THERMAL SCIENCE

International Scientific Journal

ON EXACT AND APPROXIMATE SOLUTIONS OF (2+1)-DIMENSIONAL KONOPELCHENKO-DUBROVSKY EQUATION VIA MODIFIED SIMPLEST EQUATION AND CUBIC B-SPLINE SCHEMES

ABSTRACT
This paper studies the analytical and numerical solutions for (2+1)- dimensional Konopelchenko-Dubrovsky equation. It also examines the performance of the modified simplest equation method and the cubic B-spline scheme on this model. Many explicit wave solutions are found by using the analytical technique. These solutions allow studying the physical properties of this model. The comparison between the analytical and numerical solutions are discussed to show which one of cubic B-spline scheme families is more accurate in finding the numerical solutions of this model.
KEYWORDS
PAPER SUBMITTED: 2019-01-31
PAPER REVISED: 2019-06-20
PAPER ACCEPTED: 2019-07-05
PUBLISHED ONLINE: 2019-09-15
DOI REFERENCE: https://doi.org/10.2298/TSCI190131349A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 6, PAGES [S1889 - S1899]
REFERENCES
  1. Aliyu, A.I., Inc, M., Yusuf, A., Bayram, M., Baleanu, D. "Symmetry reductions, explicit solutions, convergence analysis and conservation laws via multipliers approach to the Chen-Lee-Liu model in nonlinear optics." Modern Physics Letters B 33, no. 04 (2019): 1950035.
  2. Osman, M. S. "On complex wave solutions governed by the 2D Ginzburg-Landau equation with variable coefficients." Optik 156 (2018), pp.169-174.
  3. Osman, M. S. "One-soliton shaping and inelastic collision between double solitons in the fifthorder variable-coefficient Sawada-Kotera equation." Nonlinear Dynamics (2019), pp. 1-6.
  4. Rezazadeh, Hadi, Hira Tariq, Mostafa Eslami, Mohammad Mirzazadeh, and Qin Zhou. "New exact solutions of nonlinear conformable time-fractional Phi-4 equation." Chinese Journal of Physics 56, no. 6 (2018), pp. 2805-2816.
  5. Biswas, A, Rezazadeh, H., Mirzazadeh, M., Eslami, M., Ekici, M., Zhou, Q., Moshokoa, S.P., Belic, M. "Optical soliton perturbation with Fokas-Lenells equation using three exotic and efficient integration schemes." Optik 165 (2018), pp.288-294.
  6. Osman, M.S., Korkmaz, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Zhou, Q., "The unified method for conformable time fractional Schro¨ dinger equation with perturbation terms." Chinese Journal of Physics 56 (2018), pp.2500-2506..
  7. Rezazadeh, H., Mirzazadeh, M., Mirhosseini-Alizamini, S.M., Neirameh, A., Eslami, M., Zhou, Q. "Optical solitons of Lakshmanan-Porsezian-Daniel model with a couple of nonlinearities." Optik 164 (2018), pp.414-423.
  8. Osman, M.S., Abdel-Gawad, H.I., El Mahdy, M.A. "Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion." Results in physics 8 (2018), pp.1054-1060.
  9. Rezazadeh, H., Mirhosseini-Alizamini, S.M., Eslami, M., Rezazadeh, M., Mirzazadeh, M., Abbagari, S. "New optical solitons of nonlinear conformable fractional Schrödinger-Hirota equation." Optik 172 (2018), pp.545-553.
  10. Hereman, W., Takaoka, M. "Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA." Journal of Physics A: Mathematical and General 23 (1990), pp.4805.
  11. Aliyu, A.I., Tchier, F., Inc, M., Yusuf, A., Baleanu, D. "Dynamics of optical solitons, multipliers and conservation laws to the nonlinear schrödinger equation in (2+ 1)-dimensions with non-Kerr law nonlinearity." Journal of Modern Optics 66 (2019), pp.136-142.
  12. Aliyu, A.I., Inc, M., Yusuf, A., Baleanu, D. "Optical solitons and stability analysis with spatiotemporal dispersion in Kerr and quadric-cubic nonlinear media." Optik 178 (2019), pp.923-931.
  13. Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D., Bayram, M. "Dark-Bright Optical Soliton and Conserved Vectors to the Biswas-Arshed Equation with Third-Order Dispersions in the Absence of Self-Phase Modulation." Frontiers in Physics 7 (2019): 28.
  14. Ebaid, A. "Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method." Physics Letters A 365, no. 3 (2007), pp. 213-219.
  15. Ghanbari, B., Inc, M., Yusuf, A., Bayram, M. "Exact optical solitons of Radhakrishnan-Kundu- Lakshmanan equation with Kerr law nonlinearity." Modern Physics Letters B 33, no. 06 (2019): 1950061.
  16. Kudryashov, N.A., Loguinova, N.B.. "Extended simplest equation method for nonlinear differential equations." Applied Mathematics and Computation 205, no. 1 (2008), pp.396-402.
  17. Jawad, A.J.M., Petković, M.D., Biswas, A. "Modified simple equation method for nonlinear evolution equations." Applied Mathematics and Computation217, no. 2 (2010), pp. 869-877.
  18. Vitanov, N.K., Dimitrova, Z.I., Kantz, H. "Modified method of simplest equation and its application to nonlinear PDEs." Applied Mathematics and Computation 216, no. 9 (2010), pp. 2587-2595.
  19. Vitanov, N.K. "Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs." Communications in Nonlinear Science and Numerical Simulation 16, no. 3 (2011), pp. 1176-1185.
  20. Vitanov, N.K. "On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation." Communications in Nonlinear Science and Numerical Simulation 16, no. 11 (2011), pp. 4215-4231.
  21. Dillon, S.J., Tang, M., Carter, W.C., Harmer, M.P. "Complexion: a new concept for kinetic engineering in materials science." Acta Materialia 55, no. 18 (2007), pp. 6208-6218.
  22. Zhai, W., Chen, D.Y. "Rational solutions of the general nonlinear Schrödinger equation with derivative." Physics Letters A 372, no. 23 (2008), pp. 4217-4221.
  23. Ray, J.R., Thompson, E.L. "Spacetime symmetries and the complexion of the electromagnetic field." Journal of Mathematical Physics 16, no. 2 (1975), pp. 345-346.
  24. Höll, A., Krassnigg, A., Roberts, C.D., Wright, S.V. "On the complexion of pseudoscalar mesons." International Journal of Modern Physics A 20, no. 08n09 (2005), pp. 1778-1784.
  25. Chai, X., Liu, G., Zhao, X., Hao, Y., Zhao, Y. "Complexion between mercury and humic substances from different landfill stabilization processes and its implication for the environment." Journal of Hazardous Materials 209 (2012), pp. 59-66.
  26. Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H. "Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis." SIAM Journal on Numerical Analysis 55, no. 1 (2017), pp. 31-62.
  27. Kadapa, C., Dettmer, W.G., Perić, D. "A fictitious domain/distributed Lagrange multiplier based fluid-structure interaction scheme with hierarchical B-spline grids." Computer Methods in Applied Mechanics and Engineering 301 (2016), pp. 1-27.
  28. Amirfakhrian, M.. Approximation of 3D-Parametric Functions by Bicubic B-spline Functions. International Journal of Mathematical Modelling & Computations, 2 (2016), pp. 211- 220.
  29. Beck, J., Stiller, C. "Generalized B-spline Camera Model." In 2018 IEEE Intelligent Vehicles Symposium (IV), pp. 2137-2142. IEEE, 2018.
  30. Rashidinia, J., Sharif, Sh. "B-Spline Method for Two-Point Boundary Value Problems." International Journal of Mathematical Modelling & Computations 5, no. 02 (2015), pp. 111-125.
  31. Liu, W., Zhang, Y., Shi, D. "Lump waves, solitary waves and interaction phenomena to the (2+ 1)- dimensional Konopelchenko-Dubrovsky equation." Physics Letters A 383 (2019), pp. 97-102.
  32. Yuan, Y.Q., Tian, B., Liu, L., Wu, X.Y., Sun, Y. "Solitons for the (2+ 1)-dimensional Konopelchenko-Dubrovsky equations." Journal of Mathematical Analysis and Applications 460, no. 1 (2018), pp. 476-486.
  33. Wazwaz, A.M. "Study on a New (3+ 1)-Dimensional Extensions of the Konopelchenko- Dubrovsky Equation." Appl. Math 12, no. 6 (2018), pp. 1067-1071.
  34. Wu, P., Zhang, Y., Iqbal M., Qiqi, Y. "Complexiton and resonant multiple wave solutions to the (2+ 1)-dimensional Konopelchenko-Dubrovsky equation." Computers & Mathematics with Applications 76, no. 4 (2018), pp. 845-853.
  35. Yasar, E., Giresunlu, I.B. "‘Exact Traveling Wave Solutions and Conservation Laws of (2+ 1) Dimensional Konopelchenko-Dubrovsky System." IJNS 22 (2016), pp. 118-128.
  36. Jawad, A.J.M., Petković, M.D., Biswas, A. "Modified simple equation method for nonlinear evolution equations." Applied Mathematics and Computation 217, no. 2 (2010), pp. 869-877.
  37. Kudryashov, N.A., Loguinova, N.B. "Extended simplest equation method for nonlinear differential equations." Applied Mathematics and Computation 205, no. 1 (2008), pp. 396-402.
  38. Vitanov, N.K. "On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation." Communications in Nonlinear Science and Numerical Simulation 16, no. 11 (2011), pp. 4215-4231.
  39. Xu, X., Kargoll, B., Bureick, J., Yang, Y., Alkhatib, H., Neumann, I. "TLS-based profile model analysis of major composite structures with robust B-spline method." Composite Structures 184 (2018), pp. 814-820.
  40. Aksan, E.N. "An application of cubic B-spline finite element method for the Burgers' equation." Thermal Science 22, no. 1 (2018), pp. S195-S202.

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