problems. Appl. Math. Comput. 146: 873-878.

Kuramoto, Y. and Tsuzuki, T. 1976. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55: 356-369.

Kutluay, S., Bahadir, A.R. and Ozdes, A. 1999. Numerical solution of one dimensional Burgers equation: Explicit and exact-explicit finite difference finite difference methods. J. Comput. Appl. Math. 103: 251-261.

Page 9: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

133

Kutulay, S., Esen, A. and Dag, I. 2004. Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J. Comput. Appl. Math. 167: 21-33.

Lai, Y.L., Hadjidimos, A. and Houstis, E.N. 1996. A generalized Schwarz splitting method based on Hermite collocation for elliptic boundary value problems. Appl. Numer. Math. 21: 265-290.

Lakestani, M. and Dehghan, M. 2012. Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions. Appl. Math. Model. 36: 605-617.

Lamata, P., Niederer, S., Nordsletten, D., Barber, D.C., Roy, I., Hose, D.R. and Smith, N. 2011. An accurate, fast and robust method to generate patient-specific cubic Hermite meshes. Medical Image Ana. 15: 801-813.

Lang, A.W. and Sloan, D.M. 2002. Hermite collocation solution of near-singular problems using numerical coordinate transformations based on adaptivity. J. Comput. Appl. Math. 140: 499-520.

Lang, F.G. and Xu, X.P. 2012. Quintic B-spline collocation method for second order mixed boundary value problem. Comput. Physics Comm. 183: 913-921.

Leao, C.P. and Rodrigues, A.E. 2004. Transient and steady-state models for simulatedmoving bed processes. Numer. Sol. Comp. Chem. Eng. 28: 1725-1741.

Lefevre, L., Dochain, D., Azevedo, S.F. and Magnus, A. 2000. Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods. Comp. Chem. Eng. 24: 2571-2588.

Liao, H.T. and Shiau, C.Y. 2000. Analytical solution to an axial dispersion model forthe fixed bed adsorber. AIChE J. 46: 1168-1176.

Liao, W. 2008. An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation. Appl. Math. Comput. 206: 755-764.

Liu, C.S. 2010. A two-stage Lie-group shooting method to identify time-dependent thermal diffusivity, Int. J. Heat Mass Transfer 53: 4876-4884.

Liu, F. and Bhatia, S.K. 1999. Computationally efficient solution techniques for adsorption problems involving steep gradients bidisperse particles. Comp. Chem. Eng. 23: 933-943.

Page 10: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

134

Liu, F. and Bhatia, S.K. 2001. Application of Petrov–Galerkin methods to transient boundary value problems in chemical engineering: Adsorption with steep gradients in bidisperse solids. Chem. Eng. Sci. 56: 3727-3735.

Liu, L.B., Liu, H.W. and Chen, Y. 2011. Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions. Appl. Math. Comput. 217: 6872-6882.

Liu, Y. and Jacobsen, W.E. 2004. On the use of reduced order models in bifurcation analysis of distributed parameter systems. Comp. Chem. Eng. 28: 161-169.

Long, W.S., Bhatia, S. and Kamaruddin, A. 2003. Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester. J. Membrane Sci. 219: 69-88.

Luo, X. and Du, Q. 2013. An unconditionally stable fourth-order method for telegraph equation based on Hermite interpolation. Appl. Math. Comput. 219: 8237-8246.

Manickam, A.V., Moudgalya, K.M. and Pani, A.K. 1998. Second order splitting combined with orthogonal cubic spline collocation method for the Kuramoto- Sivashinsky equation. Comp. Math. Applic. 35: 5-25.

Mantri, P.S., Nataraj, N. and Pani, A.K. 2008. A qualocation method for Burgers’ equation. J. Comput. Appl. Math. 213: 1-13.

Miller, E.L. 1966. Predictor-corrector studies of Burgers’ model of turbulent flow. M.S. Thesis, University of Delaware, Newark, DE.

Miller, J.J.H., O’Riordan, E. and Shishkin, G.I. 1996. Fitted numerical methods for singular perturbation problems: Error estimate in the maximum norm for linear problems in one and two dimensions. World Scientific.

Mittal, R.C. and Arora, G. 2010. Quintic B-spline collocation method for numerical solution of the Kuramoto-Sivashinsky equation. Commun. Nonlinear Sci. Numer. Simulat. 15: 2798-2808.

Mittal, R.C. and Arora, G. 2011. Numerical solution of the coupled viscous Burgers’ equation. Commun. Nonlinear Sci. Numer. Simulat. 16: 1304-1313.

Mittal, R.C. and Jain, R.K. 2012. Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Appl. Math. Comput. 218: 7839-7855.

Page 11: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

135

Mohebbi, A. and Dehghan, M. 2010. High-order compact solution of the one-dimensional heat and advection-diffusion equations. Appl. Math. Model. 34: 3071-3084

Momani, S. 2005. A numerical scheme for the solution of Sivashinsky equation. Appl. Math. Comput. 168: 1273-1280.

Moon, B.S., Yoo, D.S., Lee, Y.H., Oh, I. S., Lee, J.W., Lee, D.Y. and Kwon, K.C. 2010. A non-separable solution of the diffusion equation based on the Galerkin's method using cubic splines. Appl. Math. Comput. 217(5): 1831-1837.

Omrani, K. 2003. A second-order splitting method for a finite difference scheme for the Sivashinsky equation. Appl. Math. Letters 16: 441-445.

Omrani, K. 2007. Numerical methods and error analysis for the nonlinear Sivashinsky equation. Appl. Math. Comput. 189: 949-962.

Onah, S.E. 2002. Asymptotic behavior of the Galerkin and the finite element collocation methods for a parabolic equation. Appl. Math. Comput. 127: 207-213.

Orsini, P., Power, H. and Lees, M. 2011. The Hermite radial basis function control volume method for multi-zones problems; A non-overlapping domain decomposition algorithm. Comput. Meth. Appl. Mech. Eng. 200: 477-493.

Ozis, T., Esen, A. and Kutluay, S. 2005. Numerical solution of Burgers’ equation by quadratic B-spline finite elements. Appl. Math. Comput. 165: 237-249.

Ozis, T., Aksan, E.N. and Ozdes, A. 2003. A finite element approach for solution of Burger’s equation. Appl. Math. Comput. 139: 417-428.

Pacheco, C.R.F., Paiwa, J. L. and Reynol, A.S. 2006. Operational evaluations of rotary drum vacuum filters for brownstock washing using basic filtration parameters. TAPPI J. 5(3): 15-20.

Pandey, K., Verma, L. and Verma, A.K. 2009. On a finite difference scheme for Burgers’ equation. Appl. Math. Comput. 215: 2206-2214.

Parand, K., Dehghan,M., Rezaeia, A.R. and Ghaderi, S.M. 2010. An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method. Comp. Physics Commun. 181: 1096-1108.

Peirce, A. 2010. A Hermite cubic collocation scheme for plane strain hydraulic fractures. Comp. Meth. Appl. Mech. Eng. 199: 1949-1962.

Page 12: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

136

Pellett, G.L. 1966. Longitudinal dispersion, intraparticle diffusion, and liquid- phase mass transfer during flow through multiparticle systems. TAPPI J., 49(2): 75-82.

Potucek, F. 2001. Displacement washing of pulp II: Analysis of laboratory data. Papir a Celuloza. 56(2): 49-53.

Prenter, P.M. and Russell, R. D. 1976. Orthogonal collocation for elliptic partial differential equations. SIAM J. Numer. Ana. 13(6): 923-939.

Prenter, P.M. 1975. Splines and variational methods. Wiley interscience publication, New York.

Qin, F. G.F., Yang, X. and Yang, M. 2011. An adhesion model of the axial dispersion in wash columns of packed ice beds. Sep. Purif. Tech. 79: 321-328.

Ramadan, M.A., Lashien, I.F. and Zahra, W.K. 2007. Polynomial and non polynomial spline approaches to the numerical solution of second order boundary value problems. Appl. Math. Comput. 184: 476-484.

Rashid, A. and Ismail, A.I.B. 2009. The Fourier spectral method for the Sivashinsky equation. An. St. Univ. Ovidius Constanta. 17(2): 191-202.

Raslan, K.R. 2005. Collocation method using quartic B-spline for the equal width equation. Appl. Math. Comput. 168: 795-805.

Raza, N., Sial, S. and Neuberger, J.W. 2011. Numerical solution of Burgers’ equation by the Sobolev gradient method. Appl. Math. Comput. 218: 4017-4024.

Rivera, F.F., Cruz-Díaz, M.R., Rivero, E.P. and González, I. 2010. Analysis and interpretation of residence time distribution experimental curves in FM01-LC reactor using axial dispersion and plug dispersion exchange models with closed–closed boundary conditions. Electrochimica Acta 56: 361-371.

Robinson, M.P. and Fairweather, G. 1994. Orthogonal spline collocation methods for Schrodinger type equations in one space variable. Numer. Math. 68: 355-376.

Rocca, A.L. and Power, H. 2008. A double boundary collocation Hermitian approach for the solution of steady state convection-diffusion problems. Comp. Math. Applic. 55: 1950-1960.

Rocca, A.L., Rosales, A.H. and Power, H. 2005. Radial basis function Hermite collocation approach for the solution of time dependent convection-diffusion problems. Eng. Ana. Boundary Elements 29: 359-370.

Page 13: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

137

Roininen, J. and Alopaeus, V. 2011. The moment method for one-dimensional dynamic reactor models with axial dispersion. Comp. Chem. Eng. 3: 423-433.

Roshan, T. and Bhamra, K.S. 2011. Numerical solutions of the modified Burgers’ equation by Petrov-Galerkin method. Appl. Math. Comput. 218: 3673-3679.

Rubin, S.G. and Graves, R.A. 1975. Cubic spline approximation for problems in fluid mechanics. NASA TR R-436, Washington.

Saka, B. and Dag, I. 2008. A numerical study of the Burgers’ equation. J. Franklin Inst. 345: 328-348.

Saritha, N.V. and Madras, G. 2001. Modeling the chromatographic response of inverse size-exclusion chromatography. Chem. Eng. Sci. 56: 6511-6524.

Sewell, G. 2010. Solving PDEs in non-rectangular 3D regions using a collocation finite element method. Adv. Eng. Software 41: 748-753.

Shawaqfeh, A.T. 2003. Gas holdup and liquid axial dispersion under slug flow conditions in gas/liquid bubble column. Chem. Eng. Processing 42: 767-775.

Sherman, W.R. 1964. The movement of a soluble material during the washing of a bed of packed solids, AIChE J., 10(6): 855-860.

Shirashi, F. 2001. Highly accurate solution of the axial dispersion model expressed in S-system canonical form by Taylor series method. J. Chem. Eng. 83: 175-183.

Siddiqi, S.S. and Akram, G. 2006. Solution of fifth order boundary value problems using non polynomial spline technique. Appl. Math. Comput. 175: 1574-1581.

Siddiqi, S.S., Akram, G. and Elahi, A. 2008. Quartic spline solution of linear fifth order boundary value problems. Appl. Math. Comput. 196: 214-220.

Sivashinsky, G. 1977. Nonlinear analysis of hydrodynamic instability in laminar flame I: Derivation of basic equations. Acta Astron. 4: 1117-1206.

Siyakatshana, N., Kudrna, V. and Machon, V. 2005. Incorporating Danckwerts’ boundary conditions into the solution of the stochastic differential equation. Chem. Eng. Sci. 60: 1987-1994.

Soliman, M.A. 2000. Studies on the method of orthogonal collocation IV: Leguerre and Hermite orthogonal collocation method, J. King Saud Uni.: Eng. Sci. 12(1): 1-14.

Soliman, M.A. and Ibrahim, A.A. 1999. Studies on the method of orthogonal collocation III: The use of Jacobi orthogonal polynomials for the solution of boundary value problems. J. King Saud Uni.: Eng. Sci. 11(2): 191-202.

Page 14: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

138

Soltanalizadeh, B. and Zarebnia, M. 2011. Numerical analysis of the linear and nonlinear Kuramoto-Sivashinsky equation by using differential transformation method. Int. J. of Appl. Math. and Mech. 7(12): 63-72.

Subramanian, V.R. and White, R.E. 2000. Symbolic solutions for boundary value problems using Maple. Comp. Chem. Eng. 24: 2405-2416.

Sule, Z., Lakatos, B.G. and Mihalyko, Cs. 2010. Axial dispersion/population balance model of heat transfer in gas-solid turbulent fluidized beds. Comp. Chem. Eng. 34: 753-762.

Sun, L. M. and Meunier, F. 1991. An improved finite difference method for fixed bed multicomponent adsorption. AIChE J. 37(2): 244-254.

Sun, W. 2000. Hermite cubic spline collocation methods with upwind features. ANZIAM J. 42: 1379-1397.

Szukiewicz, M.K. 2000. New approximate model for diffusion and reaction in a porous catalyst. AIChE J. 46(3): 661-665.

Tatari, M. and Dehghan, M. 2009. On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl. Math. Model. 33: 1729-1738.

Tatari, M. and Dehghan, M. 2010. A method for solving partial differential equations via radial basis functions: Application to the heat equation. Eng. Ana. Boundary Elements 34: 206-212.

Tervola, P. 2006. Fourier series solution for multistage counter current cake washing and segregated wash effluent circulation. Chem. Eng. Sci. 61: 3268-3277.

Trinidad, P., Ponce de Léon, C. and Walsh, F.C. 2006. The application of flow dispersion models to the FM01-LC laboratory filter-press reactor. Electrochimica Acta 52: 604-613.

Uddin, M., Haq, S. and Islam, S. 2009. A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations. Appl. Math. Comput. 212: 458-469.

Vianna, A.S. and Nichele, J. 2010. Modeling an annular flow tubular reactor. Chem. Eng. Sci. 65: 4261-4270.

Vikhansky, A. and Wang, W. 2011. Taylor dispersion in finite-length capillaries. Chem. Eng. Sci. 66: 642-649.

Villadsen, J.V. and Stewart, W.E. 1967. Solution of boundary value problems by orthogonal collocation. Chem. Eng. Sci. 22: 1483-1501.

Page 15: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

139

Villadsen, J. and Michelsen, M.L. 1978. Solution of differential equation models by polynomial approximation. Prentice-Hall, Englewood Cliffs.

Wazwaz, A.M. 2006. New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations. Appl. Math. Comput. 182: 1642-1650.

Wazzan, L. 2009. A modified tanh-coth method for solving the general Burgers-Fisher and the Kuramoto-Sivashinsky equations. Commun. Nonlinear Sci. Numer. Simulat. 14: 2642-2652.

Wood, W.L. 2006. An exact solution for Burger’s equation. Commun. Numer. Meth. Eng. 22: 797-798.

Xiong, Z. and Chen, Y. 2007. Finite volume element method with interpolated coefficients for two-point boundary value problem of semilinear differential equations. Comput. Methods Appl. Mech. Eng. 196: 3798-3804.

Xu, M., Wang, R.H., Zhang, J.H. and Fang, Q. 2011. A novel numerical scheme for solving Burgers’ equation. Appl. Math. Comput. 217: 4473-4482.

Zaki, S.S. 2000. A quintic B-spline finite elements scheme for the KdVB equation. Comput. Methods Appl. Mech. Eng. 188: 121-134.

Zhang, T., Zhao, B. and Wang, J. 2006. Mathematical models for macro-scale mass transfer in airlift loop reactors. Chem. Eng. J. 119: 19-26.

Page 16: REFERENCES - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/45196/17/17_reference.pdf · Numerical solution of the Burgers’ equation by automatic differentiation ... using

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LIST OF PUBLICATIONS

Research Papers in Journals

1. I.A. Ganaie, B. Gupta, N. Parumasur, P. Singh and V.K. Kukreja, Asymptotic convergence of cubic Hermite collocation method for parabolic partial differential equation, Applied Mathematics and Computation Vol. 220 (2013) 560–567.

2. I. A. Ganaie, S. Arora and V.K. Kukreja, Modelling and simulation of a packed bed of pulp fibers using mixed collocation method, International Journal of Differential Equations Vol. 2013 (2013) 1-7.

3. I.A. Ganaie, S. Arora and V.K. Kukreja, cubic Hermite collocation method for

solving boundary value problems with Dirichlet, Neumann and Robin conditions, International Journal of Engineering Mathematics Vol. 2013 (2013) 1-8.

4. I.A. Ganaie, V.K. Kukreja, N. Parumasur, P. Singh and F. Potůček, Comparative study of axial dispersion model using cubic Hermite collocation method for linear and nonlinear adsorption isotherms, Cellulose Chemistry and Technology (2013). (Accepted).

5. J. Kumar, Ishfaq Ganaie and V.K. Kukreja, Application of Mathematica software to solve pulp washing model, ISRN Chemical Engineering Vol. 2013, 1-6.

6. I.A. Ganaie and V.K. Kukreja, A novel numerical scheme of cubic Hermite spline collocation method for solving Burgers’ equation, AIP Conference Proceedings Vol. 1558 (2013) 1196-1199.

7. Ishfaq Ahmad Ganaie and V.K. Kukreja, Numerical solution of Burgers' equation by cubic Hermite collocation method, Applied Mathematics and Computation (2013). (Submitted).

8. Ishfaq Ahmad Ganaie and V.K. Kukreja, Asymptotic convergence of cubic Hermite collocation method, Applied Numerical Mathematics (2013). (Submitted).

9. Ishfaq Ganaie, Jitender Kumar and V.K. Kukreja, Simulation of packed bed of porous particles using axial dispersion model, Chinese Journal of Chemical Engineering 2013 (Submitted).

10. I.A. Ganaie, S. Arora and V. K. Kukreja, Cubic Hermite collocation solution of Kuramoto-Sivashinsky equation, International Journal of Computer Mathematics 2014 (Submitted).

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Research Papers in Conferences

1. I.A. Ganaie, S. Arora and V.K. Kukreja, Solution of boundary value problems using Mathematica, 15th Punjab Science Congress, GNDU, Amritsar, India, February, 2012.

2. I.A. Ganaie, B. Gupta and V.K. Kukreja, Solution of axial dispersion model using cubic Hermite collocation method, 8th Science Congress, University of Kashmir, Srinagar, India, September, 2012.

3. I.A. Ganaie, Ajay Mittal and V.K. Kukreja, Numerical solution of diffusion

dispersion models using cubic Hermite collocation method, National Seminar on Mathematics and Its Applications, Punjabi University Patiala, India, December, 2012.