Numerical Simulation of Time Fractional BBM-Burger Equation Using Cubic B-Spline Functions
Mohsin Kamran, Muhammad Abbas, Abdul Majeed, Homan Emadifar, Tahir Nazir
Abstract
The unidirectional propagation of long waves in certain nonlinear dispersive system is explained by the Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The purpose of this study is to investigate the BBM-Burger equation numerically using Caputo derivative and B-spline basis functions. The fractional derivative is considered in Caputo form, and <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>L</a:mi> <a:mn>1</a:mn> </a:math> formula is used for discretization of temporal derivative. The interpolation of space derivative is done with the help of B-spline functions. The effect of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi>α</c:mi> </c:math> and time on solution profile of travelling wave for different domain of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mi>x</e:mi> </e:math> is discussed in this paper. The numerical results have been presented to show that the cubic B-spline method is effective and efficient in solving the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. Moreover, the convergence and stability of the proposed scheme are analyzed. The error norms are also calculated to check the accuracy of the proposed scheme. The numerical results reflect that the proposed scheme can be used for linear and highly nonlinear models.