zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A meshless method for solving mKdV equation. (English) Zbl 1282.65123
Summary: A meshless method based on a global collocation with radial basis functions for the numerical solution of the modified Korteweg-de Vries (mKdV) equation is presented. Standard types of radial basis functions are applied in the method of collocation. The stability analysis of the method is dealt with using a linearized stability analysis. The method’s accuracy and efficiency are examined by the simulation of a single soliton and interaction of two solitary waves. The four invariants of the motion are evaluated to determine the conservation properties of the method. A comparison with some earlier reported results is also carried out.

MSC:
65M70Spectral, collocation and related methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
WorldCat.org
Full Text: DOI
References:
[1] A.H.A. Ali, Finite element studies of the Korteweg-de Vries equation, Ph.D. thesis, University of Wales, UK, 1989.
[2] Dağ, .I.; Dereli, Y.: Numerical solutions of KdV equation using radial basis functions, Appl. math. Model. 32, No. 4, 535-546 (2008) · Zbl 1132.65096 · doi:10.1016/j.apm.2007.02.001
[3] Dereli, Y.; Irk, D.; Dağ, .I.: Soliton solutions for NLS equation using radial basis functions, Chaos solitons fractals 42, No. 2, 1227-1233 (2009) · Zbl 1198.81035 · doi:10.1016/j.chaos.2009.03.030
[4] Djidjeli, K.; Price, W. G.; Twizell, E. H.; Cao, Q.: A linearized implicit pseudo-spectral method for some model equations: the regularized long wave equations, Commun. numer. Meth. eng. 19, 847-863 (2003) · Zbl 1035.65110 · doi:10.1002/cnm.635
[5] Drazin, P. G.; Johnson, R. S.: Solitons: an introduction, (1989) · Zbl 0661.35001
[6] E. Fermi, J. Pasta, S. Ulam, Studies of nonlinear problems, I. Los Alamos Report, Los Alamos, NM, 1940. · Zbl 0353.70028
[7] Gardner, G. A.; Ali, A. H. A.; Gardner, L. R. T.: Solutions for the modified Korteweg-de Vries equation, Numerical methods in engineering, vol. 1 1, 590-597 (1990)
[8] Gardner, G. A.; Gardner, L. R. T.; Ali, A. H. A.: A collocation solution for Burgers equation using cubic B-spline finite elements, Comput. methods appl. Mech. eng. 100, 325-337 (1992) · Zbl 0762.65072 · doi:10.1016/0045-7825(92)90088-2
[9] Gardner, L. R. T.; Gardner, G. A.; Geyikli, T.: Solitary wave solutions of the mkdv- equation, Comput. methods appl. Mech. eng. 124, 321-333 (1995) · Zbl 0945.65520 · doi:10.1016/0045-7825(94)00755-C
[10] L.R.T. Gardner, G.A. Gardner, T. Geyikli, U.C.N.W. Maths Preprint 28, 1991.
[11] Gardner, L. R. T.; Gardner, G. A.; Geyikli, T.: The boundary forced mkdv equation, J. comput. Phys. 113, 5-12 (1994) · Zbl 0806.65121 · doi:10.1006/jcph.1994.1113
[12] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, M. R.: Method for solving the Korteweg-de Vries equation, Phys. lett. A 19, 1095-1097 (1967) · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[13] Hon, Y. C.; Mao, X. Z.: An efficient numerical scheme for Burgers equation, Appl. math. Comput. 95, 37-50 (1998) · Zbl 0943.65101 · doi:10.1016/S0096-3003(97)10060-1
[14] Hon, Y. C.; Schaback, R.: On unsymmetric collocation by radial basis functions, Appl. math. Comput. 119, 177-186 (2001) · Zbl 1026.65107 · doi:10.1016/S0096-3003(99)00255-6
[15] Inc, M.: Exact special solutions to the nonlinear dispersive $K(2,2,1)$ and $K(3,3,1)$ equations by he’s variational iteration method, Nonlinear anal. Theor. meth. Appl. 69, 624-631 (2008) · Zbl 1159.35413 · doi:10.1016/j.na.2007.05.046
[16] Kansa, E. J.: Multiquadrics, a scattered data approximation scheme with applications to computational fluid-dynamics, I - surface approximations and partial derivative estimates, Comput. math. Appl. 19, 127-145 (1990) · Zbl 0692.76003 · doi:10.1016/0898-1221(90)90270-T
[17] Kansa, E. J.: Multiquadrics, a scattered data approximation scheme with applications to computational fluid-dynamics, II - solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. math. Appl. 19, 147-161 (1990) · Zbl 0850.76048 · doi:10.1016/0898-1221(90)90271-K
[18] Karageorghis, A.; Chen, C. S.; Smyrlis, Y. S.: A matrix decomposition RBF algorithm: approximation of functions and their derivatives, Appl. numer. Math. 57, 304-319 (2007) · Zbl 1107.65305 · doi:10.1016/j.apnum.2006.03.028
[19] Khattak, A. J.; Tirmizi, S. I. A.; Siraj-Ul-Islam: Application of meshfree collocation method to a class of nonlinear partial differential equations, Eng. anal. Bound. elem. 33, No. 5, 661-667 (2009) · Zbl 1244.65149
[20] Li, J.; Cheng, A. H. D.; Chen, C. S.: On the efficiency and exponential convergence of multiquadric collocation method compared to finite element method, Eng. anal. Bound. elem. 27, 251-257 (2003) · Zbl 1044.76050 · doi:10.1016/S0955-7997(02)00081-4
[21] Micchelli, C. A.: Interpolation of scattered data: distance matrix and conditionally positive functions, Construct. approx. 2, 11-22 (1986) · Zbl 0625.41005 · doi:10.1007/BF01893414
[22] Miura, R. M.: Korteweg-de Vries equation and generalizations, I - A remarkable explicit nonlinear transformation, J. math. Phys. 9, 1202-1204 (1968) · Zbl 0283.35018 · doi:10.1063/1.1664700
[23] Salahuddin, M.: Ion temperature effect on the propagation of ion acoustic solitary waves in a relativistic magnetoplasma, Plasma phys. Control. fusion 32, 33-41 (1990)
[24] Siraj-Ul-Islam; Khattak, A. J.; Tirmizi, Ikram A.: A meshfree method for numerical solution of KdV equation, Eng. anal. Bound. elem. 32, No. 10, 849-855 (2008) · Zbl 1244.76087
[25] Siraj-Ul-Islam; Haq, Sirajul; Ali, Arshed: A meshfree method for the numerical solution of the RLW equation, J. comput. Appl. math. 223, 997-1012 (2009) · Zbl 1156.65090 · doi:10.1016/j.cam.2008.03.039
[26] Siraj-Ul-Islam; Haq, Sirajul; Uddin, Marjan: A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations, Eng. anal. Bound. elem. 33, No. 3, 399-409 (2009) · Zbl 1244.65193
[27] Su, C. H.; Gardner, C. S.: Korteweg-de Vries equation and generalizations, III - derivation of the Korteweg-de Vries equation and Burgers equation, J. math. Phys. 10, No. 3, 536-539 (1969) · Zbl 0283.35020 · doi:10.1063/1.1664873
[28] Taha, T. R.; Ablowitz, M. J.: Analytical and numerical aspects of certain nonlinear evolution equations IV. Numerical, modified Korteweg-de Vries equation, J. comput. Phys. 77, 540-548 (1988) · Zbl 0646.65087 · doi:10.1016/0021-9991(88)90184-2
[29] Wazwaz, A. M.: Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos solitons fractals 28, 457-462 (2006) · Zbl 1084.35079 · doi:10.1016/j.chaos.2005.06.004