Numerical approximation for inverse problem of the Ostrovsky-Burgers equation. (English) Zbl 1482.65172
Summary: This article considers a nonlinear inverse problem of the Ostrovsky-Burgers equation by using noisy data. Two B-Splines with different levels, the quintic B-spline and septic B-spline, are used to study this problem. For both B-splines, the stability and convergence analysis are calculated, and results show that an excellent estimation of the unknown functions of the
nonlinear inverse problem.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B35 | Stability in context of PDEs |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Ostrovsky equation; quintic B-spline collocation; septic B-spline collocation; convergence analysis; stability analysis; noisy dataReferences:
| [1] | Akram, G. and Siddiqi, S.S.End conditions for interpolatory septic splines, Int. J. Comput. Math. 82 (2005) 1525-1540. · Zbl 1136.41002 |
| [2] | Bhatta, D.Use of modified Bernstein polynomials to solve KdV-Burgers equation numerically,Appl. Math. Comput. 206 (2008) 457-464. · Zbl 1157.65447 |
| [3] | Cabeza, J.M.G., Garcia, J.A.M. and Rodriguez, A.C.A sequential algorithm of inverse heat conduction problems using singular value decomposition,Int. J. Therm. Sci. 44 (2005) 235-244. |
| [4] | Chen, G.U. and Boyd, J.P.Analytical and numerical studies of weakly nonlocal solitary waves of the rotation-modified Korteweg-de Vries equation,Physica D 155 (2001) 201-222. · Zbl 0985.35077 |
| [5] | Chen, R. and Wu, Z.Solving partial differential equation by using multiquadric quasi-interpolation,Appl. Math. Comput. 186 (2007) 1502-10. · Zbl 1117.65134 |
| [6] | de Boor, C.On the convergence of odd degree spline interpolation,J. Approx. Theory 1 (1968) 452-463. · Zbl 0174.09902 |
| [7] | Esfahani, A. and Levandosky, S.Solitary waves of the rotation-generalized Benjamin-Ono equation,Discrete Contin. Dyn. Syst. 33 (2013) 663-700. · Zbl 1277.35289 |
| [8] | Foadian, S., Pourgholi, R. and Tabasi, S.H.Cubic B-spline method for the solution of an inverse parabolic system,Appl. Anal. 97 (2018) 438-465. · Zbl 1466.65103 |
| [9] | Galkin, V.N. and Stepanyants, Y.A.On the existence of stationary solitary waves in a rotating field,Appl. Math. Mech. 55 (1991) 939-943. · Zbl 0786.76016 |
| [10] | Gilman, O.A., Grimshaw, R. and Stepanyants, Y.A.Approximate and numerical solutions of the stationary Ostrovsky equation,Stud. Appl. Math. 95 (1995) 115-126 · Zbl 0843.76008 |
| [11] | Grimshaw, R.Internal solitary waves, environmental stratified flows, Kluwer Academic Publishers, Dordrecht, 2001, Ch. 1, pp. 1-29. |
| [12] | Haq, S., Siraj-Ul-Islam, and Uddin, M.A mesh-free method for the numerical solution of the KdV-Burgers equation,Appl. Math. Model. 33 (2009) 3442-3449. · Zbl 1205.65233 |
| [13] | Hall, C.A.On error bounds for spline interpolation,J. Approx. Theory 1 (1968) 209-218. · Zbl 0177.08902 |
| [14] | Helal, M.A. and Mehanna, M.S.A comparison between two different methods for solving KdV-Burgers equation,Chaos Soliton Fract. 28 (2006) 320-326. · Zbl 1084.65079 |
| [15] | Khater, A.H., Temsah, R.S. and Hassan, M.M.A Chebyshev spectral collocation method for solving Burgers’-type equations,J. Comput. Appl. Math. 222 (2008) 333-350. · Zbl 1153.65102 |
| [16] | Mitsudera, H. and Grimshaw, R.Effects of friction on a localized structure in a baroclinic current,J. Physical Oceanography 23 (1993) 2265- 2292. |
| [17] | Obregon, M.A. and Stepanyants, Y.A.Oblique magneto-acoustic solitons 460 in rotating plasma,Phys. Lett. A 249 (1998) 315-323. |
| [18] | Ostrovsky, L.A.Nonlinear internal waves in a rotating ocean, Oceanologia 18 (1978) 181-191 |
| [19] | Ott, E. and Sudan, R.N.Damping of solitary waves,Phys. Fluids 13 (1970) 1432-1434. |
| [20] | Rudin, W.Principles of mathematical analysis,Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. · Zbl 0346.26002 |
| [21] | Smith, G.D.Numerical solution of partial differential equation: finite difference method,Learendom Press, Oxford 1978. · Zbl 0389.65040 |
| [22] | Wang, H. and Esfahani, A.Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation,Evol. Equ. Control Theory 8 (2019) 709-735. · Zbl 1428.35469 |
| [23] | Xu, Y. and Shu, C-W.Local discontinuous Galerkin methods for the Kuramoto Sivashinsky equations and the Ito-type coupled equations,Comput. Methods Appl. Mech. 195 (2006) 3430-3447. · Zbl 1124.76035 |
| [24] | Zhan, J.M. and Li, Y.S.Generalized finite spectral method for 1D burgers and KdV equations,Appl. Math. Mech. (English Ed.) 27 (2006) 1635- 1643 · Zbl 1165.76040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
