
تعداد نشریات | 21 |
تعداد شمارهها | 620 |
تعداد مقالات | 9,098 |
تعداد مشاهده مقاله | 67,289,835 |
تعداد دریافت فایل اصل مقاله | 7,788,295 |
The operational matrices of two dimensional Bernstein polynomials for solving the hyperbolic partial differential equation with boundary conditions | ||
International Journal of Nonlinear Analysis and Applications | ||
مقاله 9، دوره 16، شماره 7، مهر 2025، صفحه 107-120 اصل مقاله (657.74 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22075/ijnaa.2024.32159.4777 | ||
نویسندگان | ||
Mohsen Mohamadi* ؛ Amir Shahmari؛ Hassan Eshaghi Kenari | ||
Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran | ||
تاریخ دریافت: 03 آبان 1402، تاریخ بازنگری: 20 بهمن 1402، تاریخ پذیرش: 15 اردیبهشت 1403 | ||
چکیده | ||
The wave equations are one of the most important equations in engineering and physics, which are usually formulated as hyperbolic partial differential equations with special boundary conditions. In this paper, a numerical method for solving these equations based on Bernstein polynomials is introduced. The properties of Bernstein polynomial operational matrices turn this differential equation and its boundary conditions into a system of algebraic equations. Some numerical examples illustrate the accuracy, validity, and applicability of the new technique. | ||
کلیدواژهها | ||
Bernstein polynomial؛ two dimensions Bernstein polynomial؛ Best approximation؛ Operational matrices؛ Kronecker products؛ Hyperbolic Partial differential equation of second order | ||
مراجع | ||
[1] P.F. Antonietti, A. Cangiani, J. Collis, Z. Dong, E. H. Georgoulis, S. Giani, and P. Houston, Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains, Building bridges: connections and challenges in modern approaches to numerical partial differential equations, 2016, pp. 281–310. [2] A.G. Atta, W.M. Abd-Elhameed, G.M. Moatimid, and Y.H. Youssri, Shifted fifth-kind Chebyshev Galerkin treat[1]ment for linear hyperbolic first-order partial differential equations, Appl. Numer. Math. 167 (2021), 237–256. [3] K. Batselier and N. Wong, A constructive arbitrary-degree Kronecker product decomposition of tensors, Numer. Linear Algebra Appl. 24 (2017), no. 5, e2097. [4] M.J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys. 53 (1984), no. 3, 484–512. [5] S. Bernstein, D´emonstration du theoreme de Weierstrass fondee sur le calcul des probabilites (Proof of the theorem of Weierstrass based on the calculus of probabilities), Comm. Kharkov Math. Soc. 13 (1912), no. 1, 1-–2. [English Translation] [6] R. Bojanic and F. Cheng, Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation, J. Math. Anal. Appl. 141 (1989), no. 1, 136–151. [7] T. Bui-Thanh, From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations, J. Comput. Phys. 295 (2015), 114–146. [8] P.L. Butzer, On two-dimensional Bernstein polynomials, Canad. J. Math. 5 (1953), 107–113. [9] B. Cockburn, G.E. Karniadakis, and C.W. Shu, Discontinuous Galerkin Methods: Theory, Computation and Applications, Vol. 11, Springer Science & Business Media, 2012. [10] A.J. Davies, The Finite Element Method: An Introduction with Partial Differential Equations, Oxford University Press, USA, 2011. [11] M. Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Meth. Partial Differ. Equ.: Int. J. 21 (2005), no. 1, 24–40. [12] E.H. Doha, R.M. Hafez, and Y.H. Youssri, Shifted Jacobi spectral-Galerkin method for solving hyperbolic partial differential equations, Comput. Math. Appl. 78 (2019), no. 3, 889–904. [13] J. Douglas, P.P. Leme, J.E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numerische Math. 65 (1993), 95–108. [14] M. Foupouagnigni and M. Mouafo Wouodji´e, On multivariate Bernstein polynomials, Mathematics 8 (2020), no. 9, 1397. [15] M.Y. Hussaini, C.L. Streett, and T.A. Zang, Spectral methods for partial differential equations, AD-P002980, Army Res. Office Trans. of the 1st Army Conf. Appl. Math. Comput., 1983. [16] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Courier Corporation, 2012. [17] J. Jost, Partial Differential Equations, Vol. 214, Springer Science & Business Media, 2012. [18] F. Khan, M. Omar, and Z. Ullah, Discretization method for the numerical solution of 2D Volterra integral equation based on two-dimensional Bernstein polynomial, AIP Adv. 8 (2018), no. 12. [19] H.O. Kreiss and G. Scherer, Finite element and finite difference methods for hyperbolic partial differential equations, Mathematical aspects of finite elements in partial differential equations, Elsevier Academic Press, 1974, pp. 195–212. [20] E. Kreyszig, Introductory Functional Analysis with Applications, Vol. 17, John Wiley & Sons, 1991. [21] S. Kumar, R.C. Mittal, and R. Jiwari, A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations, Int. J. Comput. Math. 100 (2023), no. 7, 1580–1600. [22] L. Lapidus and G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley & Sons, 2011. [23] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. 2, Cambridge University Press, 2000. [24] R.J. LeVeque and J. Oliger, Numerical methods based on additive splittings for hyperbolic partial differential equations, Math. Comput. 40 (1983), no. 162, 469–497. [25] F. Liu, New Kronecker product decompositions and its applications, Res. Inventy: Int. J. Engin. Sci. 1 (2012), 25–30. [26] R. Lohner, Gauge theory techniques in quantum cohomology, Ph.D. thesis, University College of Swansea, 1984. [27] G.G. Lorentz, Bernstein Polynomials, American Mathematical Society, 2012. [28] F.L. Martinez, Some properties of two-dimensional Bernstein polynomials, J. Approx. Theory 59 (1989), no. 3, 300–306. [29] F. Mirzaee, S. Rezaei, and N. Samadyar, Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and mesh free methods, Engin. Anal. Boundary Elements 127 (2021), 53–63. [30] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, 1973. [31] M. Mohamadi, E. Babolian, and S.A. Yousefi, Bernstein multiscaling polynomials and application by solving Volterra integral equations, Math. Sci. 11 (2017), 27–37. [32] M. Mohamadi, E. Babolian, and S.A. Yousefi, A solution for Volterra integral equations of the first kind based on Bernstein polynomials, Int. J. Ind. Math. 10 (2018), no. 1, 19–27. [33] M. Mohamadi, E. Babolian, and S.A. Yousefi, Bernstein multi-scaling operational matrix method for nonlinear matrix differential models of second-order, Int. J. Ind. Math. 11 (2019), no. 3, 223–227. [34] K.W. Morton and D.F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, 2nd edition, Cambridge University Press, 2005. [35] G.M. Phillips, Interpolation and Approximation by Polynomials, Vol. 14, Springer Science & Business Media, 2003. [36] M. Ramezani, M. Dehghan, and M. Razzaghi, Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition, Numer. Meth. Partial Differ. Equ.: Int. J. 24 (2008), no. 1, 1–8. [37] F.H. Shekarabi, K. Maleknejad, and R. Ezzati, Application of two-dimensional Bernstein polynomials for solving mixed Volterra–Fredholm integral equations, Afr. Mate. 26 (2015), no. 7-8, 1237–1251. [38] C.W. Shu, Discontinuous Galerkin methods: General approach and stability, Numer. Solutions Partial Differ. Equ. 201 (2009). [39] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edition, Oxford University Press, 1985. [40] P. Solın, Partial Differential Equations and the Finite Element Method, John Wiley & Sons, 2005. [41] V. Totik, Approximation by Bernstein polynomials, Amer. J. Math. 116 (1994), no. 4, 995–1018. [42] A.M. Wazwaz, Partial Differential Equations, CRC Press, 2002. [43] A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer Science & Business Media, 2010. [44] S.A. Yousefi, Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation, Numer. Meth. Partial Differ. Equ.: Int. J. 26 (2010), no. 3, 535–543. [45] S.A. Yousefi and M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int. J. Syst. Sci. 41 (2010), no.6, 709–716. [46] S.A. Yousefi, M. Behroozifar, and M. Dehghan, The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass, J. Comput. Appl. Math. 235 (2011), no. 17, 5272–5283. [47] T.H. Youssri and R.M. Hafez, Exponential Jacobi spectral method for hyperbolic partial differential equations, Math. Sci. 13 (2019), no. 4, 347–354. | ||
آمار تعداد مشاهده مقاله: 38 تعداد دریافت فایل اصل مقاله: 122 |