[1] |
Korteweg, D.; De Vries, G., On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave, Philos. Mag., 39, 422-443, 1895 · JFM 26.0881.02 |
[2] |
Zhang, J.; Wu, F., A simple method to construct soliton-like solution of the general KdV equation with external force, Commun. Nonlinear Sci. Numer. Simul., 5, 04, 170-173, 2000 · Zbl 0986.35102 |
[3] |
Flamarion, M. V., Complex flow structures beneath rotational depression solitary waves in gravity-capillary flows, Wave Motion, 117, Article 103108 pp., 2023 · Zbl 1524.76108 |
[4] |
Zhu, D.; Zhu, X., Exact multi-soliton solutions of the KdV equation with a source: Riemann-Hilbert formulation, Appl. Math. Lett., 149, Article 108919 pp., 2023 · Zbl 1529.35458 |
[5] |
Hou, S.; Zhang, R.; Zhang, Z.; Yang, L., On the Quartic Korteweg-de Vries hierarchy of nonlinear Rossby waves and its dynamics, Wave Motion, 124, Article 103249 pp., 2024 · Zbl 07825043 |
[6] |
Zhang, S., Exact solutions of a KdV equation with variable coefficients via Exp-function method, Nonlinear Dyn., 52, 1, 11-17, 2008 · Zbl 1173.35670 |
[7] |
Ganji, Z. Z.; Ganji, D. D.; Rostamiyan, Y., Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique, Appl. Math. Model., 33, 7, 3107-3113, 2009 · Zbl 1205.35251 |
[8] |
Liu, H.; Li, J.; Liu, L., Lie symmetry analysis, optimal systems and exact solutions to the fifth-order KdV types of equations, J. Math. Anal. Appl., 368, 2, 551-558, 2010 · Zbl 1192.35011 |
[9] |
Wang, G.; Kara, A. H.; Fakhar, K.; Vega-Guzman, J.; Biswas, A., Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation, Chaos Solitons Fract., 86, 8-15, 2016 · Zbl 1360.35232 |
[10] |
Selima, E. S.; Yao, X.; Wazwaz, A., Multiple and exact soliton solutions of the perturbed Korteweg-de Vries equation of long surface waves in a convective fluid via Painlevé analysis, factorization, and simplest equation methods, Phys. Rev. Lett., 95, 06, Article 062211 pp., 2017 |
[11] |
Kumar, V.; Kaur, L.; Kumar, A.; Koksal, M. E., Lie symmetry based-analytical and numerical approach for modified Burgers-KdV equation, Results Phys., 8, 1136-1142, 2018 |
[12] |
Wang, Z.; Liu, X., Bifurcations and exact traveling wave solutions for the KdV-like equation, Nonlinear Dyn., 95, 1, 465-477, 2019 · Zbl 1439.35435 |
[13] |
Huang, J.; Wang, M., New lower bounds on the radius of spatial analyticity for the KdV equation, J. Differ. Equ., 266, 9, 5278-5317, 2019 · Zbl 1412.35298 |
[14] |
Liu, J. G.; Yang, X. J.; Feng, Y. Y.; Cui, P., A new perspective to study the third-Order modified KdV equation on fractal set, Fractals, 28, 6, Article 2050110 pp., 2020 · Zbl 1445.28012 |
[15] |
Ismael, H. F.; Murad, M. A.; Bulut, H., Various exact wave solutions for KdV equation with time-variable coefficients, J. Ocean Eng. Sci., 7, 5, 409-418, 2021 |
[16] |
Karakoc, S. B.G.; Ali, K. K., New exact solutions and numerical approximations of the generalized kdv equation, Comput. Methods Differ. Equ., 9, 3, 670-691, 2021 · Zbl 1499.65668 |
[17] |
Boral, S.; Meylan, M. H.; Sahoo, T., Time-dependent wave propagation on a variable Winkler foundation with compression, Wave Motion, 106, Article 102792 pp., 2021 · Zbl 1524.74198 |
[18] |
Zhang, L.; Wang, J.; Shchepakina, E.; Sobolev, V., New type of solitary wave solution with coexisting crest and trough for a perturbed wave equation, Nonlinear Dyn., 106, 4, 3479-3493, 2021 |
[19] |
Shakeel, M.; Alaoui, M. K.; Zidan, A. M.; Shah, N. A., Closed form solutions for the generalized fifth-order KDV equation by using the modified exp-function method, J. Ocean Eng. Sci., 6, 37, 1-22, 2022 |
[20] |
Akbulut, A.; Kaplan, M.; Kaabar, M. K.A., New conservation laws and exact solutions of the special case of the fifth-order KdV equation, J. Ocean Eng. Sci., 7, 4, 377-382, 2022 |
[21] |
Cai, N.; Qiao, Z.; Zhou, Y., Wave solutions to an integrable negative order KdV equation, Wave Motion, 116, Article 103072 pp., 2023 · Zbl 1524.35518 |
[22] |
Vliegenthart, A. C., On finite-difference methods for the Korteweg-de Vries equation, J. Eng. Math., 5, 2, 137-155, 1971 · Zbl 0221.76003 |
[23] |
Zheng, Y. G.; Liu, Z. R.; Huang, D. B., Discrete soliton-like for KdV prototypes, Chaos Solit. Fract., 14, 7, 989-994, 2002 · Zbl 1038.35108 |
[24] |
Başhan, A., A mixed algorithm for numerical computation of soliton solutions of the coupled KdV equation: Finite difference method and differential quadrature method, Appl. Math. Comput., 360, 42-57, 2019 · Zbl 1429.65234 |
[25] |
Geyikli, T.; Kaya, D., Comparison of the solutions obtained by B-spline FEM and ADM of KdV equation, Appl. Math. Comput., 169, 1, 146-156, 2005 · Zbl 1121.65353 |
[26] |
Zaki, S. I., A quintic B-spline finite elements scheme for the KdVB equation, Comput. Meth. Appl. Mech. Eng., 188, 1-3, 121-134, 2000 · Zbl 0957.65088 |
[27] |
Karczewska, A.; Rozmej, P.; Szczeciński, M.; Boguniewicz, B., Finite element method for extended KdV equations, Int. J. Appl. Math. Comput. Sci., 26, 3, 555-567, 2016 · Zbl 1347.65153 |
[28] |
Bai, D.; Zhang, L., The finite element method for the coupled Schrödinger-KdV equations, Phys. Lett. A, 373, 26, 2237-2244, 2009 · Zbl 1231.35224 |
[29] |
Liu, H.; Yi, N., A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation, J. Comput. Phys., 321, 776-796, 2016 · Zbl 1349.65462 |
[30] |
Yi, N.; Huang, Y.; Liu, H., A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: energy conservation and boundary effect, J. Comput. Phys., 242, 351-366, 2013 · Zbl 1297.65122 |
[31] |
Boyd, J. P., Trouble with Gegenbauer reconstruction for defeating Gibbs’ phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations, J. Comput. Phys., 204, 1, 253-264, 2005 · Zbl 1071.65189 |
[32] |
O. Dubrule, C. Kostov, An interpolation method taking into account inequality constraints: I. Methodology, Math. Geol. 18 (1) (1986) 33-51. https://doi.org/0882 8121/86/0100-0033505.00/0. |
[33] |
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 · Zbl 0247.65061 |
[34] |
Başhan, A., An effective application of differential quadrature method based on modified cubic B-splines to numerical solutions of the KdV equation, Turk. J. Math., 42, 373-394, 2018 · Zbl 1424.65179 |
[35] |
Başhan, A.; Yağmurlu, N.; Uçar, Y.; Esen, A., A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method, Int. J. Mod. Phys. C, 2018 |
[36] |
Başhan, A., An efficient approximation to numerical solutions for the Kawahara equation via modified cubic B-Spline differential quadrature method, Mediterr. J. Math., 16, 2019 · Zbl 1530.65136 |
[37] |
Başhan, A., A novel approach via mixed Crank-Nicolson scheme and differential quadrature method for numerical solutions of solitons of mKdV equation, Pramana - J. Phys., 92, 6, 84, 2019 |
[38] |
Başhan, A., An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods, Comput. Appl. Math., 39, 2, 80, 2020 · Zbl 1449.65296 |
[39] |
Başhan, A., Bell-shaped soliton solutions and travelling wave solutions of the fifth-order nonlinear modified Kawahara equation, Int. J. Nonlinear Sci. Numer. Simul., 22, 6, 781-795, 2021 · Zbl 07486822 |
[40] |
Başhan, A., Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods, Math. Comput. Simul., 179, 111-125, 2021 · Zbl 1524.65314 |
[41] |
Başhan, A., Modification of quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments, Appl. Numer. Math., 167, 356-374, 2021 · Zbl 1466.41005 |
[42] |
Başhan, A., A novel outlook to the mKdV equation using the advantages of a mixed method, Appl. Anal., 102, 1, 65-87, 2023 · Zbl 1509.65021 |
[43] |
Krige, D. G., A statistical approach to some basic mine valuation problems on the Wit-watersrand, J. S. Afr. Inst. Min. Metall., 52, 6, 119-139, 1951 |
[44] |
Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76, 8, 1905-1915, 1971 |
[45] |
Duchon, J., Splines minimizing rotation-invariant semi-norms in Sobolev spaces, Const. Theory Funct. Sev. Variabl.: Proc. Conf. Held Oberwolfach, 85-100, 1977 · Zbl 0342.41012 |
[46] |
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, 8-9, 147-161, 1990 · Zbl 0850.76048 |
[47] |
Wang, L., Radial basis functions methods for boundary value problems: performance comparison, Eng. Anal. Bound. Elem., 84, 191-205, 2017 · Zbl 1403.65179 |
[48] |
Wang, L.; Wang, Z.; Qian, Z., A meshfree method for inverse wave propagation using collocation and radial basis functions, Comput. Meth. Appl. Mech. Eng., 322, 1, 311-350, 2017 · Zbl 1439.74492 |
[49] |
Yang, J. P.; Chen, Y. C., Gradient enhanced localized radial basis collocation method for inverse analysis of cauchy problems, Int. J. Appl. Mech., 12, 09, Article 2050107 pp., 2020 |
[50] |
Wang, L.; Qian, Z.; Zhou, Y.; Peng, Y., A weighted meshfree collocation method for incompressible flows using radial basis functions, Int. J. Mech. Sci., 401, Article 108964 pp., 2020 · Zbl 1453.65364 |
[51] |
Wang, L.; Liu, Y.; Zhou, Y.; Yang, F., Static and dynamic analysis of thin functionally graded shell with in-plane material inhomogeneity, Int. J. Mech. Sci., 193, Article 106165 pp., 2021 |
[52] |
Hosseini, S.; Rahimi, G., Nonlinear bending analysis of hyperelastic plates using FSDT and meshless collocation method based on radial basis function, Int. J. Appl. Mech., 13, 01, Article 2150007 pp., 2021 |
[53] |
Kumar, S.; Jiwari, R.; Mittal, R. C., Radial basis functions based meshfree schemes for the simulation of non-linear extended Fisher-Kolmogorov model, Wave Motion, 109, Article 102863 pp., 2022 · Zbl 1525.65105 |
[54] |
Dehghan, M.; Shokri, A., A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dyn., 50, 1, 111-120, 2007 · Zbl 1185.76832 |
[55] |
Seydaoğlu, M., A meshless two-stage scheme for the fifth-order dispersive models in the science of waves on water, Ocean Eng., 250, Article 111014 pp., 2022 |
[56] |
Dağ, İ.; Dereli, Y., Numerical solutions of KdV equation using radial basis functions, Appl. Math. Model., 32, 4, 535-546, 2008 · Zbl 1132.65096 |
[57] |
Shen, Q., A meshless method of lines for the numerical solution of KdV equation using radial basis functions, Eng. Anal. Bound. Elem., 33, 10, 1171-1180, 2009 · Zbl 1253.76101 |
[58] |
Mohyud-Din, S. T.; Negahdary, E.; Usman, M., A meshless numerical solution of the family of generalized fifth-order Korteweg-de Vries equations, Int. J. Numer. Methods Heat Fluid Flow, 22, 5, 641-658, 2012 · Zbl 1357.65264 |
[59] |
Uddin, M.; Haq, S., Numerical solution of complex modified Korteweg-de Vries equation by mesh-free collocation method, Comput. Math. Appl., 58, 3, 566-578, 2009 · Zbl 1189.65239 |
[60] |
Kaya, D.; Gülbahar, S.; Yokuş, A.; Gülbahar, M., Solutions of the fractional combined KdV-mKdV equation with collocation method using radial basis function and their geometrical obstructions, Adv. Differ. Equ., 2018, 1, 1-16, 2018 · Zbl 1445.65033 |
[61] |
Sagar, B.; Ray, S. Saha, Numerical solution of fractional Kersten-Krasil’shchik coupled KdV-mKdV system arising in shallow water waves, Comput. Appl. Math., 41, 6, 1-24, 2022 · Zbl 1513.35137 |
[62] |
Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math., 11, 2, 193-210, 1999 · Zbl 0943.65017 |
[63] |
Gherlone, M.; Iurlaro, L.; Di Sciuva, M., A novel algorithm for shape parameter selection in radial basis functions collocation method, Compos. Struct., 94, 2, 453-461, 2012 |
[64] |
Wang, L.; Chu, F.; Zhong, Z., Study of radial basis collocation method for wave propagation, Eng. Anal. Bound. Elem., 37, 2, 453-463, 2013 · Zbl 1352.65402 |
[65] |
Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, International Journal for Numerical Methods in Fluids Int. J. Numer. Methods Fluids, 20, 8-9, 1081-1106, 1995 · Zbl 0881.76072 |
[66] |
Liu, W. K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. Numer. Methods Eng., 38, 10, 1655-1679, 1995 · Zbl 0840.73078 |
[67] |
Liu, W. K.; Li, S.; Belytschko, T., Moving least-square reproducing kernel methods (I) methodology and convergence, Comput. Meth. Appl. Mech. Eng., 143, 1-2, 113-154, 1997 · Zbl 0883.65088 |
[68] |
Li, S.; Liu, W. K., Moving least square reproducing kernel method part II: Fourier analysis, Comput. Meth. Appl. Mech. Eng., 139, 159-194, 1996 · Zbl 0883.65089 |
[69] |
Li, S.; Liu, W. K., Synchronized reproducing kernel interpolant via multiple wavelet expansion, Comput. Mech., 21, 28-47, 1998, 50281 · Zbl 0912.76057 |
[70] |
Wang, D.; Wang, J.; Wu, J., Arbitrary order recursive formulation of meshfree gradients with application to superconvergent collocation analysis of Kirchhoff plates, Comput. Mech., 65, 3, 877-903, 2020 · Zbl 1477.74123 |
[71] |
Deng, L.; Wang, D.; Qi, D., A least squares recursive gradient meshfree collocation method for superconvergent structural vibration analysis, Comput. Mech., 68, 5, 1063-1096, 2021 · Zbl 1479.74134 |
[72] |
Chen, J. S.; Wu, C. T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods, Int. J. Numer. Methods Eng., 50, 2, 435-466, 2001, 10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A · Zbl 1011.74081 |
[73] |
Chen, J. S.; Zhang, X.; Belytschko, T., An implicit gradient model by a reproducing kernel strain regularization in strain localization problems, Comput. Meth. Appl. Mech. Eng., 193, 27-29, 2827-2844, 2004 · Zbl 1067.74564 |
[74] |
Li, S.; Liu, W. K., Reproducing kernel hierarchical partition of unity Part I: formulation and theory, Int. J. Numer. Methods Eng., 45, 251-288, 1999, 10.1002/(SICI)1097-0207(19990530)45:3<251::AID-NME583>3.0.CO;2-I |
[75] |
Li, S.; Liu, W. K., Reproducing kernel hierarchical partition of unity Part II: Applications, Int. J. Numer. Methods Eng., 45, 1999, 10.1002/(SICI)1097-0207(19990530)45:3<289::AID-NME584>3.0.CO;2-P |
[76] |
Chi, S. W.; Chen, J. S.; Hu, H. Y.; Yang, J. P., A gradient reproducing kernel collocation method for boundary value problems, Int. J. Numer. Methods Eng., 93, 13, 1381-1402, 2013 · Zbl 1352.65562 |
[77] |
Mahdavi, A.; Chi, S.; Zhu, H., A Gradient reproducing kernel collocation method for high order differential equations, Comput. Mech., 64, 5, 1421-1454, 2019 · Zbl 1470.74074 |
[78] |
Wang, L.; Liu, Y.; Zhou, Y.; Yang, F., A gradient reproducing kernel based stabilized collocation method for the static and dynamic problems of thin elastic beams and plates, Comput. Mech., 68, 4, 709-739, 2021 · Zbl 1478.74087 |
[79] |
Liu, Y.; Wang, L.; Zhou, Y.; Yang, F., A stabilized collocation method based on the efficient gradient reproducing kernel approximations for the boundary value problems, Eng. Anal. Bound. Elem., 132, 446-459, 2021 · Zbl 1521.65130 |
[80] |
Tan, M.; Cheng, J.; Shu, C., Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations, J. Comput. Phys., 464, Article 1111314 pp., 2022 · Zbl 07540358 |
[81] |
Habibe, R. A.; Mohammad, K.; Mohammadreza, Y., Leave-Two-Out Cross Validation to optimal shape parameter in radial basis functions, Eng. Anal. Bound. Elem., 100, 204-210, 2019 · Zbl 1464.65011 |
[82] |
Qian, Z.; Wang, L.; Gu, Y.; Zhang, C., An efficient meshfree gradient smoothing collocation method (GSCM) using reproducing kernel approximation, Comput. Meth. Appl. Mech. Eng., 374, Article 113573 pp., 2021 · Zbl 1506.74501 |
[83] |
Wang, L.; Qian, Z., A meshfree stabilized collocation method (SCM) based on reproducing kernel approximation, Comput. Meth. Appl. Mech. Eng., 371, Article 113303 pp., 2020 · Zbl 1506.65237 |
[84] |
Wang, L.; Hu, M.; Zhong, Z.; Yang, F., Stabilized Lagrange interpolation collocation method: A meshfree method incorporating the advantages of finite element method, Comput. Meth. Appl. Mech. Eng., 404, Article 115780 pp., 2023 · Zbl 1536.65156 |
[85] |
Q. Tao, L. Ji, J.K. Ryan, Y. Xu, Accuracy-enhancement of discontinuous Galerkin methods for PDEs containing high order spatial derivatives, J. Sci. Comput. 93 (22) (2022). 10.1007/s10915-022-01967-9. · Zbl 1497.65183 |
[86] |
Rubin, S. G.; Graves, R. A., Viscous flow solutions with a cubic spline approximation, Comput. Fluids, 3, 1, 1-36, 1975 · Zbl 0347.76020 |
[87] |
Pandit, S., Local radial basis functions and scale-3 Haar wavelets operational matrices based numerical algorithms for generalized regularized long wave model, Wave Motion, 109, Article 102846 pp., 2022 · Zbl 1524.65675 |
[88] |
Salas, A., Some solutions for a type of generalized Sawada-Kotera equation, Appl. Math. Comput., 196, 2, 812-817, 2008 · Zbl 1132.35461 |
[89] |
Zhao, F.; Wang, X.; Jie, O., An improved element-free Galerkin method for solving the generalized fifth-order Korteweg-de Vries equation, Chin. Phys. B, 22, Article 074704 pp., 2013 |
[90] |
A. Parker, On soliton solutions of the Kaup-Kupershmidt equation. I. Direct bilinearisation and solitary wave, Physica D. 137 (1-2) (2000) 25-33. 10.1016/S0167-2789(99)00166-9. · Zbl 0943.35088 |
[91] |
Shen, J., A new Dual-Petrov-Galerkin method for third and higher odd-order differential equations: application to the KDV equation, SIAM J. Num. Anal., 41, 1595-1619, 2003 · Zbl 1053.65085 |
[92] |
Yuan, J.; Wu, J., A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations, Discret. Contin. Dyn. Syst., 26, 4, 1525-1536, 2010 · Zbl 1184.35287 |
[93] |
Zhou, D., Non-homogeneous initial-boundary-value problem of the fifth-order Korteweg-de Vries equation with a nonlinear dispersive term, J. Math. Anal. Appl., 491, 1, Article 124848 pp., 2021 · Zbl 1462.35343 |
[94] |
Wang, X.; Dai, W.; Usman, M., A high-order accurate finite difference scheme for the KdV equation with time-periodic boundary forcing, Appl. Numer. Math., 160, 102-121, 2021 · Zbl 1459.76094 |
[95] |
Flamarion, M. V.; Ribeiro-Jr, R., Gravity-capillary flows over obstacles for the fifth-order forced Korteweg-de Vries equation, J. Eng. Math., 129, 17, 2021 · Zbl 1497.76020 |