Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions

References

• S. Atluri and S. Shen, The Meshless Local Petrov-Galerkin (MLPG) method, Tech Science Press, Encino, CA, 2002.
   Google Scholar
• S. Atluri and T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22 (1998), pp. 117–127. doi: 10.1007/s004660050346
   Web of Science ®Google Scholar
• S. Atluri and T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation, Comp. Model. Simul. Eng. 3(3) (1998), pp. 187–196.
   Google Scholar
• S. Atluri and T. Zhu, New concepts in meshless methods, Comput. Mech. 22 (2000), pp. 117–127. doi: 10.1007/s004660050346
   Web of Science ®Google Scholar
• S. Atluri and T. Zhu, The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elsto-statics, Comput. Mech. 25 (2000), pp. 169–179. doi: 10.1007/s004660050467
   Web of Science ®Google Scholar
• J. Awrejcewicz, Numerical Analysis – Theory and Application, InTech, 2011.
   Google Scholar
• T. Belytschko, Y.Y. Lu, and L. Gu, Element-free Galerkin methods, Int. J. Numer. Methods Engrg. 37 (1994), pp. 229–256. doi: 10.1002/nme.1620370205
   Web of Science ®Google Scholar
• X.Y. Cui, H. Feng, G.Y. Li, and S.Z. Feng, A cell-based smoothed radial point interpolation method (CS-RPIM) for three-dimensional solids, Eng. Anal. with Boundary Elem. 50 (2015), pp. 474–485. doi: 10.1016/j.enganabound.2014.09.017
   Web of Science ®Google Scholar
• W. Dai, H. Song, S. Su, and R. Nassar, A stable finite difference scheme for solving a hyperbolic two-step model in a 3D micro sphere exposed to ultra short-pulsed lasers, Int. J. Numer. Meth. H 16 (2006), pp. 693–717. doi: 10.1108/09615530610679066
   Web of Science ®Google Scholar
• M. Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Meth. Partial Differ. Eq. 21 (2005), pp. 24–40. doi: 10.1002/num.20019
   Web of Science ®Google Scholar
• M. Dehghan and A. Ghesmati, Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation, Eng. Anal. With Boundary Elem. 34 (2010), pp. 324–336. doi: 10.1016/j.enganabound.2009.10.010
   Web of Science ®Google Scholar
• M. Dehghan and A. Ghesmati, Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM), Comput. Phys. Commun. 181 (2010), pp. 772–786. doi: 10.1016/j.cpc.2009.12.010
   Web of Science ®Google Scholar
• M. Dehghan and A. Mohebbi, The combination of collocation, finite difference, and multigrid methods for solution of the two-dimensional wave equation, Numer. Meth. Partial Differ. Eq. 24 (2008), pp. 897–910. doi: 10.1002/num.20295
   Web of Science ®Google Scholar
• M. Dehghan and A. Shokri, A numerical method for solution of the two dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Simul. 79 (2008), pp. 700–715. doi: 10.1016/j.matcom.2008.04.018
   Web of Science ®Google Scholar
• M. Dehghan and A. Shokri, A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions, Numer. Algor. 52 (2009), pp. 461–477. doi: 10.1007/s11075-009-9293-0
   Web of Science ®Google Scholar
• S.T. Grilli, P. Guyenneb, and F. Diasc, A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom, Int. J. Numer. Meth. Fluids 35 (2001), pp. 829–867. doi: 10.1002/1097-0363(20010415)35:7<829::AID-FLD115>3.0.CO;2-2
   Web of Science ®Google Scholar
• R.L. Hardy, Theory and applications of the multiquadrics – biharmonic method (20 years of discovery 1968–1988), Comput. Math. Appl. 19 (1990), pp. 163–208. doi: 10.1016/0898-1221(90)90272-L
   Web of Science ®Google Scholar
• A. Hirose, Introduction to Wave Phenomena, Wile, New York, NY, 1985.
   Google Scholar
• E. Kansa, Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. I. surface approximations and partial derivative estimates, Comput. Math. Appl. 19(8–9) (1990), pp. 127–145. doi: 10.1016/0898-1221(90)90270-T
   Web of Science ®Google Scholar
• F. Kremer, An existence and uniqueness theorem for wave equations with scalar nonlinearities, Appl. Math. Phys. 38(1) (1987), pp. 46–64. doi: 10.1007/BF00944920
   Web of Science ®Google Scholar
• K. Liew and R. Cheng, Numerical study of the three-dimensional wave equation using the meshfree kp-Ritz method, Eng. Anal. Boundary. Elem. 37 (2013), pp. 977–989. doi: 10.1016/j.enganabound.2013.04.001
   Web of Science ®Google Scholar
• G. Liu, Meshfree Methods: Moving Beyond the Finite Element Method, Taylor & Francis Group, Boca Raton, FL, 2010.
   Google Scholar
• G. Liu and Y. Gu, A truly meshless method based on the strong-weak form. Advances in Meshfree and X-FEM Methods, Proceeding of the 1st Asian Workshop in Meshfree Methods (Ed. Liu GR), Singapore, (2002), pp. 259–261.
   Google Scholar
• G. Liu and Y. Gu, A meshfree method: Meshfree weak-strong (MWS) form method, for 2-D solids, Comput. Mech. 33(1) (2003), pp. 2–14. doi: 10.1007/s00466-003-0477-5
   Web of Science ®Google Scholar
• G. Liu and Y. Gu, An Introduction to Meshfree Methods and their Programing, Berlin, Springer, 2005.
   Google Scholar
• G. Liu, L. Yan, J. Wang, and Y. Gu, Point interpolation method based on local residual formulation using radial basis functions, Struct. Eng. Mech. 14(6) (2002), pp. 713–732. doi: 10.12989/sem.2002.14.6.713
   Web of Science ®Google Scholar
• K. Masuda, A unique continuation theorem for solutions of wave equations with variable coefficients, Math. Anal. Appl. 21 (1968), pp. 369–376. doi: 10.1016/0022-247X(68)90219-9
   Web of Science ®Google Scholar
• R. Mohanty and V. Gopal, A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations, Appl. Math. Model. 37 (2013), pp. 2802–2815. doi: 10.1016/j.apm.2012.06.021
   Web of Science ®Google Scholar
• S. Nettel, Wave Physics: Oscillations-Solitons-Chaos, Springer, Berlin, 2003.
   Google Scholar
• E. Oñate, S. Idelsohn, O.Z. Zienkiewicz, and R.L. Taylor, A finite point method in computational mechanics. Applications to convective transport and fluid flow, Int. J. Numer. Methods Eng. 39 (1996), pp. 3839–3867. doi: 10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R
   Web of Science ®Google Scholar
• E. Oñate, F. Perazzo, and J. Miquel, A finite point method for elasticity problems, Comput. Struct. 79 (2001), pp. 2151–2163. doi: 10.1016/S0045-7949(01)00067-0
   Web of Science ®Google Scholar
• A. Selvadurai, Partial Differential Equations in Mechanics, Springer, Berlin, 2000.
   Google Scholar
• U. Senturk, Modeling nonlinear waves in a numerical wave tank with localized meshless RBF method, Comput. Fluids 44 (2011), pp. 221–228. doi: 10.1016/j.compfluid.2011.01.004
   Web of Science ®Google Scholar
• F. Shakeri and M. Dehghan, The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition, Comput. Math. Appl. 56 (2008), pp. 2175–2188. doi: 10.1016/j.camwa.2008.03.055
   Web of Science ®Google Scholar
• F. Shakeri and M. Dehghan, A hybrid Legendre tau method for the solution of a class of nonlinear wave equations with Nonlinear dissipative terms, Numer. Meth. Partial Differ. Eq. 27 (2011), pp. 1055–1071. doi: 10.1002/num.20569
   Web of Science ®Google Scholar
• D. Soares Jr, Time-domain electromagnetic wave propagation analysis by edge-based smoothed point interpolation methods, J. Comput. Phys. 234 (2013), pp. 472–486. doi: 10.1016/j.jcp.2012.10.009
   Web of Science ®Google Scholar
• J. Wang and G. Liu, Radial point interpolation method for elastoplastic problems. Proc. of the 1st Int. Conf. On Structural Stability and Dynamics, Dec. 7–9 (2000) Taipei, Taiwan (2000), pp. 703–708.
   Google Scholar
• J. Wang and G. Liu, A point interpolation meshless method based on radial basis functions, Int. J. Numer. Meth. Eng. 54(11) (2002), pp. 1623–1648. doi: 10.1002/nme.489
   Web of Science ®Google Scholar
• A. Wazwaz, Partial Differential Equations: Methods and Applications, Swets & Zeitlinger B.V, Lisse, Netherlands, 2002.
   Google Scholar
• Z. Yan and V.V. Konotop, Exact solutions to three-dimensional generalized nonlinear schrdinger equations with varying potential and nonlinearities, Phys. Rev. E 80 (2009), 036607, pp. 1–9.
   Web of Science ®Google Scholar