Solving the Nonlinear PDE Problems Numerically Using the Fast Fourier Transform
University of Zielona Góra
Institute of Physics
ul. Szafrana 4a, 65-069 Zielona Góra, Poland
E-mail: M.Dudek@if.uz.zgora.pl
Received:
Received: 17 November 2024; accepted: 25 November 2024; published online: 7 December 2024
DOI: 10.12921/cmst.2024.0000019
Abstract:
A novel, highly efficient numerical method for solving nonlinear partial differential equations (PDEs) is proposed. This method involves a symbolic polynomial of a given degree N defined by a formal variable λ. The sole purpose of λ is to collect expressions that approximate the exact solution of the PDE in terms of the coefficients of this polynomial. It is a formal parameter which is set to 1 afterwards. What is crucial to the method is that the partial derivatives contained in these coefficients of the polynomial are computed using the Fast Fourier Transform. The method can be seen as a complement to finite difference methods. It can be applied to a variety of PDE problems, including those with higher order PDEs. The robustness of the simple iterative scheme that converts the solution of PDEs to the polynomial problem makes this method also promising for engineering applications.
Key words:
Fast Fourier Transform, Fisher equation, inviscid Burgers equation, KdV equation, non-linear PDE
References:
[1] J.H.E. Cartwright, O. Piro, The dynamics of Runge–Kutta methods, Int. J. Bifurcat. Chaos 2, 427–449 (1992).
[2] M.J. Ablowitz, B.M. Herbst, C. Schober, On the Numerical Solution of the sine–Gordon Equation: I. Integrable Discretizations and Homoclinic Manifolds, J. Comput. Phys. 126, 299–314 (1996).
[3] M.J. Ablowitz, B.M. Herbst, C.M. Schober, Discretizations, integrable systems and computation, J. Phys. A-Math. Gen. 34, 10671 (2001).
[4] D.J. Kouri, D.S. Zhang, G.W. Wei, T. Konshak, D.K. Hoffman, Numerical solutions of nonlinear wave equations, Phys. Rev. E 59, 1274–1277 (1999).
[5] E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press (2000).
[6] R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press (1997).
[7] H. Schaeffer, Learning partial differential equations via data discovery and sparse optimization, Proc. R. Soc. A 473, 20160446 (2016).
[8] J.-H. He, Homotopy perturbation technique, Comput. Method. Appl. M. 178, 257–262 (1999).
[9] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Springer (1994).
[10] A. Bratsos, M. Ehrhardt, I.T. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations, Appl. Math. Comput. 197, 190–205 (2008).
[11] H.N.A. Ismail, K.R. Raslan, G.S.E. Salem, Solitary wave solutions for the general KdV equation by Adomian decomposition method, Appl. Math. Comput. 154, 17–29 (2008).
[12] T.A. Abassy, M.A. El-Tawil, H.K. Saleh, The Solution of KdV and mKdV Equations Using Adomian Pade Approximation, Int. J. Nonlin. Sci. Num. 5, 327–340 (2004).
[13] D. Kaya, A numerical solution of the sine-Gordon equation using the modified decomposition method, Appl. Math. Comput. 143, 0096–3003 (2003).
[14] J.-H. He, Homotopy Perturbation Method for Bifurcation of Nonlinear Problems, Int. J. Nonlin. Sci. Num. 6, 207–208 (2005).
[15] M.S.H. Chowdhury, I. Hashim, O. Abdulaziz, Application of homotopy-perturbation method to nonlinear population dynamics models, Phys. Letters A 368, 251–258 (2007).
[16] A.C. Loyinmi, T.K. Akinfe, Exact solutions to the family of Fisher’s reaction-diffusion equation using Elzaki homotopy transformation perturbation method, Eng. Rep. 2, e12084 (2020).
[17] E. Valseth, C. Dawson, An unconditionally stable space–time FE method for the Korteweg–de Vries equation, Comput. Methods Appl. Mech. Engrg. 371, 0045–7825 (2020).
[18] B.G. Chen, N. Upadhyaya, V. Vitelli, Nonlinear conduction via solitons in a topological mechanical insulator, PNAS 111, 13004–13009 (2014).
[19] B. Deng, M. Zanaty, A.E. Forte, K. Bertoldi, Topological solitons make metamaterials crawl, Phys. Rev. Appl. 17, 014004 (2022).
[20] Y. Shen, I. Dierking, Dynamics of electrically driven solitons in nematic and cholesteric liquid crystals, Comm. Phys. 3, 14 (2020).
[21] B.A. Malomed, D. Mihalache, F. Wise, L. Torner, Spatiotemporal optical solitons, J. Opt. B-Quantum S. O. 7, R53 (2005).
[22] M. Izadi, H.M. Srivastava, Numerical treatments of nonlinear Burgers-Fisher equation via a combined approximation technique, Kuwait J. Sci. 51, 100163 (2024).
[23] A.V. Slunyaev, A.V. Kokorina, E.N. Kokorina, Nonlinear waves, modulations and rogue waves in the modular Korteweg-de Vries equation, Commun. Nonlinear Sci. 127, 107527 (2023).
[24] L. Ju, J. Zhou, Y. Zhang, Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures, Commun. Anal. Mech. 15, 24–49 (2023).
[25] M. Meylan, Nonlinear PDE’s course, https://wikiwaves.org/ Numerical_Solution_of_the_KdV.
[26] X.-K. Wang, G.-B. Wang, A Hybrid Method Based on the Iterative Fourier Transform and the Differential Evolution for Pattern Synthesis of Sparse Linear Arrays, Int. J. Antenn. Propag. 2018, 6309192 (2018).
[27] B. Brzostowski, M.R. Dudek, B. Grabiec, T. Nadzieja, Non-finite-difference algorithm for integrating Newton’s motion equations, Phys. Status Solidi B 244, 851–858 (2005).
[28] C. Peterson, The Radial Equation for Hydrogen-Like Atoms, J. Chem. Educ. 52, 92–94 (1975).
[29] D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 39, 422–443 (1895).
[30] A.C. Scott, F.Y.F. Chu, D.W. McLaughlin, The soliton: A new concept in applied science, Proceedings of the IEEE 61, 1443–1483 (1973).
[31] M. Izadi, S¸ . Yüzbas¸i, A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems, Math. Commun. 27, 47–62 (2022).
[32] A.B. Downey, Think DSP. Digital Signal Processing in Python, Green Tea Press (2014).
[33] F. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings of the IEEE 66, 51–83 (1978).
[34] J. Chambarel, C. Kharif, O. Kimmoun, Generation of two-dimensional steep water waves on finite depth with and without wind, Eur. J. Mech. B-Fluid. 29, 132–142 (2010).
[35] J.G. Verwer, W.H. Hundsdorfer, B.P Sommeijer, Convergence properties of the Runge-Kutta-Chebyshev method, Math. Sci. 57, 157–178 (1990).
[36] C.R. Mittal, J.R. Kumar, Numerical solutions of nonlinear Fisher’s reaction–diffusion equation with modified cubic B-spline collocation method, Math. Sci. 7, 12 (2013).
[37] M. Izadi, Applications of the Newton-Raphson method in a SDFEM for inviscid Burgers equation, Comput. Meth. Diff. Eq. 8, 708–732 (2020).
A novel, highly efficient numerical method for solving nonlinear partial differential equations (PDEs) is proposed. This method involves a symbolic polynomial of a given degree N defined by a formal variable λ. The sole purpose of λ is to collect expressions that approximate the exact solution of the PDE in terms of the coefficients of this polynomial. It is a formal parameter which is set to 1 afterwards. What is crucial to the method is that the partial derivatives contained in these coefficients of the polynomial are computed using the Fast Fourier Transform. The method can be seen as a complement to finite difference methods. It can be applied to a variety of PDE problems, including those with higher order PDEs. The robustness of the simple iterative scheme that converts the solution of PDEs to the polynomial problem makes this method also promising for engineering applications.
Key words:
Fast Fourier Transform, Fisher equation, inviscid Burgers equation, KdV equation, non-linear PDE
References:
[1] J.H.E. Cartwright, O. Piro, The dynamics of Runge–Kutta methods, Int. J. Bifurcat. Chaos 2, 427–449 (1992).
[2] M.J. Ablowitz, B.M. Herbst, C. Schober, On the Numerical Solution of the sine–Gordon Equation: I. Integrable Discretizations and Homoclinic Manifolds, J. Comput. Phys. 126, 299–314 (1996).
[3] M.J. Ablowitz, B.M. Herbst, C.M. Schober, Discretizations, integrable systems and computation, J. Phys. A-Math. Gen. 34, 10671 (2001).
[4] D.J. Kouri, D.S. Zhang, G.W. Wei, T. Konshak, D.K. Hoffman, Numerical solutions of nonlinear wave equations, Phys. Rev. E 59, 1274–1277 (1999).
[5] E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press (2000).
[6] R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press (1997).
[7] H. Schaeffer, Learning partial differential equations via data discovery and sparse optimization, Proc. R. Soc. A 473, 20160446 (2016).
[8] J.-H. He, Homotopy perturbation technique, Comput. Method. Appl. M. 178, 257–262 (1999).
[9] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Springer (1994).
[10] A. Bratsos, M. Ehrhardt, I.T. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations, Appl. Math. Comput. 197, 190–205 (2008).
[11] H.N.A. Ismail, K.R. Raslan, G.S.E. Salem, Solitary wave solutions for the general KdV equation by Adomian decomposition method, Appl. Math. Comput. 154, 17–29 (2008).
[12] T.A. Abassy, M.A. El-Tawil, H.K. Saleh, The Solution of KdV and mKdV Equations Using Adomian Pade Approximation, Int. J. Nonlin. Sci. Num. 5, 327–340 (2004).
[13] D. Kaya, A numerical solution of the sine-Gordon equation using the modified decomposition method, Appl. Math. Comput. 143, 0096–3003 (2003).
[14] J.-H. He, Homotopy Perturbation Method for Bifurcation of Nonlinear Problems, Int. J. Nonlin. Sci. Num. 6, 207–208 (2005).
[15] M.S.H. Chowdhury, I. Hashim, O. Abdulaziz, Application of homotopy-perturbation method to nonlinear population dynamics models, Phys. Letters A 368, 251–258 (2007).
[16] A.C. Loyinmi, T.K. Akinfe, Exact solutions to the family of Fisher’s reaction-diffusion equation using Elzaki homotopy transformation perturbation method, Eng. Rep. 2, e12084 (2020).
[17] E. Valseth, C. Dawson, An unconditionally stable space–time FE method for the Korteweg–de Vries equation, Comput. Methods Appl. Mech. Engrg. 371, 0045–7825 (2020).
[18] B.G. Chen, N. Upadhyaya, V. Vitelli, Nonlinear conduction via solitons in a topological mechanical insulator, PNAS 111, 13004–13009 (2014).
[19] B. Deng, M. Zanaty, A.E. Forte, K. Bertoldi, Topological solitons make metamaterials crawl, Phys. Rev. Appl. 17, 014004 (2022).
[20] Y. Shen, I. Dierking, Dynamics of electrically driven solitons in nematic and cholesteric liquid crystals, Comm. Phys. 3, 14 (2020).
[21] B.A. Malomed, D. Mihalache, F. Wise, L. Torner, Spatiotemporal optical solitons, J. Opt. B-Quantum S. O. 7, R53 (2005).
[22] M. Izadi, H.M. Srivastava, Numerical treatments of nonlinear Burgers-Fisher equation via a combined approximation technique, Kuwait J. Sci. 51, 100163 (2024).
[23] A.V. Slunyaev, A.V. Kokorina, E.N. Kokorina, Nonlinear waves, modulations and rogue waves in the modular Korteweg-de Vries equation, Commun. Nonlinear Sci. 127, 107527 (2023).
[24] L. Ju, J. Zhou, Y. Zhang, Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures, Commun. Anal. Mech. 15, 24–49 (2023).
[25] M. Meylan, Nonlinear PDE’s course, https://wikiwaves.org/ Numerical_Solution_of_the_KdV.
[26] X.-K. Wang, G.-B. Wang, A Hybrid Method Based on the Iterative Fourier Transform and the Differential Evolution for Pattern Synthesis of Sparse Linear Arrays, Int. J. Antenn. Propag. 2018, 6309192 (2018).
[27] B. Brzostowski, M.R. Dudek, B. Grabiec, T. Nadzieja, Non-finite-difference algorithm for integrating Newton’s motion equations, Phys. Status Solidi B 244, 851–858 (2005).
[28] C. Peterson, The Radial Equation for Hydrogen-Like Atoms, J. Chem. Educ. 52, 92–94 (1975).
[29] D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 39, 422–443 (1895).
[30] A.C. Scott, F.Y.F. Chu, D.W. McLaughlin, The soliton: A new concept in applied science, Proceedings of the IEEE 61, 1443–1483 (1973).
[31] M. Izadi, S¸ . Yüzbas¸i, A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems, Math. Commun. 27, 47–62 (2022).
[32] A.B. Downey, Think DSP. Digital Signal Processing in Python, Green Tea Press (2014).
[33] F. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings of the IEEE 66, 51–83 (1978).
[34] J. Chambarel, C. Kharif, O. Kimmoun, Generation of two-dimensional steep water waves on finite depth with and without wind, Eur. J. Mech. B-Fluid. 29, 132–142 (2010).
[35] J.G. Verwer, W.H. Hundsdorfer, B.P Sommeijer, Convergence properties of the Runge-Kutta-Chebyshev method, Math. Sci. 57, 157–178 (1990).
[36] C.R. Mittal, J.R. Kumar, Numerical solutions of nonlinear Fisher’s reaction–diffusion equation with modified cubic B-spline collocation method, Math. Sci. 7, 12 (2013).
[37] M. Izadi, Applications of the Newton-Raphson method in a SDFEM for inviscid Burgers equation, Comput. Meth. Diff. Eq. 8, 708–732 (2020).
