Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method

Applied Mathematical Modelling

Volumn 37, Issue 18-19, 2013, Pages 8578-8599

  Dehghan, Mehdi ^a   Nikpour, Ahmad ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Applications; Local RBF based differential quadrature method; Multiquadric (MQ); Non equispaced meshes; Radial Basis Functions (RBFs); System of second order boundary value problems

Indexed keywords

DIFFERENTIAL QUADRATURE METHODS; MULTIQUADRICS; NON-EQUISPACED MESHES; RADIAL BASIS FUNCTIONS; SECOND-ORDER BOUNDARY VALUE PROBLEMS;

APPLICATIONS; DIFFERENTIATION (CALCULUS); IMAGE SEGMENTATION; RADIAL BASIS FUNCTION NETWORKS;

NUMERICAL METHODS;

EID: 84883456372     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2013.03.054     Document Type: Article

References (79)

1. Tomantschger K.W. Series solutions of coupled differential equations with one regular singular point. J. Comput. Appl. Math. 2002, 140:773-783.
2. Wellstead P.E. Introduction to Physical System Modelling 1979, Academic Press, London.
3. Clarence W.D.S. Modeling and Control of Engineering Systems 2009, CRC Press.
4. H. Jack, Dynamic System Modeling and Control, http://claymore.engineer.gvsu.edu/~jackh/books.html, 2006, pp. 110-158.
5. Kazmierczak B., Lipniacki T. Homoclinic solutions in mechanical systems with small dissipation. Application to DNA dynamics. J. Math. Biol. 2002, 44:309-329.
6. Cheng X., Zhong C. Existence of positive solutions for a second-order ordinary differential system. J. Math. Anal. Appl. 2005, 312:14-23.
7. Thompson H.B., Tisdell C.C. Systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations. J. Math. Anal. Appl. 2000, 248:333-347.
8. Thompson H.B., Tisdell C.C. Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations. Appl. Math. Lett. 2002, 15(6):761-766.
9. Thompson H.B., Tisdell C.C. The nonexistence of spurious solutions to discrete two-point boundary value problems. Appl. Math. Lett. 2003, 16(1):79-84.
10. Tisdell C.C. Existence of solutions to first-order periodic boundary value problems. J. Math. Anal. Appl. 2006, 323(2):1325-1332.
11. Geng F., Cui M. Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 2007, 327:1167-1181.
12. Geng F., Cui M. Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems. J. Comput. Appl. Math. 2011, 235:2405-2411.
13. Lu J. Variational iteration method for solving a nonlinear system of second-order boundary value problems. Comput. Math. Appl. 2007, 54:1133-1138.
14. Caglar N., Caglar H. B-spline method for solving linear system of second-order boundary value problems. Comput. Math. Appl. 2009, 57:757-762.
15. Dehghan M., Lakestani M. Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions. Int. J. Comput. Math. 2008, 85:1455-1461.
16. Dehghan M., Saadatmandi A. The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method. Math. Comput. Model. 2007, 46:1434-1441.
17. Saadatmandi A., Dehghan M., Eftekhari A. Application of He's homotopy perturbation method for nonlinear system of second-order boundary value problems, Nonlinear Analysis. Real World Appl. 2009, 10:1912-1922.
18. Shu C., Ding H., Yeo K.S. Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 2003, 192:941-954.
19. Fasshauer G.E. Newton iteration with multiquadrics for the solution of the nonlinear PDEs. Comput. Math. Appl. 2002, 43:423-438.
20. G.B. Wright, Radial basis function interpolation: numerical and analytical developments, Ph.D. thesis, University of Colorado, Boulder, 2003.
21. Tolstykh A.I., Shirobokov D.A. On using radial basis functions in a finite difference mode with applications to elasticity problems. Comput. Mech. 2003, 33:68-79.
22. Shen Q. Local RBF-based differential quadrature collocation method for the boundary layer problems. Eng. Anal. Bound. Elem. 2010, 34:213-228.
23. Shen Q. Numerical solution of the Sturm-Liouville problem with local RBF-based differential quadrature collocation method. Int. J. Comput. Math. 2011, 88:285-295.
24. Roque C.M.C., Cunha D., Shu C., Ferreira A.J.M. A local radial basis functions-finite differences technique for the analysis of composite plates. Eng. Anal. Bound. Elem. 2011, 35:363-374.
25. Soleimani S., Jalaal M., Bararnia H., Ghasemi E., Ganji D.D., Mohammadi F. Local RBF-DQ method for two dimensional transient heat conduction problems. Int. Commun. Heat Mass Transfer 2010, 37:1411-1418.
26. Flyer N., Lehto E., Blaise S., Wright G.B., St-Cyr A. A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere. J. Comput. Phys. 2012, 231:4078-4095.
27. Stevens D., Power H., Cliffe K.A. A solution to linear elasticity using locally supported RBF collocation in a generalised finite-difference mode. Eng. Anal. Bound. Elem. 2013, 37:32-41.
28. Bayona V., Moscoso M., Carretero M., Kindelan M. RBF-FD formulas and convergence properties. J. Comput. Phys. 2010, 229:8281-8295.
29. Bayona V., Moscoso M., Kindelan M. Optimal constant shape parameter for multiquadric based RBF-FD method. J. Comput. Phys. 2011, 230:7384-7399.
30. Bayona V., Moscoso M., Kindelan M. Optimal variable shape parameter for multiquadric based RBF-FD method. J. Comput. Phys. 2012, 231:2466-2481.
31. Bayona V., Moscoso M., Kindelan M. Gaussian RBF-FD weights and its corresponding local truncation errors. Eng. Anal. Bound. Elem. 2012, 36:1361-1369.
32. B. Fornberg, E. Lehto, C. Powell, Stable calculation of Gaussian-based RBF-FD stencils, Comput. Math. Appl. doi:10.1016/j.camwa.2012.11.006.
33. Wright G.B., Fornberg B. Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 2006, 212:99-123.
34. Dehghan M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simulat. 2006, 71:16-30.
35. Lee C.K., Liu X., Fan S.C. Local multiquadric approximation for solving boundary value problems. Comput. Mech. 2003, 30:396-409.
36. Sanyasiraju Y.V.S.S., Chandhini G. Local radial basis function based gridfree scheme for unsteady incompressible viscous flows. J. Comput. Phys. 2008, 227:8922-8948.
37. Fornberg B., Larsson E., Flyer N. Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 2011, 33:869-892.
38. Kansa E.J. Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-I. Comput. Math. Appl. 1990, 19:127-145.
39. Kansa E.J. Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-II. Comput. Math. Appl. 1990, 19:147-161.
40. Wu Z.M. Hermite-Bikhoff interpolation of scattered data by radial basis function. Approx. Theory Appl. 1992, 8:1-10.
41. Schaback R. Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 1995, 3:251-264.
42. Dehghan M., Shokri A. Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. J. Comput. Appl. Math. 2009, 230:400-410.
43. Shokri A., Dehghan M. Meshless method using radial basis functions for the numerical solution of two-dimensional complex Ginzburg-Landau equation. Comput. Model. Eng. Sci., CMES 2012, 34:333-358.
44. Ling L., Kansa E.J. A least-square preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 2005, 23:31-54.
45. Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 1995, 4:389-396.
46. Wong A.S.M., Hon Y.C., Li T.S., Chug S.L., Kansa E.J. Multizone decomposition of time dependent problems using the multiquadric scheme. Comput. Math. Appl. 1999, 37:23-43.
47. Wu Y.L., Liu G.R. A meshfree formulation of local radial point interpolation method (LRPIM) for incompressible flow simulation. Comput. Mech. 2003, 30:355-365.
48. Bellman R.E., Kashef B.G., Casti J. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 1972, 10:40-52.
49. Civan F., Sliepcevich C.M. Application of differential quadrature to transport processes. J. Math. Anal. Appl. 1983, 93:206-221.
50. Zong Z., Lam K.Y. A localized differential quadrature method and its application to the 2D wave equation. Comput. Mech. 2002, 29:382-391.
51. Liew K.M., Huang Y.Q., Reddy J.N. Moving least squares differential quadrature method and its application to the analysis of shear deformable plates. Int. J. Numer. Meth. Eng. 2003, 56:2331-2351.
52. Liew K.M., Huang Y.Q., Reddy J.N. A hybrid moving least squares and differential quadrature meshfree method. Int. J. Comput. Eng. Sci. 2002, 3:1-12.
53. Micchelli C.A. Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 1986, 2:11-22.
54. Driscoll T.A., Fornberg B. Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl. 2002, 43:413-422.
55. Fornberg B., Wright G.B., Larsson E. Some observations regarding interpolants in the limit of flat radial basis functions. Comput. Math. Appl. 2004, 47:37-55.
56. Franke R. Scattered data interpolation: test of some methods. Math. Comput. 1982, 38:181-200.
57. B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, paperback edition, 1996.
58. Sleeman B.D., Grinrod P. Homoclinic solutions for coupled systems of differential equations. Proc. Royal Soc. Edinb. 1985, 99A:319-328.
59. Cheng A.H.D. Multiquadric and its shape parameter-A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation. Eng. Anal. Bound. Elem. 2012, 36:220-239.
60. Dehghan M., Shokri A. A numerical method for solution of the two-dimensional Sine-Gordon equation using the radial basis functions. Math. Comput. Simul. 2008, 79:700-715.
61. Dehghan M., Shokri A. A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions. Comput. Math. Appl. 2007, 54:136-146.
62. Ding H., Shu C., Tang D.B. Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments. Int. J. Numer. Meth. Eng. 2005, 63:1513-1529.
63. Li K., Huang Q.B., Wang J.L., Lin L.G. An improved localized radial basis function meshless method for computational aeroacoustics. Eng. Anal. Bound. Elem. 2011, 35:47-55.
64. Liu G.R., Wu T.Y. Differential quadrature solutions of eighth-order boundary-value differential equations. J. Comput. Appl. Math. 2002, 145:223-235.
65. Shan Y.Y., Shu C., Lu Z.L. Application of local MQ-DQ method to solve 3D incompressible viscous flows with curved boundary. Comput. Model. Eng. Sci. 2008, 25:99-113.
66. Shokri A., Dehghan M. A Not-a-Knot meshless method using radial basis functions and predictor-corrector scheme to the numerical solution of improved Boussinesq equation. Comput. Phys. Commun. 2010, 181:1990-2000.
67. Shu C., Ding H., Yeo K.S. Solution of partial differential equations by a global radial basis function-based differential quadrature method. Eng. Anal. Bound. Elem. 2004, 28:1217-1226.
68. Shu C., Ding H., Chen H.Q., Wang T.G. An upwind local RBF-DQ method for simulation of inviscid compressible flows. Comput. Meth. Appl. Mech. Eng. 2005, 194:2001-2017.
69. Shu C., Wu Y.L. Integrated radial basis functions-based differential quadrature method and its performance. Int. J. Numer. Meth. Fluids 2007, 53:969-984.
70. Tatari M., Dehghan M. On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl. Math. Model. 2009, 33:1729-1738.
71. Tatari M., Dehghan M. A method for solving partial differential equations via radial basis functions: Application to the heat equation. Eng. Anal. Bound. Elem. 2010, 34:206-212.
72. Wu W.X., Shu C., Wang C.M. Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method. J. Sound Vib. 2007, 306:252-270.
73. Wu T.Y., Wang Y.Y., Liu G.R. A generalized differential quadrature rule for bending analysis of cylindrical barrel shells. Comput. Meth. Appl. Mech. Eng. 2003, 192:1629-1647.
74. Yao G., Sarler B., Chen C.S. A comparison of three explicit local meshless methods using radial basis functions. Eng. Anal. Bound. Elem. 2011, 35:600-609.
75. Zong Z. A variable order approach to improve differential quadrature accuracy in dynamic analysis. J. Sound Vib. 2003, 266:307-323.
76. Zong Z., Li Z., Dong J. Solving the sod shock tube problem using localized differential quadrature (LDQ) method. J. Mar. Sci. Appl. 2011, 10:41-48.
77. Z. Zong, Y.Y. Zhang, Advanced Differential Quadrature Methods, Chapman and Hall/CRC Appl. Math. Nonlinear Sci. Series, 2009, pp. 241-256.
78. Shokri A., Dehghan M. Meshless method using radial basis functions for the numerical solution of two-dimensional complex Ginzburg-Landau equation. Comput. Mod. Eng. Sci. CMES 2012, 34:333-358.
79. Salehi R., Dehghan M. A moving least square reproducing polynomial meshless method. Appl. Numer. Math. 2013, 69:34-58.