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Multipoint boundary value problems by differential quadrature method. (English) Zbl 0999.65074
Summary: This paper extends the application of the differential quadrature method (DQM) to high order (\(\geq 3^{\text{rd}}\)) ordinary differential equations with the boundary conditions specified at multiple points (\(\geq\) three different points). Explicit weighting coefficients for higher order derivatives have been derived using interpolating trigonometric polynomials. A three-point, linear third-order differential equation governing the shear deformation of sandwich beams is examined.
Two examples of four-point nonlinear fourth-order systems are also presented. Accurate results are obtained for the example problems. Since boundary conditions are usually specified only at two extreme ends and not at intermediate boundary points, the present work opens new areas of application of the DQM.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34B05 Linear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
74S25 Spectral and related methods applied to problems in solid mechanics
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[1] Bellman, R.; Casti, J., Differential quadrature and long term integration, J. math. anal. appl., 34, 235-238, (1971) · Zbl 0236.65020
[2] Bellman, R.; Kashef, B.G.; Casti, J., Differential quadrature: A technique for the rapid solution of non-linear partial differential equations, J. comp. phys., 10, 40-52, (1972) · Zbl 0247.65061
[3] Bellman, R.; Kashef, B.; Vasudevan, R., The inverse problem of estimating heart parameters from cardiograms, Math. biosci., 19, 221-230, (1974) · Zbl 0276.92011
[4] Civan, F.; Sliepcevich, C.M., Application of differential quadrature to transport processes, J. math. anal. appl., 93, 206-221, (1983) · Zbl 0538.65084
[5] Civan, F.; Sliepcevich, C.M., Differential quadrature for multi-dimensional problems, J. math. anal. appl., 101, 423-443, (1984) · Zbl 0557.65084
[6] Civan, F., Solving multivariable mathematical models by the quadrature and cubature methods, Numer. meth. partial diff. eq., 10, 545-567, (1994) · Zbl 0810.65141
[7] Civan, F., A theoretically derived transfer function for oil recovery from fractured reservoirs by waterflooding, Proceedings of the 1994 improved oil recovery symposium, tulsa, OK, 87-98, (1994), SPE 27745
[8] Bellomo, N., Nonlinear models and problems in applied sciences from differential quadrature to generalized collocation methods, Mathl. comput. modelling, 26, 4, 13-34, (1997) · Zbl 0898.65074
[9] Jang, S.K.; Bert, C.W.; Striz, A.G., Application of differential quadrature to static analysis of structural components, Int. J. numer. meth. eng., 28, 561-577, (1989) · Zbl 0669.73064
[10] Bert, C.W.; Malik, M., Differential quadrature method in computational mechanics: A review, Appl. mech. rev., 49, 1-27, (1996)
[11] Wu, T.Y.; Liu, G.R., (), 119
[12] Wu, T.Y.; Liu, G.R., The differential quadrature as a numerical method to solve the differential equation, Comput. mech., 24, 197-205, (1999) · Zbl 0976.74557
[13] Wu, T.Y.; Liu, G.R., The generalized differential quadrature rule for initial-value differential equations, J. sound. vib., 233, 195-213, (2000) · Zbl 1237.65018
[14] Liu, G.R.; Wu, T.Y., Numerical solution for differential equations of a Duffing type nonlinearity using the generalized differential quadrature rule, J. sound vib., 237, 5, 805-817, (2000) · Zbl 1237.65073
[15] Wu, T.Y.; Liu, G.R., Axisymmetric bending solution of shells of revolution by the generalized differential quadrature rule, Int. J. pres. vessels piping, 77, 149-157, (2000)
[16] Wu, T.Y.; Liu, G.R., A generalized differential quadrature rule for analysis of thin cylindrical shells, (), 223-228 · Zbl 1001.65085
[17] Wu, T.Y.; Liu, G.R., The generalized differential quadrature rule for fourth order differential equations, Int. J. numer. meth. eng., 50, 8, 1907-1929, (2001) · Zbl 0999.74120
[18] Wu, T.Y.; Liu, G.R., Application of the generalized differential quadrature rule to sixth-order differential equations, Commun. numer. methods engng., 16, 11, 777-784, (2000) · Zbl 0969.65070
[19] Wu, T.Y.; Liu, G.R., Application of the generalized quadrature rule to eighth-order differential equations, Commun. numer. methods engng., 17, 5, 355-364, (2001) · Zbl 0985.65090
[20] G.R. Liu and T.Y. Wu, An application of the generalized differential quadrature rule in Blasius and Onsager equations, Int. J. Numer. Meth. Eng. (to appear). · Zbl 0996.76072
[21] Wang, X.; Gu, H., Static analysis of frame structures by the differential quadrature element method, Int. J. numer. meth. eng., 40, 759-772, (1997) · Zbl 0888.73078
[22] Bellman, R.; Kashef, B.; Vasudevan, R., The inverse problem of estimating heart parameters from cardiograms, Math. biosci., 19, 221-230, (1974) · Zbl 0276.92011
[23] Shu, C.; Chew, Y.T., Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems, Commun. numer. meth. engng., 13, 643-653, (1997) · Zbl 0886.65109
[24] Shu, C.; Xue, H., Explicit computation of weighting coefficients in the harmonic differential quadrature, J. sound vib., 204, 549-555, (1997)
[25] Quan, J.R.; Chang, C.T., New insights in solving distributed system equations by the quadrature method, I: analysis, Comput. chem. eng., 13, 779-788, (1989)
[26] Shu, C.; Richards, B.E., Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, Int. J. numer. meth. fluids, 15, 791-798, (1992) · Zbl 0762.76085
[27] Chang, C.T.; Tsai, C.S.; Lin, T.T., The modified differential quadrature and their applications, Chem. eng. commun., 123, 135-164, (1993)
[28] Striz, A.G.; Wang, X.; Bert, C.W., Harmonic differential quadrature method and applications to analysis of structural components, ACTA mech., 111, 85-94, (1995) · Zbl 0854.73080
[29] Stoer, J.; Bulirsch, B., Introduction to numerical analysis, (1992), Springer-Verlag New York
[30] Haque, M.; Baluch, M.H.; Mohsen, M.F.N., Solution of multiple point, nonlinear boundary value problems by method of weighted residuals, Int. J. comput. math., 19, 69-84, (1986) · Zbl 0653.65059
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