A One-Step Block Hybrid Integrator for Solving Fifth Order Korteweg-de Vries Equations
Keywords:Korteweg-de Vries (KdV) equations, Fifth-order PDE, Linear multistep, Block Method, Convergence
Fifth-order Korteweg-de Vries (KdV) equations, arise in modeling waves phenomena such as the propagation of shallow water waves over a flat surface, gravity-capillary waves and sound waves in plasmas. In this work, a one-step block hybrid linear multistep method was derived using the collocation technique, to solve fifth-order KdV models via the Method of Line (MoL). The consistency, stability and convergence of the method were established. The efficiency of the method can be seen from comparison of the exact solutions of problems and other methods cited from literature.
M. Kazeminia, S. Soleimani-Amiri & S.A. Zahedi, “Exact and numerical solutions for nonlinear higher order modified KdV equations by using variational iteration method”, Adv. Stud. Theor. Phys. 4 (2010) 437.
G. Amit, J. Singh & D. Kumar, “A reliable algorithm for KdV equations arising in warm plasma”, Nonlin. Eng. 5 (1) (2016) 7. DOI: https://doi.org/10.1515/nleng-2015-0024
D. Kumar, J. Singh & S. Kumar, “A fractional model of Navier-Stokes equation arising in unsteady flow of a viscous fluid“, J. Assoc. Arab Univ. Basic Appl. Sci. 17 (2015) 14. DOI: https://doi.org/10.1016/j.jaubas.2014.01.001
G. Amit, J. Singh & D. Kumar, “Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves”, Ain. Shams. Eng. J (2017), http://dx.doi.org/10.1016/j.asej.2017.03.004. DOI: https://doi.org/10.1016/j.asej.2017.03.004
T. Kakutani. & H. Ono, “Weak non-linear hydromagnetic waves in a cold collision free plasma”, J. Phys. Soc. Jpn. 26 (130) (1969) 5. DOI: https://doi.org/10.1143/JPSJ.26.1305
P. Saucez, A. Vande Wouwer, W.E. Schiesser & P. Zegeling, “Method of lines study of nonlinear dispersive waves”, Journal of Computational and Applied Mathematics 168 (2004) 413. DOI: https://doi.org/10.1016/j.cam.2003.12.012
P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Commun. Pure Appl. Math. 21 (1968) 467. DOI: https://doi.org/10.1002/cpa.3160210503
P. Caudrey, R. Dodd, & J. Gibbon, “A new hierarchy of Korteweg-de Vries equations”, Proc. R. Soc. Lond. A Math. Phys. Sci. 351 (1976) 407. DOI: https://doi.org/10.1098/rspa.1976.0149
S. Handibag & B. Karande, “Existence the solutions of some fifth-order KdV equation by Laplace decomposition method”, Am. J. Comput. Math. 3 (2013) 80, https://doi.org/10.4236/ajcm.2013.31013. DOI: https://doi.org/10.4236/ajcm.2013.31013
K. Sawada & T. Kotera, “A method for finding n-soliton solutions of the KdV equation and KdV-like equation”, Prog. Theor. Phys. 51 (5) (1355). DOI: https://doi.org/10.1143/PTP.51.1355
H. Ahmad, T. A. Khan, P. S. Stanimirovic & I. Ahmad, “Modified Variational Iteration Technique for the Numerical Solution of Fifth Order KdV-type Equations”, J. Appl. Comput. Mech. 6 (2020) 1220, https://doi.org/10.22055/JACM.2020.33305.2197
H. Ahmad, T. A. Khan, & Shao-Wen Yao, “An efficient approach for the numerical, solution of fifth-order KdV equations”, De Gruyter Open Mathematics 18 (2020) 738. DOI: https://doi.org/10.1515/math-2020-0036
Kawahara, T., “Oscillatory solitary waves in dispersive media”, Journal of the Physical Society of Japan 33 (1972) 260. DOI: https://doi.org/10.1143/JPSJ.33.260
K. A. Gorshkov, L. A. Ostrovsky, V. V. Papko & A. S. Pikovsky, “On the existence of stationary multisolitons”, Physics Letters 74 (1979) 177. DOI: https://doi.org/10.1016/0375-9601(79)90763-1
Y. Yamamoto, & E. I. Takizawa, “On a solution on non-linear timeevolution equation of fifth order”, Journal of the Physical Society of Japan 50 (1981) 1421. DOI: https://doi.org/10.1143/JPSJ.50.1421
M. O. Miansari, M. E. Miansari, A. Barari & D. D. Ganji, “Application of He’s Variational Iteration Method to nonlinear Helmholtz and fifth-order KdV equations”, J. Appl. Math. Stat. Inform. 5 (2009) 5.
S. Abbasbandy & F. S. Zakaria, “Soliton solutions for the fifth-order KdV equation with the homotopy analysis method”, Nonlinear Dyn. 2008 (2008) 83. DOI: https://doi.org/10.1007/s11071-006-9193-y
G. Wang, A. H. Kara, K. Fakhar, J. V. Guzman & A. Biswas, “Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation”, Chaos, Solitons Fractals 86 (2016) 8. DOI: https://doi.org/10.1016/j.chaos.2016.02.013
T. A. Biala & S. N. Jator, “Block Unification Algorithm for 2D and 3D Elliptic PDES”, Journal of the Nig. Math. Soc. 36 (2017) 319.
M. I. Modebei, R. B. Adeniyi & S. N. Jator, “Numerical Approximations of Fourth-order PDEs Using Block Unification Method”, Journal of the Nigerian Mathematical Society 39 (2020) 47.
M. I. Modebei & R. B. Adeniyi, “A six-step Block Unification Integrator for Numerical Solution of Fourth Order Boundary Value Problems”, General Letters in Mathematics 5 (2018) 71. DOI: https://doi.org/10.31559/glm2018.5.2.2
O. O. Olaiya, R. A. Azeez & M. I. Modebei, “Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines”, Journal of the Nigerian Society of Physical Sciences 3 (2021) 26. DOI: https://doi.org/10.46481/jnsps.2021.140
M. I. Modebei, O. O. Olaiya & I. P. Ngwongwo, “Computational study of some three-step hybrid integrators forsolution of third order ordinary differential equations”, Journal of the Nigerian Society of Physical Sciences 3 (2021) 459. DOI: https://doi.org/10.46481/jnsps.2021.323
M. I. Modebei, “Optimized hybrid block integrator for Numerical solution of general third order Ordinary di erential equations”, Journal of the Nigerian MAthematical Society 41 (1) (2022) 49.
M. I. Modebei, O. O. Olaiya&A. C. Onyekonwu, “A 3-step fourth derivatives method for numerical integration of third order ordinary differential equations”, International Journal of Mathematical Analysis and Optimization: Theory and Applications 7 (2021) 32.
J. Sunday, G. M. Kumleng, N. M. Kamoh, J. A. Kwanamu, Y. Skwame & O. Sarjiyus, “Implicit Four-Point Hybrid Block Integrator for the Simulations of Stiff Models”, Journal of the Nigerian Society of Physical Sciences 4 (2022) 287. DOI: https://doi.org/10.46481/jnsps.2022.777
V. O. Atabo & S. O. Adee, “A New Special 15-Step Block Method for Solving General Fourth Order Ordinary Di erential Equations”, Journal of the Nigerian Society of Physical Sciences 3 (2021) 308. DOI: https://doi.org/10.46481/jnsps.2021.337
M. Kida, S. Adamu, O. O. Aduroja & T. P. Pantuvo, “Numerical Solution of Stiff and Oscillatory Problems using Third Derivative Trigonometrically Fitted Block Method”, Journal of the Nigerian Society of Physical Sciences 4 (2022) 34. DOI: https://doi.org/10.46481/jnsps.2022.271
J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley, New York (1973).
P. Henrici, Discrete Variable Methods for ODEs, John Wiley, New York, USA (1962).
M. A. Rufai & H. Ramos, “Numerical solution of second-order singular problems arising in astrophysics by combining a pair of one-step hybrid block Nystr¨om methods”, Astrophys Space Sci. 365 (2020) 96, https://doi.org/10.1007/s10509-020-03811-8 DOI: https://doi.org/10.1007/s10509-020-03811-8
Y. Cheng & C Shu, “A Discontinuous Galerkin Finite Element Method for Time Dependent Partial Differential Equations with Higher Order Derivatives”, Mathematics of Computation 77 (2008) 699. DOI: https://doi.org/10.1090/S0025-5718-07-02045-5
S. T. Mohyud-Din, E. Negahdary & M. Usman, “A Meshless Numerical Solution of the Family of Generalized Fifth-order Korteweg-de Vries Equations”, Mathematics Faculty Publications Paper 10 (2012), http://ecommons:udayton:edu=mthf acpub=10.
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