NUMERICAL SOLUTIONS OF BOUSSINESQ TYPE EQUATIONS BY MESHLESS METHODS

Yıl 2024, Cilt: 25 Sayı: 3, 471 - 484, 30.09.2024

Murat Arı , Yılmaz Dereli

https://doi.org/10.18038/estubtda.1485966

Öz

In this paper, two different meshfree method with radial basis functions (RBFs) is proposed to solve Boussinesq-type (Bq) equations. The basic conservative properties of the equation are investigated by computing the numerical values of the motion’s invariants. The accuracy of the method is tested using computational tests to simulate solitary waves in terms of L_∞ error norm. The outcomes are contrasted with analytical solution and a few other earlier studies in the literature. The results show that meshless methods are very effective and accurate.

Anahtar Kelimeler

Radial basis function, Collocation method, Method of lines, Soliton, Boussinesq equation

Kaynakça

• [1] Boussinesq J. Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans un canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 1872; 7: 55–108.
• [2] Manoranjan VS, Mitchell AR, Morris JL. Numerical solutions of the good Boussinesq equation. SIAM Journal on Scientific and Statistical Computing 1984; 5(4): 946-957.
• [3] Manoranjan VS, Ortega T, Sanz-Serna JM. Soliton and antisoliton interactions in the “good” Boussinesq equation. J. Math. Phys. 1988; 29: 1964–1968.
• [4] Bratsos AG. The solution of the Boussinesq equation using the method of lines. Comput. Methods Appl. Mech. Engrg. 1998; 157: 33–44.
• [5] Pani AK, Saranga H. Finite element Galerkin method for the “good” Boussinesq equation. Nonlinear Analysis: Theory, Methods and Applications, 1997; 29: 937-956.
• [6] Ortega T, Sanz-Serna JM. Nonlinear stability and convergence of finite difference methods for the “good” Boussinesq equation. Numerische Mathematik, 1990; 58: 215-229. doi: 10.1007/BF01385620
• [7] El-Zoheiry H. Numerical investigation for the solitary waves interaction of the “good” Boussinesq equation. Applied Numerical Mathematics, 2003; 45: 161-173.
• [8] Wazwaz AM. Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, Solitons and Fractals, 2001; 12: 549-1556.
• [9] Ismail MS, Bratsos AG. A predictor-corrector scheme for the numerical solutions of the Boussinesq equation. Journal of Applied Mathematics and Computing, 2003; 13: 11-27.
• [10] Bratsos AG, A second-order numerical scheme for the solution of the one-dimensional Boussinesq equation. Numer. Algorithms 46, 2007; 45–58.
• [11] Bratsos AG. Solitary-wave propagation and interactions for the ‘good’ Boussinesq equation. Int. J. Comput. Math. 85, 2008; 1431–1440.
• [12] Dehghan M, Salehi RA. Meshless based numerical technique for traveling solitary wave solution of Boussinesq equation. Applied Mathematical Modelling, 2012; 36(5): 1939-1956.
• [13] Ucar Y, Esen A, Karaagac B. Numerical solutions of Boussinesq equation using Galerkin finite element method. Numerical Methods for Partial Differential Equations, 2020; 3782: 1612-1630.
• [14] Ismail MS, Mosally F. A fourth order finite difference method for the good Boussinesq equation. Abstract and Applied Analysis, 2014; 323260: 10 pages.
• [15] Kirli E, Irk D. A Fourth Order One Step Method for Numerical Solution of Good Boussinesq Equation. Turkish Journal of Mathematics, 2021; 45(5): 2693-2703.
• [16] Zhijian Y. Existence and non-existence of global solutions to a generalized modification of the improved Boussinesq equation. Math. Methods Appl. Sci. 1998; 21: 1467.
• [17] Lin Q, Wu YH, Loxton R. Linear B-Spline Finite Element Method for the Improved Boussinesq Equation. J. Comput. Appl. Math. 2009; 224: 658.
• [18] Iskandar L, Jain PC, Proc. Indian Acad. Sci. Math. Sci. 89, 1980, 171.
• [19] Irk D, Dağ I, Numer. Methods Partial Differential Equations, 2009, doi:10.1002/num.20492.
• [20] Kansa EJ. Multiquadrics–A scattered data approximation scheme with applications to computational fluid dynamics. I., Comput. Math. Appl. 1990; 19: 127–145.
• [21] Kansa EJ. Multiquadrics– A scattered data approximation scheme with applications to computational fluid dynamics. II., Comput. Math. Appl. 1990; 19: 147–161.
• [22] Wendland H. Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, Cambridge: Cambridge University Press, 17, 2005.
• [23] Rubin SG, Graves RA. A Cubic Spline Approximation for Problems in Fluid Mechanics (NASA TR R-436, Washington, DC, 1975).
• [24] Karaagac B, Ucar Y, Esen A. Numerical Solutions of the Improved Boussinesq Equation by the Galerkin Quadratic B-Spline Finite Element Method. Filomat 2018; 32:16: 5573–5583.