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A Meshless Runge–Kutta Method for Some Nonlinear PDEs Arising in Physics

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This paper deals with the numerical solutions of a general class of one-dimensional nonlinear partial differential equations (PDEs) arising in different fields of science. The nonlinear equations contain, as special cases, several PDEs such as Burgers equation, nonlinear-Schrödinger equation (NLSE), Korteweg–De Vries (KDV) equation, and KdV–Schrödinger equations. Inspired by the method of lines, an RBF-FD approximation of the spatial derivatives in terms of local unknown function values, converts the nonlinear governing equations to a system of nonlinear ordinary differential equations(ODEs). Then, a fourth-order Runge–Kutta method is proposed to solve the resulting nonlinear system of first-order ODEs. For the RBF-FD approximation of derivatives, three kinds of different basis are investigated, and it is shown that the polynomial basis gives the highest accuracy. Solitary wave solutions form a special class of solutions of the model equations considered in this paper. The results reveal that the proposed method simulates this kind of solutions with high accuracy. Also, the method is able to simulate the collision of two soliton solutions, provided they are initially located far enough from each other.

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References

  1. A. Shirzadi and L. Ling, “Convergent overdetermined-rbf-mlpg for solving second order elliptic pdes,” Adv. Appl. Math. Mech., 5, No. 1, 78–89 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Schaback, “Direct discretizations with applications to meshless methods for PDEs,” Dolomites Res. Notes Approx., 6, 37–50 (2013).

    Google Scholar 

  3. G. Fasshauer and M. McCourt, Kernel-Based Approximation Methods Using MATLAB, World Scientific, Singapore (2015).

    Book  MATH  Google Scholar 

  4. A. Shirzadi, “Solving 2d reaction-diffusion equations with nonlocal boundary conditions by the rbf-mlpg method,” Comput. Math. Model., 25, No. 4, 521–529 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  5. C. C. Hsiang, C. K. Chou, D. L. Young, J. Sladek, and V. Sladek, “Applying the method of characteristics and the meshless localized radial basis function collocation method to solve shallow water equations,” J. Eng. Mech., 144, No. 7, 04018047 (2018).

  6. H. R. Ghehsareh, A. Zaghian, and A. Majlesi, “The method of approximate particular solutions to simulate an anomalous mobile-immobile transport process,” Math. Methods Appl. Sci., 43, No. 6, 3637–3649 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Safarpoor, F. Takhtabnoos, and A. Shirzadi, “A localized RBF-MLPG method and its application to elliptic pdes,” Eng. Comput., 36, No. 1, 171–183 (2020).

    Article  Google Scholar 

  8. M. Safarpoor and A. Shirzadi, “A localized RBF-MLPG method for numerical study of heat and mass transfer equations in elliptic fins,” Eng. Anal. Bound. Elem., 98, 35–45 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. N. O. Sadiku and C. N. Obiozor, “A simple introduction to the method of lines,” Int. J. Electr. Eng. Educ., 37, No. 3, 282–296 (2000).

    Article  Google Scholar 

  10. H. Wendland, Scattered Data Approximation, Cambridge University Press, United Kingdom (2004).

    Book  MATH  Google Scholar 

  11. G. E. Fasshauer and Meshfree, Meshfree Approximation Methods with Matlab, World Scientific, Singapore (2007).

  12. O. Davydov and D. T. Oanh, “Adaptive meshless centres and RBF stencils for Poisson equation,” J. Comput. Phys., 230, No. 2, 287–304 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Safarpoor and A. Shirzadi, “Numerical investigation based on radial basis function-finite-difference (RBF-FD) method for solving the stokes-darcy equations,” Eng. Comput., 37, No. 2, 909–920 (2021).

    Article  Google Scholar 

  14. A. H. Khater, R. S. Temsah, and M. Hassan, “A Chebyshev spectral collocation method for solving Burgers’-type equations,” J. Comput. Appl. Math., 222, No. 2, 333–350 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Mirzazadeh, M. Eslami, and A. H. Arnous, “Dark optical solitons of Biswas–Milovic equation with dual-power law nonlinearity,” Eur. Phys. J. Plus, 130, 1–7 (2015).

    Article  Google Scholar 

  16. K. K. Sabirov, M. E. Akramov, R. S. Otajonov, and D. U. Matrasulov, “Soliton generation in optical fiber networks,” Chaos. Solit. Fractals., 133, 109636 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. H. Arnous, M. Mirzazadeh, Q. Zhou, M. F. Mahmood, A. Biswas, and M. Belic, “Optical solitons with resonant nonlinear Schrödingers equation using G′/G-expansion scheme,” Adv. Mater. Rapid Commun., 9, No. 9-10, 1214–1220 (2015).

    Google Scholar 

  18. M. Younis and S. T. R. Rizvi, “Optical solitons for ultrashort pulses in nano fibers,” J. Nanoelectron. Optoelectron., 10, No. 2, 179–182 (2015).

    Article  Google Scholar 

  19. H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E, 72, No. 4, 046610 (2005).

  20. E. C. Aslan and M. Inc, “Optical soliton solutions of the NLSE with quadratic-cubic-Hamiltonian perturbations and modulation instability analysis,” Optik, 196, 162661 (2019).

    Article  Google Scholar 

  21. M. Savescu, S. Johnson, A. H. Kara, S. H. Crutcher, R. Kohl, and A. Biswas, “Conservation laws for optical solitons with spatio-temporal dispersion,” J. Electromagn. Waves Appl., 28, No. 2, 242–252 (2014).

    Article  Google Scholar 

  22. M. Savescu, K. R. Khan, P. Naruka, H. Jafari, L. Moraru, and A. Biswas, “Optical solitons in photonic nano waveguides with an improved nonlinear Schrödinger’s equation,” J. Comput. Theor. Nanosci., 10, No. 5, 1182–1191 (2013).

    Article  Google Scholar 

  23. Y. Zhou, M. Wang, and T. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Phys. Lett. A, 323, No. 1-2, 77–88 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Habibirad, E. Hesameddini, and A. Taleei, “An efficient meshless method for solving multi-dimensional nonlinear Schrödinger equation,” Iran. J. Sci. Technol. Trans. A: Sci., 44, No. 3, 749–761 (2020).

    Article  MathSciNet  Google Scholar 

  25. W. X. Ma and M. Chen, “Direct search for exact solutions to the nonlinear Schrödinger equation,” Appl. Math. Comput., 215, No. 8, 2835–2842 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  26. M. M. El-Borai, H. M. El-Owaidy, H. M. Ahmed, and A. H. Arnous, “ Soliton solutions of the nonlinear Schrödinger equation by three integration schemes,”Nonlinear Sci. Lett. A, 8, No. 1, 32–40 (2017).

    Google Scholar 

  27. M. Mirzazadeh, A. H. Arnous, M. F. Mahmood, E. Zerrad, and A. Biswas, “Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach,”Nonlinear Dyn., 81, No. 1, 277–282 (2015).

    Article  MATH  Google Scholar 

  28. A. M. Wazwaz, “A study on linear and nonlinear schrödinger equations by the variational iteration method,” Chaos. Solit. Fractals., 37, No. 4, 1136–1142 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  29. N. Mahak and G. Akram, “Extension of rational sine-cosine and rational sinh-cosh techniques to extract solutions for the perturbed NLSE with Kerr law nonlinearity,” Eur. Phys. J. Plus, 134, No. 4, 1–10 (2019).

    Article  Google Scholar 

  30. S. A. El-Tantawy, S. A. Shan, N. Akhtar, and A. T. Elgendy, “Impact of electron trap** in degenerate quantum plasma on the ion-acoustic breathers and super freak waves,” Chaos. Solit. Fractals, 113, 356–364 (2018).

    Article  MathSciNet  Google Scholar 

  31. M. S. Ruderman, T. Talipova, and E. Pelinovsky, “Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions,” J. Plasma Phys., 74, No. 5, 639–656 (2008).

    Article  Google Scholar 

  32. H. I. Abdel-Gawad, “Waves in deep water based on the nonlinear Schrödinger equation with variable coefficients,” Can. J. Phys., 92, No. 10, 1158–1165 (2014).

    Article  Google Scholar 

  33. K. Trulsen and K. B. Dysthe, “A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water,” Wave Motion, 24, No. 3, 281–289 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  34. D. J. Korteweg and G. D. Vries, “XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Lond. Edinb. Dublin Philos. Mag., 39, No. 240, 422–443 (1895).

    Article  MATH  Google Scholar 

  35. M. Wang, “Exact solutions for a compound KdV–Burgers equation,” Phys. Lett. A, 213, No. 5-6, 279–287 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  36. M. K. Brun and H. Kalisch, “Convective wave breaking in the KdV equation,” Anal. Math. Phys., 8, No. 1, 57–75 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  37. F. Mirzaee and N. Samadyar, “Numerical solution of time fractional stochastic Korteweg-de Vries equation via implicit meshless approach,” Iran. J. Sci. Technol. Trans. A: Sci., 43, No. 6, 2905–2912 (2019).

    Article  MathSciNet  Google Scholar 

  38. G. Rowlands, P. Rozmej, E. Infeld, and A. Karczewska, “Single soliton solution to the extended KdV equation over uneven depth,” Eur. Phys. J. E, 49, No. 8, 1–5 (2017).

    MATH  Google Scholar 

  39. M. Lakestani, “Numerical solutions of the KdV equation using B-Spline functions,” Iran. J. Sci. Technol. Trans. A: Sci., 41, No. 2, 409–417 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  40. X. Li and M. Wang, “A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms,” Phys. Lett. A, 361, No. 1-2, 115–118 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  41. A. M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,” Appl. Math. Comput., 184, No. 2, 1002–1014 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  42. H. X. Ge, R. J. Cheng, and S. Q. Dai, “KdV and kink-antikink solitons in car-following models,” Phys. A: Stat. Mech. Appl. 357, No. 3-4, 466–476 (2005).

    Article  MathSciNet  Google Scholar 

  43. T. Yoshinaga, M. Wakamiya, and T. Kakutani, “Recurrence and chaotic behavior resulting from nonlinear interaction between long and short waves,” Phys. Fluids. A, 3, No. 3, 83–89 (1991).

    Article  MATH  Google Scholar 

  44. N. N. Rao, “Nonlinear wave modulations in plasmas,” Pramana, 49, No. 1, 109–127 (1997).

    Article  Google Scholar 

  45. S. V. Singh, N. N. Rao, and P. K. Shukla, “Nonlinearly coupled langmuir and dustacoustic waves in a dusty plasma,” J. Plasma Phys., 60, No. 3, 551–567 (1998).

    Article  Google Scholar 

  46. A. Filiz, M. Ekici, and A. Sonmezoglu, “F-expansion method and new exact solutions of the Schrödinger–KdV equation,” Sci. World J., 2014, No. 3 (2014).

  47. S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Phys. Lett. A, 289 No. 1-2, 69–74 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  48. D. Kaya and S. M. El-Sayed, “On the solution of the coupled Schrödinger–KdV equation by the decomposition method,” Phys. Lett. A, 313, No. 1-2, 82–88 (2003).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Persian Gulf University, Bushehr, Iran

    Mohammad Mohammadi & Ahmad Shirzadi

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  1. Mohammad Mohammadi
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  2. Ahmad Shirzadi
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Correspondence to Mohammad Mohammadi.

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Mohammadi, M., Shirzadi, A. A Meshless Runge–Kutta Method for Some Nonlinear PDEs Arising in Physics. Comput Math Model 33, 375–387 (2022). https://doi.org/10.1007/s10598-023-09579-0

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