Abstract
Two new efficient algorithms are developed to approximate the derivatives of sufficiently smooth functions. The new techniques are based on differential quadrature method with quartic B-spline bases as test functions. To obtain the weighting coefficients of differential quadrature method (DQM), we use the midpoints of a uniform partition mixed with near-boundary grid points. This enables us to obtain the weighting coefficients without adding the new extra relations. By obtaining the error bounds, it is proved that the method in its classic form is non-optimal. Then, some new weighting coefficients are constructed to obtain higher accuracy. By obtaining the error bounds, it is proved that the new algorithm is superconvergent. Afterwards, by defining some new symbols, we find a way to approximate the partial derivatives of multivariate functions. Also, some approximations are constructed to the mixed derivatives of multivariate functions. Finally, the applicability of the methods is examined by solving some well-known problems of partial differential equations. Some examples of 2D and 3D biharmonic, Poisson, and convection-diffusion equations are solved and compared to the existing methods to show the efficiency of the proposed algorithms.
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Bellman, R., Kashef, B.G., Casti, J.: Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 10, 40–52 (1972)
Quan, J.R., Chang, C.T.: New sightings in involving distributed system equations by the quadrature methods—I. Comput. Chem. Eng. 13, 779–788 (1989)
Quan, J.R., Chang, C.T.: New sightings in involving distributed system equations by the quadrature methods-II. Comput. Chem. Eng. 13, 717–724 (1989)
Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 15, 791–798 (1992)
Shu, C., Xue, H.: Explicit computation of weighting coefficients in the harmonic differential quadrature. J. Sound Vib. 204, 549–555 (1997)
Shu, C., Wu, Y.L.: Integrated radial basis functions-based differential quadrature method and its performance. Int. J. Numer. Methods Fluids 53, 969–984 (2007)
Bonzani, I.: Solution of nonlinear evolution problems by parallelized collocation interpolation methods. Comput. Math. Appl. 34, 71–79 (1997)
Tomasiello, S.: Stability and accuracy of the iterative differential quadrature method. Int. J. Numer. Meth Eng. 58, 1277–1296 (2003)
Tomasiello, S.: DQ based methods: Theory and application to engineering and physical sciences. In: Leng, J., Sharrock, W. (eds.) Handbook of Research on Computational Science and Engineering: Theory and Practice, pp. 316–346. IGI Global, Hershey (2011)
Tomasiello, S.: Some remarks on a new DQ-based method for solving a class of Volterra integro-differential equations. Appl. Math. Comput. 219, 399–407 (2012)
Tomasiello, S.: Stability and accuracy of DQ,-based step-by-step integration methods for structural dynamics. Appl. Math Model. 37(5), 3426–3435 (2013)
Tomasiello, S.: A differential quadrature-based approach à la Picard for systems of partial differential equations associated to fuzzy differential equations. J. Comput. Appl. Math. 299, 15–23 (2016)
Bellman, R., Kashef, B.G., Lee, E.S., Udevan, R.V.: Differential quadrature and splines. Comput. Math. Appl. 1, 371–376 (1975)
Mittal, R.C., Dahiya, S: Numerical solutions of differential equations using modified B-spline differential quadrature method. Math Anal Appl Springer Proc Math Stat 143, 509–523 (2015)
Arora, G., Singh, B.K.: Numerical solution of Burgers equation with modified cubic B-spline differential quadrature method. Appl. Math. Comput. 224, 166–177 (2013)
Korkmaz, A., Dag, I.: Cubic B-spline differential quadrature methods for the advection-diffusion equation. Int. J. Numer. Methods Heat Fluid Flow 22, 1021–1036 (2012)
Korkmaz, A., Dag, I.: Cubic B-spline differential quadrature methods and stability for Burgers equation. Eng. Comput. 30(3), 320–344 (2013)
Korkmaz, A., Dag, I.: Numerical simulations of boundary-forced RLW equation with cubic B-spline based differential quadrature methods. Arab. J. Sci. Eng. 38, 1151–1160 (2013)
Krowiak, A.: The application of the differential quadrature method based on a piecewise polynomial to the vibration analysis of geometrically nonlinear beams. Comput. Assist. Mech. Eng. Sci. 15, 1–13 (2008)
Krowiak, A.: Modified spline-based differential quadrature method applied to vibration analysis of truncated conical shells. Eng. Comput. 29, 856–874 (2012)
Barrera, D., González, P., Ibáñez, F., Ibáñez, M.J.: A general spline differential quadrature method based on quasi-interpolation. J. Comput. Appl. Math. 275, 465–479 (2015)
Barrera, D., González, P., Ibáñez, F., Ibáñez, M.J.: On spline-based differential quadrature. J. Comput. Appl. Math. 275, 272–280 (2015)
Ghasemi, M.: High order approximations using spline-based differential quadrature method: Implementation to the multi-dimensional PDEs. Appl. Math. Model. 46, 63–80 (2017)
Ghasemi, M.: Spline-based DQM for multi-dimensional PDEs: Application to biharmonic and Poisson equations in 2D and 3D. Comput. Math. Appl. 73(7), 1576–1592 (2017)
De Boor, C.: A Practical Guide to Splines. Springer, New York (2001)
Zhu, Y.: Quartic-spline Collocation Methods for Fourth-order Two-point Boundary Value Problems. Master’s Thesis, Department of Computer Science University of Toronto (2001)
Shi, Z., Cao, Y., Chen, Q.: Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method. Appl. Math. Modell. 36, 5143–5161 (2012)
Shi, Z., Cao, Y.: A spectral collocation method based on Haar wavelets for Poisson equations and biharmonic equations. Appl. Math. Modell. 54, 2858–2868 (2011)
Gupta, M.M., Kouatchou, J.: Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation. Numer. Meth Partial Diff. Eq. 14(5), 593–606 (1998)
Romao, E.C., Campos, M.D., Moura, L.F.M.: Application of the Galerkin and least-squares finite element methods in the solution of 3D poisson and Helmholtz equations. Comp. Math. Appl. 62, 4288–4299 (2011)
Mohanty, R.K., Setia, N.: A new high order compact off-step discretization for the system of 3D, quasi-linear elliptic partial differential equations. Appl. Math Modell. 37, 6870–6883 (2013)
Mohanty, R.K., Jain, M.K.: Technical note: the numerical solution of the system of 3D nonlinear elliptic equations with mixed derivatives and variable coefficients using fourth order difference methods. Numer. Methods Partial Differ. Equ. 11, 187–197 (1995)
Roscoe, D.F.: The solution of the three-dimensional Navier-Stokes equation using a new finite difference approach. Int. J. Numer. Methods Eng. 10, 1299–1308 (1979)
Krishnaiah, U.A., Manohar, R.P., Stephenson, J.W.: Fourth -order finite difference methods for three-dimensional general linear elliptic problems with variable coefficients. Numer. Methods Partial Differ. Equ. 3, 229–240 (1987)
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Ghasemi, M. A new efficient DQ algorithm for the solution of elliptic problems in higher dimensions. Numer Algor 77, 809–829 (2018). https://doi.org/10.1007/s11075-017-0341-x
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DOI: https://doi.org/10.1007/s11075-017-0341-x