Wen Chen's References for the Differential Quadrature Method and Its applications

(Dec., 1996)




  1. Atkinson, K. E. (1978), An introduction to numerical analysis, John Wiley & Sons.

  2. Bartels, R. H.; Stewart, G. W. (1972), "A solution of the equation AX+XB=C", Communications in AM 15, 820-826.

  3. Bathe, K. J.; Wilson, E. L. (1976), Numerical Methods in Finite Element Analysis, Englewood Cliffs, New Jersy: Prentice-Hall, Inc.

  4. Bathe, K. J. (1982), Finite Element Procedures in Engineering Analysis, Prentice-Hall Inc., Enlewood Cliffs, NJ.

  5. Bazeley, G. P.; Cheung, Y. K.; Irens, B. M.; Zienkiewicz, O. C. (1965), "Triangular elements in bending-conforming and non-conforming solutions", Proc. Conf. Matrix Methods in Structural Mechanics, J. S. Przemieniecki et al. eds. Chio: Air Force Ins. Tech. Wright-Patterson A F Base. 547-576.

  6. Bellman, R. E.; Casti, J. (1971), "Differential quadrature and long-term integration". Journal of Mathematical Analysis and Applications 34, 235-238.

  7. Bellman, R. E.; Kashef, B. G.; Casti, J (1972), "Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations", J. Comput. Phys.10, 40-52.

  8. Bellman, R. E. (1973), Methods of Nonlinear Analysis, Vol. 2, chap. 16, Academic Press, New York.

  9. Bellman, R. E.; Kashef, B. G. (1974a), "A scanning technique for systems identification", Journal of Mathematical Analysis and Applications 71, 403-411.

  10. Bellman, R. E.; Kashef, B. G.; Vasudevan, R. (1974b), "The inverse problem of estimating heart parameters from cardiograms", Math. Biosci. 19, 221-230.

  11. Bellman, R.; Kashef, B.; Lee, E. S.; Vasudevan, R. (1975), "Differential quadrature and spline", Comput. & Math. with Appl. 1, 371-376.

  12. Bellman, R.; Roth, R. S. (1979), "System identification with partial differential", J. Math. Anal. Appl., 68, 321-333.

  13. Bellman, R; Adomian, G. (1985), Partial differential equations, D Reidel Publishing, Dordrecht, The Netherlands.

  14. Bellman, R; Roth, R. S. (1986), Methods in approximation, D Reidel Publishing, Dordrecht, Netherlands.

  15. Bert, C. W.; Jang S. K.; Striz, A. G. (1988a), "Two new methods for analyzing free vibration of structural components", AIAA Journal, 26, 612-618.

  16. Bert, C. W.; Cho, K. N. (1988b), Private communication. The University of Oklahoma.

  17. Bert, C. W.; Striz, A. G.; Jang, S. K. (1989), "Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature", Comput. Mech. 5, 217-226.

  18. Bert, C. W.; Wang, X.; Striz, A. G. (1993), "Differential quadrature for static and free vibrational analysis of anisotropic plates", Int. J. Solids Structures 30, 1737-1744.

  19. Bert, C. W.; Wang, X.; Striz, A. G. (1994a), "Convergence of the DQ method in the analysis of anisotropic plates", J. Sound & Vibr. 170, 140-144.

  20. Bert, C. W., Wang, X, Striz, A. G. (1994b), "Static and free vibrational analysis of beams and plates by differential quadrature method," Acta Mech. 102, 11-24.

  21. Bert, C. W.; Malik, M. (1996a), "Free vibration analysis of thin cylindrical shells by the differential quadrature method", ASME J. Pressure Vessel Tech. 118, 1-12.

  22. Bert, C. W.; Malik, M. (1996b), "Free vibration analysis of tapered rectangular plates by differential quadrature method: A semi-analytical approach", J. Sound & Vib. 190, 41-63.

  23. Bert, C. W.; Malik, M. (1996c), "The differential quadrature method for irregular domains and applications to plate vibration", Int. J. Mech. Sci., 38, 589-606.

  24. Bert, C. W.; Malik, M. (1996d), "Differential quadrature method in computational mechanics: A review", Appl. Mech. Rev. 49, 1-28.

  25. Bjorck, A. A.; Pereyra, V. (1970), "Solution of Vandermonde system of equation", Math. Comput. 24, 893-903.

  26. Burden, R. L.; Faires, J. D.; Reynolds, A. C. (1978), Numerical Analysis, 2nd Eds., Prindle, Weber & Schmilt, Boston.

  27. Burka, M. K. (1982) "Solution of stiff ordinary differential equations by decomposition and orthogonal collocation", AIChE Journal 28, 11-20.

  28. Cantoni, A and Butler, P (1976), "Eigenvalues and eigenvectors of symmetric centrosymmetric matrices", Linear Algebra & its Applications, 13, 275-288.

  29. Canuto, C., Hussaini, M. Y.; Quarteroni, A., Zang, T. A. (1988), Spectral Methods in Fluid Dynamics, Springer, Berlin.
  30. Cao, Zhiyuan (1989), Vibration Theory of Plates and Shells (in Chinese), Chinese Railway Press, Beijing.

  31. Cao, Zhiyuan and Cheung, Y. (1992), Semi-analytical Numerical Methods (in Chinese), Defense Industry Press, Beijing.

  32. Chang, C. T.; Tsai, C. S.; Lin, T. T. (1993), "Modified differential quadratures and their applications", Chem. Eng. Commun. 123, 135-164.

  33. Cheng, Y. P. (1987), Matrix theory, West-North Univ. Tech. Press, Xian.

  34. Civan, F.; Sliepcevich, C. M. (1983a), "Application of differential quadrature to transport processes," J. Math. Anal. Appl. 93, 206-221.

  35. Civan, F.; Sliepcevich, C. M. (1983b), "Solution of the Poisson equation by differential quadrature, Int. J. Numer. Methods Eng. 19, 711-724.

  36. Civan, F.; Sliepcevich, C. M. (1984a), "On the solution of the Thomas-Fermi equation by differential quadrature," J. Comput. Phys., 56, 343-348.

  37. Civan F.; Sliepcevich, C. M. (1984b), "Differential quadrature for multidimensional problems", J. Math. Anal. Appl., 101, 423-443.

  38. Civan F.; Sliepcevich, C. M. (1985), "Application of differential quadrature to solution of pool boiling in cavities", Proc. Okla. Acad. Sci. 65, 73-78.

  39. Civan F.; Sliepcevich, C. M. (1986), "Solving integro-differential equations by the quadrature method", Proc. of the 1st Int. Conference on Integral Methods in Sci. & Engrg., UT Arlington, Mar. 18-21, 1985, Hemisphere Publ. Co., New York, 106-111.

  40. Civan, F. (1989), "Differential cubature for multidimensional problems", in Proc. of the 20th Annual Pittsburgh Conference on Modeling and Simulation, W. G. Vogt and M. H. Mickle, Eds., Instrument Society of American, North Carolina, p. 1843-1847.

  41. Civan, F. (1993), "Comments on application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations by C. Shu and B. E. Richards", Int. J. Numer. Methods Fluids, 17, 921-922.

  42. Civan, F. (1994a), "Solving multivariable mathematical models by the quadrature and cubature methods", Numer. Methods for P. D. E., 10, 545-567.

  43. Civan, F. (1994b), "Rapid and accurate solutions of reactor models by the quadrature method", Computers & Chemical Engineering, 18, 10005-10009.

  44. Cook, R. D. (1974), Concepts and Applications of Finite Element Analysis, John Wiley & Sons, Inc.

  45. Dahlquist, G. (1963), "A special stability problem for linear multistep methods", BIT, 3, 27-43.

  46. Datta, L.; Morgera, S. D. (1989), "On the reducibility of centrosymmetric matrices applications in engineering problems", Circuits Systems Signal Process, 8, 71-96.

  47. Diggelen, F. V.; Glover, K. (1994), "A Hadamard weighted loop shaping design procedure for robust decoupling", Automatica, 30, 831-845.

  48. Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C. (1982), Solitions and nonlinear wave equations, Academic Press, Orlando.

  49. Dokainish, M. A.; Subbaraj, K.(1989a), "A survey of direct time-integration methods in computational structural dynamics-I", Computer & Structures 32, 1371-1386.

  50. Dokainish, M. A.; Subbaraj, K. (1989b), "A survey of direct time-integration methods in computational structural dynamics-II", Computer & Structures 32, 1387-1401.

  51. D'souza, F.; Garg, V. K. (1984), Advanced Dynamics - Modeling and Analysis, Chapt. 7, Prentice-Hall, Inc., New Jersey.
  52. Du. H.; Lim, M. K.; Lin, R. M. (1994), "Application of generalized differential quadrature method to structural problems", Int. J. Numer. Eng. 37, 1881-1896.

  53. Engles, H. (1980), Numerical quadrature and cubature, Academic Press, San Diego.

  54. Farsa, J. Kukreti A. R.; Bert, C. W. (1991), "Fundamental frequency of laminated rectangular plates by differential quadrature method", Comput. Struct. 46, 465-477.

  55. Feng, Y.; Bert, C. W. (1992), "Application of the quadrature to flexural vibration analysis of a geometrically nonlinear beam", Nonlinear Dynamics, 3, 13-18.

  56. Finlayson, B. A. (1972), The Method of Weighted Residuals and Variational Principles, Academic Press, New York.
  57. Finlayson, B. A. (1980), Nonlinear Analysis in Chemical Engineering, McGraw-Hill, New York.

  58. Gear, C. W. (1967), "The numerical integration of ODE", Math. Comp. 21, 146-156.

  59. Gear, C. W. (1971a), "Simultaneous numerical solution of differential -algebraic equations", IEEE Transaction Circuit Theory 18, 89-95.

  60. Gear, C. W. (1971b), Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc.

  61. Golub, H.; Nash, S.; Vanloan, C. (1979), "A Hessenberg-Schur method for the problem AX+XB=C", IEEE Transactions on Automatic Control AC-24(6), 909-913.

  62. Good, I. J. (1970), "The inverse of a centrosymmetric matrix," Technometrics 12, 925-928.

  63. Gui, Bing (1992), "A Numerical Solution of Lyapunov Matrix Equation and a Partly Parallel Treatment (in Chinese)", Journal of Nanjing University of Aeronautics and Astronautics 24, 449-455.

  64. Gutierrez, R. H.; Laurra, P. A. A. (1994), "Solution of the Helmholtz equation in a parallelogrammic domain with mixed boundary conditions using the differential quadrature method", J. Sound & Vib. 178, 269-271.

  65. Haltiner, G. J.; Williams, R. T. (1980), Numerical weather prediction and dynamic meteorology, 2nd edition, John Wiley & Sons, Inc., New York.

  66. Hamming, R. W. (1973), Numerical methods for scientists and engineers, McGraw-Hill, New York.

  67. Holden, J. T. (1972), "On the finite deflections of thin beams", Int. J. Solids Struct. 8, 1051-1055.

  68. Horn, R. A.; Johnson, C. R. (1985), Matrix Analysis, Cambridge Univ. Press.

  69. Horn, R. A. (1990), "The Hadamard product", in Matrix theory and applications, Series: Proc. of Symposia in applied mathematics, V. 40, C. R. Johnson Ed., American Mathematical Society, Providence, Rhode Island, , p. 87-169.

  70. Hu, L. C.; Hu, C. (1974), "Identification of rate constants by differential quadrature in partly measurable compartmental models," Math. Biosci. 21, 71-76.

  71. Huebner, K. H. (1975), The Finite Element Method for Engineers, John Wiley & Sons Inc., New York.

  72. Jang, S. K.; Bert, C. W.; Striz, A. G. (1989), "Application of differential quadrature to static analysis of structural components", Inter. J. Numer. Methods Eng., 28, 561-577.

  73. Jarrand, M.; Simmons, A. J. (1983), "The spectral techniques. Numerical methods for weather prediction", ECMWF Seminar, Vol. 2, 1-59.

  74. Kang, K.; Bert, C. W.; Striz, A. G. (1995), "Vibration analysis of shear deformable circular arches by the differential quadrature method", J. Sound & Vibr. 183, 353-360.

  75. Kang, K; Bert, C. W.; Striz, A. G. (1996), "Static analysis of a curved shaft subjected to end torques", Int. J. Solids and Structures, 33, 1587-1596.

  76. Kashef, B. G.; Bellman, R. E.; (1974), "Solution of the partial differential equation of the Hodgkins-Huxley model using differential quadrature", Math. Biosci. 19, 1-8.

  77. Kukretic, A. R. Farsa, J. Bert, C. W. (1992), "Fundamental frequency of tapered plates by differential quadrature", J. Eng. Mech. 118, 1221-1238.

  78. Lam, S. S. E. (1993), "Application of the differential quadrature method to two-dimensional problems with arbitrary geometry", Computers & Structures, 47, 459-464.

  79. Lancaster, P.; Timenetsky, M. (1985), The Theory of Matrices with Applications, 2nd edition, Academic Press. Orlando.

  80. Lapidus, L.; Pinder, G. F. (1982), Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley & Sons Inc., New York.

  81. Laurra, P. A.; Saffell, B. F. Jr. (1967), "Small-amplitude vibration of clamped rectangular plates using polynomial approximations", J. of Acoustic Society of America, 41, 836-839.

  82. Laurra, P. A. A.; Gutierrez, R. H. (1993), "Analysis of vibrating Timoshenko beams using the method of differential quadrature", Shock Vib. 1, 89-93.

  83. Laurra, P. A. A.; Gutierrez, R. H. (1994a), "Analysis of vibrating rectangular plates with non-uniform boundary conditions by using the differential quadrature method", J. Sound & Vibr. 173, 702-706.

  84. Laura, P. A. A.; Rossi, R. E. (1994b), "The method of differential quadrature and its application to the approximate solution of ocean engineering problems", Ocean Eng. 21, 57-66.

  85. Laura, P. A. A.; Gutierrez, R. H.; Rossi, R. E. (1996), "Free vibrations of beams of bilinearly varying thickness", Ocean Eng. 23, 1-6.

  86. Leissa, A. W. (1973), "The Free Vibration of Rectangular Plates", J. Sound & Vibr., 31, 257-293.

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  88. Levy, S. (1942b), "Bending of rectangular plates with large deflections", NACA Report No. 737. 139-157.

  89. Liang S.; Li H.; Chen Wen (1997), "The DQ solution of buckling of circular arches under water loading", J. Huazhong Univ. of Sci. & Tech., 25, 81-82, 1997.

  90. Liew, K. M.; Han, J. K.; Xiao, Z. M.; Du, H. (1996), "Differential quadrature method for mindlin plates of Winkler foundations", Int. J. Mech. Sci., 38, 405-421.

  91. Lin R. M.; Lim M. K.; Du, H. (1994), "Deflection of plates with nonlinear boundary supports using generalized differential quadrature", Comput. Struct. 53, 993-999.

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  93. Lª, D. (1980), Numerical recipes in Fortran (in Chinese), Defense Industry Press.

  94. Malik, M; Bert, C. W.; Kukretic A. R. (1993), "Differential quadrature solution of uniformly loaded circular plate resting on elastic half-space", in Aliabadi, M. H. and Brebbia, C. A. (eds), Contact Mechanics: Computational Technique, 385-396, Computational Mechanics Publications, Southampton, UK.

  95. Malik, M.; Civan, F. (1994a), "A comparative study of differential quadrature and cubature methods vis-ˆ-vis some conventional techniques in context of convection-diffusion-reaction problems", Chemical Engineering Science, 50, 531-547.

  96. Malik, M.; Bert, C. W. (1994b), "Differential quadrature solutions of steady-state incompressible and compressible lubrication problems", Journal of Tribology 116, 296-302.

  97. Malik, M.; Bert, C. W. (1996), "Implementing multiple boundary conditions in the DQ solution of higher order PED's: Application to free vibration of plates", Int. J. Numer. Methods Eng. 39, 1237-1258.

  98. Mansell, G.; Merryfield, W.; Shizgal, B.; Weinerl, U. (1993), "A comparison of differential quadrature methods for the solution of partial differential equations", Computer Methods in Applied Mechanics and Engineering, 104, 295-316.

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  103. Morgera, S. D. (1982), "On the reducibility of finite Toeplitz matrices-applications in speech analysis and pattern recognition", Signal Processing, 4, 425-443.

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