Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 2667-419X Eskisehir Technical University Solitary wave simulations of Complex Modified Korteweg-de Vries (CMKdV) Equation using Quintic Trigonometric B-Spline Hepson Özlem Ersoy 06 01 2018 6 2 193 205 Copyright © 2010, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 2010 Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler

The Modified form of the Complex Korteweg-de Vries (CMKdV) Equation is solved numerically using collocation method based on quintic trigonometric B-Splines. A Crank Nicolson rule is used to discretize in time. The well-known examples, propagation of bell-shaped initial pulse and collision of multi solitary waves are simulated using Matlab programme language. Computational results are examined by calculation of the accuracy of the method in terms of maximum error norm and the three conservation laws I1, I2 and I3. Because the absolute changes of the lowest three laws are also good indicators of valid results even when the analytical solutions do not exist. A comparison with some earlier works is given

 Complex Modified Korteweg-de Vries Equation trigonometric B-Splines finite elements method solitary waves conservation laws [1] Wazwaz AM. The tanh and the sine–cosine methods for the complex modified KdV and the generalized KdV equations. Comput. Math Appl 2005;49: 1101–1112 Muslu GM, Erbay, HA. A spliet step fourier Method for the Complex Modified Korteweg De Vries Equation. Computers and Mathematics with Aplplications, 2003; 45: 503–514 Taha, TR, Numerical simulations of the complex modified Korteweg-de Vries equation. Mathematics and Computers in Simulation 1994; 37: 461-467. Irk, D. B-spline Finite Element Solutions of the Some Partial Differential Equation Systems, Department of Mathematics. Doctoral Dissertation, (2007). Ismail, MS. Numerical solution of complex modified Korteweg-de Vries equation by collocation method. Communications in Nonlinear Science and Numerical Simulation. 2009;14: 749–759. Ismail, MS. Numerical solution of complex modified Korteweg–de Vries equation by Petrov–Galerkin method. Applied Mathematics and Computation. 2008; 202: 520–531. Korkmaz A, Dag, I. Solitary wave simulations of Complex Modified Korteweg–de Vries Equation using differential quadrature method. Computer Physics Communications. 2009; 180: 1516–1523. Korkmaz, A, Dag, I. Numerical Simulations of Complex Modified KdV Equation using Polynomial Differential Quadrature Method. J of Mathematics and Statistics. 2011; 10: 1-13. Korkmaz, A, Dag, I. Quartic and quintic B-spline methods for advection–diffusion equation. Applied Mathematics and Computation. 2016; 274: 208-219. Ersoy, O, Korkmaz, A, Dag, I. Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves. Mediterranean Journal of Mathematics. 2016; 13(6): 4975–4994. Korkmaz, A, Ersoy, O, Dag, I. Motion of Patterns Modeled by the Gray-Scott Autocatalysis System in One Dimension. MATCH Communications in Mathematical and in Computer Chemistry. 2017; 77(2), 507-526. Uddin, M, Haq S, Islam, S. Numerical solution of complex modified Korteweg de Vries equation by mesh-free collocation method. Computers and Mathematics with Applications. 2009; 58: 566-578. Nikolis, A. Numerical Solutions of Ordinary Differential Equations with Quadratic Trigonometric Splines. Applied Mathematics E-Notes. 1995; 4:142-149. Nikolis, A, Seimenis, I. Solving Dynamical Systems with Cubic Trigonometric Splines. Applied Mathematics E-notes. 2005; 5: 116-123. Abbas, M, Majid, AA, Ismail, Md, Rashid, A. The Application of Cubic Trigonometric B-spline to the Numerical Solution of the Hyperbolic Problems. Applied Mathematica and Computation. 2014; 239: 74-88. Abbas, M, Majid, AA, Ismail, Md, Rashid, A. Numerical Method Using Cubic Trigonometric B-spline Technique for non-classical Diffusion Problems. Abstract and applied analysis. (2014). Dag, I, Irk, D, Kacmaz, O, Adar, N. Trigonometric B-spline collocation algortihm for solving the RLW equation. Applied and Computational Mathematics. 2016; 15(1): 96-105. Hepson, OE, Dag, I. The Numerical Approach to the Fisher’s Equation via Trigonometric Cubic B-spline Collocation Method. Communications on Nonlinear Analysis. 2017; 2: 91-100. Dag, I, Ersoy, O, Kacmaz, O. The Trigonometric Cubic B-spline Algorithm for Burgers’ Equation. International Journal of Nonlinear Science, 2017; 24(2): 120-128. Keskin, P. Trigonometric B-spline solutions of the RLW equation. Department of Mathematics & Computer. Doctoral Dissertation. (2017). Rubin SG, Graves RA. Cubic spline approximation for problems in fluid mechanics. Nasa TR R-436 Washington DC (1975). Karney, CFF, Sen, A, Chu, FYF. Nonlinear evolution of lower hybrid waves. Phys. Fluids. 1979; 22; 940–952.