Wavelet-based Numerical Approaches for Solving the Korteweg-de Vries (KdV) Equation
Yıl 2022,
Cilt: 14 Sayı: 1, 44 - 55, 30.06.2022
Neslihan Özdemir
,
Aydın Seçer
Öz
In this research work, we examine the Korteweg-de Vries equation (KdV), which is utilized to formulate the propagation of water waves and occurs in different fields such as hydrodynamics waves in cold plasma acoustic waves in harmonic crystals. This research presents two efficient computational methods based on Legendre wavelets to solve the Korteweg-de Vries. The three-step Taylor method is first applied to the Korteweg-de Vries equation for time discretization. Then, the Galerkin and collocation methods are used for spatial discretization. With these approaches, bringing the approximate solutions of the Korteweg-de Vries equation turns into getting the solution of the algebraic equation system. The solution of this system gives the Legendre wavelet coefficients. The approximate solution can be obtained by substituting the obtained coefficients into the Legendre wavelet series expansion. The presented wavelet methods are tested by studying different problems at the end of this study.
Destekleyen Kurum
Yildiz Technical University Scientific Research Projects Coordination Unit
Proje Numarası
Project Number: FBA-2020-4005.
Kaynakça
- Akdi, M., Sedra, M.B., Numerical KdV equation by the adomian decomposition method, American Journal of Modern Physics[Online], 2(3)(2013), 111-115.
- Alotaibi, F., Ismail, M.S., Numerical Solution of Kortweg-de Vries Equation, Applied Mathematics, 11(4)(2020), 344-362.
- Bahmanpour, M., Tavassoli-Kajani, M., Maleki, M., A Müntz wavelets collocation method for solving fractional differential equations, Computational and Applied Mathematics, 37(4)(2018), 5514-5526.
- BV Rathish, K., Mani, M., Time-accurate solutions of Korteweg-de Vries equation using wavelet Galerkin method, Applied Mathematics and Computation, 162(1)(2005), 447-460.
- Canıvar, A., Sari, M., Dag, I., A Taylor-Galerkin finite element method for the KdV equation using cubic B-splines, Physica B: Condensed Matter, 405(16)(2010), 3376-3383.
- Çelik, İ, Gegenbauer wavelet collocation method for the extended Fisher-Kolmogorov equation in two dimensions, Mathematical Methods in the Applied Sciences, 843(8)(2020), 5615-5628.
- Dag, I., Dereli, Y., Numerical solutions of KdV equation using radial basis functions, Applied Mathematical Modelling, 32(4)(2008), 535-546.
- Das, G.C., Sarma, J., Response to ``Comment on `A new mathematical approach for finding the solitary waves in dusty plasma''' [Phys. Plasmas 6, 4392 (1999)], Physics of Plasmas, 6(11)(1999), 4394-4397.
- Dehestani H., Ordokhani Y., Razzaghi, M., On the applicability of Genocchi wavelet method for different kinds of fractional-order differential equations with delay, Numerical Linear Algebra with Applications, 26(5)(2019), e2259.
- Gardner, L.R.T., Gardner, G.A., Ali, A.H.A., Simulations of solitons using quadratic spline finite elements, Computer Methods in Applied Mechanics and Engineering, 92(2)(1991), 231-243.
- Hafiz, K.M.E.B., Andallah, L.S., Second order scheme for Korteweg-de Vries (KDV) equation, Journal of Bangladesh Academy of Sciences, 43(1)(2019), 85-93.
- Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M., A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type0, Appl. Math. Model., 38(2014), 1597-1606.
- Kong, D., Xu, Y., Zheng, Z., A hybrid numerical method for the KdV equation by finite difference and sinc collocation method, Applied Mathematics and Computation, 355(2019), 61-72.
- Korteweg, D.J., De Vries, G., XLI. 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(240)(1895), 422-443.
- Kutluay, S., Bahadir, A.R., Ozdes, A., A small time solutions for the Korteweg-de Vries equation, Applied Mathematics and Computation, 107(2-3)(2000), 203-210.
- Ma, H., Sun, W., A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations, SIAM Journal on Numerical Analysis, 38(5)(2000), 1425-1438.
- Mirzaee, F., Rezaei, S., Samadyar, N., Application of combination schemes based on radial basis functions and finite difference to solve stochastic coupled nonlinear time fractional sine-Gordon equations, Computational and Applied Mathematics, 41(1)(2022), 1-16.
- Mirzaee, F., Rezaei, S., Samadyar, N., Solution of time-fractional stochastic nonlinear sine-Gordon equation via finite difference and meshfree techniques, Mathematical Methods in the Applied Sciences, (2021).
- Mirzaee, F., Rezaei, S., Samadyar, N., Solving one-dimensional nonlinear stochastic Sine-Gordon equation with a new meshfree technique, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 34(4)(2021), e2856.
- Mirzaee, F., Rezaei, S., Samadyar, N., Numerical solution of two-dimensional stochastic time-fractional Sine-Gordon equation on non-rectangular domains using finite difference and meshfree methods, Engineering Analysis with Boundary Elements, 127(2021), 53-63.
- Mirzaee, F., Sayevand, K., Rezaei, S., Samadyar, N., Finite difference and spline approximation for solving fractional stochastic advection-diffusion equation, Iranian Journal of Science and Technology, Transactions A: Science, 45(2)(2021), 607-617.
- Mirzaee, F., Sayevand, K., Rezaei, S., Samadyar, N., Finite difference and spline approximation for solving fractional stochastic advection-diffusion equation, Iranian Journal of Science and Technology, Transactions A: Science, 45(2)(2021), 607-617.
- Mirzaee, F., Samadyar, N., Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection-diffusion equations, Engineering with Computers, 36(4)(2020), 1673-1686.
- Mirzaee, F., Samadyar, N., Numerical solution of time fractional stochastic Korteweg-de Vries equation via implicit meshless approach, Iranian Journal of Science and Technology, Transactions A: Science, 43(6)(2019), 2905-2912.
- Oruc, O., Bulut, F., Esen, A., Numerical solution of the KdV equation by Haar wavelet method, Pramana, 87(2016), 94.
- Ozdemir, N., Secer, A., Bayram, M., The Gegenbauer wavelets-based computational methods for the coupled System of Burgers' equations with time-fractional derivative, Mathematics, 7(2019), 486.
- Ozdemir, N., Secer, A., Bayram, M., An algorithm for numerical solution of some nonlinear multi-dimensional parabolic partial differential equations, Journal of Computational Science, (2021), 101487.
- Ozlem, H., Idris, D. , The exponential cubic B-spline algorithm for Korteweg-de Vries equation, Advances in Numerical Analysis, (2015).
- Samadyar, N., Ordokhani, Y., Mirzaee, F., Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulation, 90(2020), 105346.
- Samadyar, N., Ordokhani, Y., Mirzaee, F., The couple of Hermite-based approach and Crank-Nicolson scheme to approximate the solution of two dimensional stochastic diffusion-wave equation of fractional order, Engineering Analysis with Boundary Elements, 118(2020), 285-294.
- Secer, A., Ozdemir, N., An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation, Advances in Difference Equations, (2019), 386.
- Secer, A., Cinar, M., A Jacobi wavelet collocation method for fractional Fisher's equation in time, Thermal Science, 24(2020)(Suppl. 1), 119-129.
- Secer, A., Altun, S., A new operational matrix of fractional derivatives to solve systems of fractional differential equations via legendre wavelets, Mathematics, 6(2018), 238.
- Soliman, A.A., Collocation solution of the Korteweg-de Vries equation using septic splines, International Journal of Computer Mathematics, 81(2004), 325-331.
- Torabi, M., Hosseini, M.M., A new efficient method for the numerical solution of linear time-dependent partial differential equations, Axioms, 7(2018), 70.
- Usman, M., Hamid, M., Haq, R.U.,Wang, W., An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations, The European Physical Journal Plus, 133(2018), 327.
- Wazwaz, A.M. , Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos Soliton Fract., (28)(2006), 457-462.
- Yan, J., Shu, C.W., A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis, (40)(2002), 769-791.
- Zheng, X, Wei, Z., Discontinuous Legendre wavelet Galerkin method for one-dimensional advection-diffusion equation, Applied Mathematics, (6)(9)(2015), 1581-1591.
Yıl 2022,
Cilt: 14 Sayı: 1, 44 - 55, 30.06.2022
Neslihan Özdemir
,
Aydın Seçer
Proje Numarası
Project Number: FBA-2020-4005.
Kaynakça
- Akdi, M., Sedra, M.B., Numerical KdV equation by the adomian decomposition method, American Journal of Modern Physics[Online], 2(3)(2013), 111-115.
- Alotaibi, F., Ismail, M.S., Numerical Solution of Kortweg-de Vries Equation, Applied Mathematics, 11(4)(2020), 344-362.
- Bahmanpour, M., Tavassoli-Kajani, M., Maleki, M., A Müntz wavelets collocation method for solving fractional differential equations, Computational and Applied Mathematics, 37(4)(2018), 5514-5526.
- BV Rathish, K., Mani, M., Time-accurate solutions of Korteweg-de Vries equation using wavelet Galerkin method, Applied Mathematics and Computation, 162(1)(2005), 447-460.
- Canıvar, A., Sari, M., Dag, I., A Taylor-Galerkin finite element method for the KdV equation using cubic B-splines, Physica B: Condensed Matter, 405(16)(2010), 3376-3383.
- Çelik, İ, Gegenbauer wavelet collocation method for the extended Fisher-Kolmogorov equation in two dimensions, Mathematical Methods in the Applied Sciences, 843(8)(2020), 5615-5628.
- Dag, I., Dereli, Y., Numerical solutions of KdV equation using radial basis functions, Applied Mathematical Modelling, 32(4)(2008), 535-546.
- Das, G.C., Sarma, J., Response to ``Comment on `A new mathematical approach for finding the solitary waves in dusty plasma''' [Phys. Plasmas 6, 4392 (1999)], Physics of Plasmas, 6(11)(1999), 4394-4397.
- Dehestani H., Ordokhani Y., Razzaghi, M., On the applicability of Genocchi wavelet method for different kinds of fractional-order differential equations with delay, Numerical Linear Algebra with Applications, 26(5)(2019), e2259.
- Gardner, L.R.T., Gardner, G.A., Ali, A.H.A., Simulations of solitons using quadratic spline finite elements, Computer Methods in Applied Mechanics and Engineering, 92(2)(1991), 231-243.
- Hafiz, K.M.E.B., Andallah, L.S., Second order scheme for Korteweg-de Vries (KDV) equation, Journal of Bangladesh Academy of Sciences, 43(1)(2019), 85-93.
- Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M., A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type0, Appl. Math. Model., 38(2014), 1597-1606.
- Kong, D., Xu, Y., Zheng, Z., A hybrid numerical method for the KdV equation by finite difference and sinc collocation method, Applied Mathematics and Computation, 355(2019), 61-72.
- Korteweg, D.J., De Vries, G., XLI. 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(240)(1895), 422-443.
- Kutluay, S., Bahadir, A.R., Ozdes, A., A small time solutions for the Korteweg-de Vries equation, Applied Mathematics and Computation, 107(2-3)(2000), 203-210.
- Ma, H., Sun, W., A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations, SIAM Journal on Numerical Analysis, 38(5)(2000), 1425-1438.
- Mirzaee, F., Rezaei, S., Samadyar, N., Application of combination schemes based on radial basis functions and finite difference to solve stochastic coupled nonlinear time fractional sine-Gordon equations, Computational and Applied Mathematics, 41(1)(2022), 1-16.
- Mirzaee, F., Rezaei, S., Samadyar, N., Solution of time-fractional stochastic nonlinear sine-Gordon equation via finite difference and meshfree techniques, Mathematical Methods in the Applied Sciences, (2021).
- Mirzaee, F., Rezaei, S., Samadyar, N., Solving one-dimensional nonlinear stochastic Sine-Gordon equation with a new meshfree technique, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 34(4)(2021), e2856.
- Mirzaee, F., Rezaei, S., Samadyar, N., Numerical solution of two-dimensional stochastic time-fractional Sine-Gordon equation on non-rectangular domains using finite difference and meshfree methods, Engineering Analysis with Boundary Elements, 127(2021), 53-63.
- Mirzaee, F., Sayevand, K., Rezaei, S., Samadyar, N., Finite difference and spline approximation for solving fractional stochastic advection-diffusion equation, Iranian Journal of Science and Technology, Transactions A: Science, 45(2)(2021), 607-617.
- Mirzaee, F., Sayevand, K., Rezaei, S., Samadyar, N., Finite difference and spline approximation for solving fractional stochastic advection-diffusion equation, Iranian Journal of Science and Technology, Transactions A: Science, 45(2)(2021), 607-617.
- Mirzaee, F., Samadyar, N., Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection-diffusion equations, Engineering with Computers, 36(4)(2020), 1673-1686.
- Mirzaee, F., Samadyar, N., Numerical solution of time fractional stochastic Korteweg-de Vries equation via implicit meshless approach, Iranian Journal of Science and Technology, Transactions A: Science, 43(6)(2019), 2905-2912.
- Oruc, O., Bulut, F., Esen, A., Numerical solution of the KdV equation by Haar wavelet method, Pramana, 87(2016), 94.
- Ozdemir, N., Secer, A., Bayram, M., The Gegenbauer wavelets-based computational methods for the coupled System of Burgers' equations with time-fractional derivative, Mathematics, 7(2019), 486.
- Ozdemir, N., Secer, A., Bayram, M., An algorithm for numerical solution of some nonlinear multi-dimensional parabolic partial differential equations, Journal of Computational Science, (2021), 101487.
- Ozlem, H., Idris, D. , The exponential cubic B-spline algorithm for Korteweg-de Vries equation, Advances in Numerical Analysis, (2015).
- Samadyar, N., Ordokhani, Y., Mirzaee, F., Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulation, 90(2020), 105346.
- Samadyar, N., Ordokhani, Y., Mirzaee, F., The couple of Hermite-based approach and Crank-Nicolson scheme to approximate the solution of two dimensional stochastic diffusion-wave equation of fractional order, Engineering Analysis with Boundary Elements, 118(2020), 285-294.
- Secer, A., Ozdemir, N., An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation, Advances in Difference Equations, (2019), 386.
- Secer, A., Cinar, M., A Jacobi wavelet collocation method for fractional Fisher's equation in time, Thermal Science, 24(2020)(Suppl. 1), 119-129.
- Secer, A., Altun, S., A new operational matrix of fractional derivatives to solve systems of fractional differential equations via legendre wavelets, Mathematics, 6(2018), 238.
- Soliman, A.A., Collocation solution of the Korteweg-de Vries equation using septic splines, International Journal of Computer Mathematics, 81(2004), 325-331.
- Torabi, M., Hosseini, M.M., A new efficient method for the numerical solution of linear time-dependent partial differential equations, Axioms, 7(2018), 70.
- Usman, M., Hamid, M., Haq, R.U.,Wang, W., An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations, The European Physical Journal Plus, 133(2018), 327.
- Wazwaz, A.M. , Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos Soliton Fract., (28)(2006), 457-462.
- Yan, J., Shu, C.W., A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis, (40)(2002), 769-791.
- Zheng, X, Wei, Z., Discontinuous Legendre wavelet Galerkin method for one-dimensional advection-diffusion equation, Applied Mathematics, (6)(9)(2015), 1581-1591.