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 L 1 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 α and time on solution profile of travelling wave for different domain of x 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.
Highlights
The fractional calculus has a long history, beginning on 30 September, 1695 [1]
Fractional derivatives are an excellent method for explaining the memory and genetic characteristics of various materials and processes
A great deal of attention has been paid to managing fractional differential equations (FDEs)
Summary
The fractional calculus has a long history, beginning on 30 September, 1695 [1]. Fractional derivatives are an excellent method for explaining the memory and genetic characteristics of various materials and processes. G′/G expansion method has been implemented to investigate the exact solution of time fractional model of BBM-Burger equation in [24]. Cubic B-spline method along with Caputo derivative will be used to solve time fractional BBM-Burger equation numerically. The advantages of B-spline collocation method over the existing schemes are that it efficiently delivers a piecewise-continuous, closed form solution and it is simpler and can be used to a wide range of problems involving partial and fractional partial differential equations. The advantage of this method over others is that once the solution has been computed, the information needed for interpolation at different locations in the interval is available.
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