| تعداد نشریات | 32 |
| تعداد شمارهها | 856 |
| تعداد مقالات | 8,306 |
| تعداد مشاهده مقاله | 52,804,500 |
| تعداد دریافت فایل اصل مقاله | 9,235,556 |
A new approach to numerical solution of the time-fractional KdV-Burgers equations using least squares support vector regression | ||
| Journal of Mathematical Modeling | ||
| مقاله 1، دوره 12، شماره 4، اسفند 2024، صفحه 583-602 اصل مقاله (561.71 K) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.22124/jmm.2024.26733.2358 | ||
| نویسندگان | ||
| Abumoslem Mohammadi؛ Abolfazl Tari* | ||
| Department of Mathematics, Shahed University, Tehran, Iran | ||
| چکیده | ||
| The evolution of the waves on shallow water surfaces is described by a mathematical model given by nonlinear KdV and KdV-Burgers equations. These equations have many other applications and have been simulated by classical numerical methods in recent decades. In this paper, we develop a machine learning algorithm for the time-fractional KdV-Burgers equations. The proposed method implements a linearization of the problem and a time reduction by a Crank-Nicolson scheme. The least squares support vector regression (LS-SVR) is proposed to seek the approximate solution in a finite-dimensional polynomial kernel space. The Bernstein polynomials are used as the kernel of the proposed algorithm to handle the homogeneous boundary conditions easily in the framework of the Petrov-Galerkin spectral method. The proposed LS-SVR implements the orthogonal system of Bernstein-dual polynomials in the learning process, which gives quadratic programming in the primal form and provides a linear system of equations in dual variables with sparse positive definite matrices. It is shown that the involving mass and stiffness matrices are sparse. Some new theorems for the introduced basis are provided. Also, numerical results are presented to support the spectral convergence and accuracy of the method. | ||
| کلیدواژهها | ||
| Fractional KdV equation؛ machine learning؛ support vector machines؛ Petrov-Galerkin؛ least squares support vector regression | ||
| مراجع | ||
|
[1] E. Bas, R. Ozarslan, Real world applications of fractional models by Atangana–Baleanu fractional derivative, Chaos Solit. Fractals 116 (2018) 121–125. [2] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015) 73–85. [3] M. Dehghan, M. Abbaszadeh, The use of proper orthogonal decomposition (POD) meshless RBF- FD technique to simulate the shallow water equations, J. Comput. Phys. 351 (2017) 478–510. [4] R.T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Comput. Aided Geom. Des. 29, (2012) 379–419. [5] A.K. Gupta, S.S. Ray, On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water, Chaos Solit. Fractals 116 (2018) 376–380. [6] A.H. Hadian-Rasanan, N. Bajalan, K. Parand, J.A. Rad, Simulation of nonlinear fractional dynam- ics arising in the modeling of cognitive decision making using a new fractional neural network, Math. Methods Appl. Sci. 43 (2020) 1437–1466. [7] A.H. Hadian-Rasanan, D. Rahmati, S. Gorgin, K. Parand, A single layer fractional orthogonal neural network for solving various types of LaneEmden equation, New Astron. 1 (2020) 101307. [8] M. Jani, E. Babolian, S. Javadi, Bernstein modal basis: Application to the spectral Petrov-Galerkin method for fractional partial differential equations, Math. Methods Appl. Sci. 40 (2017) 7663– 7672. [9] M. Jani, E. Babolian, S. Javadi, D. Bhatta, Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation, Numer. Algorithms 75 (2017) 1041–1063. [10] B. Jin, R. Lazarov, Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput. 38 (2016) A146–A170. [11] B. Juttler, The dual basis functions for the Bernstein polynomials, Adv. Comput. Math. 8 (1998) 345–352. [12] D. Kaya, S. Glbahar, A. Yokus¸, M. Glbahar, Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions, Adv. Differ. Equ. 1 (2018) 77. [13] NR. Kevlahan, R. Khan, B. Protas, On the convergence of data assimilation for the one- dimensional shallow water equations with sparse observations, Adv. Comput. Math. 45 (2019) 3195–3216. [14] M.M. Khader, K.M. Saad, Z. Hammouch, D. Baleanu, A spectral collocation method for solving fractional KdV and KdV-Burgers equations with non-singular kernel derivatives, Appl. Numer. Math. 161 (2021) 137–146. [15] A. Kurganov, Finite-volume schemes for shallow-water equations, Acta Numer. 27 (2018) 289– 351. [16] J. Li, J. Chen, B. Li, Gradient-optimized physics-informed neural networks (GOPINNs): a deep learning method for solving the complex modified KdV equation, Nonlinear Dyn. 107 (2022) 781– 792. [17] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007) 1533–1552. [18] G.G. Lorentz, Approximation of Functions, American Mathematical Scociety, 2023. [19] F. Mainardi, Why the Mittag-Leffler function can be considered the Queen function of the Frac- tional Calculus? Entropy, 22 (2020) 1359. [20] S. Mehrkanoon, T. Falck, J.A. Suykens, Approximate solutions to ordinary differential equations using least squares support vector machines, IEEE Trans. Neural Netw. Learn. Syst. 23 (2012) 1356–1367. [21] S. Mehrkanoon, S. Mehrkanoon, J.A. Suykens, Parameter estimation of delay differential equa- tions: an integration-free LS-SVM approach, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 830–841. [22] S. Mehrkanoon, J.A. Suykens, Learning solutions to partial differential equations using LS-SVM, Neurocomputing 159 (2015) 105–116. [23] V.H. Moghaddam, J. Hamidzadeh, New Hermite orthogonal polynomial kernel and combined kernels in support vector machine classifier, Pattern Recognit. 60 (2016) 921–935. [24] A. Pakniyat, K. Parand, M. Jani, Least squares support vector regression for differential equations on unbounded domains, Chaos Solit. Fractals. 151 (2021) 111232. [25] K. Parand, A.A. Aghaei, M. Jani, A. Ghodsi, A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression, Math. Comput. Simul. 180 (2021) 114–128. [26] K. Parand, A.A. Aghaei, M. Jani, A. Ghodsi, Parallel LS-SVM for the numerical simulation of fractional Volterra’s population model, Alex. Eng. J. 60 (2021) 5637–5647. [27] K. Parand, M. Razzaghi, R. Sahleh, M. Jani, Least squares support vector regression for solving Volterra integral equations, Eng. Comput. 9 (2020) 789–796. [28] J.A. Rad, K. Parand, S. Chakraverty, Learning with Fractional Orthogonal Kernel Classifiers in Support Vector Machines: Theory, algorithms and applications, Springer, 2023. [29] Z. Yu, J. Sun, B. Wu, A space–time spectral Petrov–Galerkin method for nonlinear time-fractional Korteweg–de Vries–Burgers equations, Math. Methods Appl. Sci. 44 (2021) 4348–4365. [30] H. Zhang, SJ. Cai, JY. Li, Y. Liu, ZY. Zhang, Enforcing generalized conditional symmetry in physics-informed neural network for solving the KdV-like equation with Robin initial/boundary conditions, Nonlinear Dyn.111 (2023) 10381–10392. | ||
|
آمار تعداد مشاهده مقاله: 584 تعداد دریافت فایل اصل مقاله: 882 |
||