A computational approach for numerical solution of time fractional Burgers’ equation via hybrid B-spline technique

Due to the complexity arising from fractional-order and nonlinearity, the existing articles offer a limited set of efficient numerical methods to solve the Caputo time fractional nonlinear Burgers’ equation. In this paper, the authors have developed a numerical method that can efficiently perform in both fractional and integer orders. We have used a reliable L1 finite difference method to handle the fractional time component. For spatial discretization, we have implemented a mixed uniform hyperbolic polynomial B-spline function that combines cubic and uniform hyperbolic polynomial B-spline functions. Since the equation contains a nonlinear term, the linearization technique proposed by Rubin and Graves is adopted. This method turns the problem into a linear equation, which can be solved easily. We have tested our method with four different examples and found that the obtained results are better than those given in the literature. To confirm the accuracy of the proposed method, the L∞ norms are calculated, and linear profiles for different α values are plotted. The graphs indicate the perfect alignment of the numerical and exact solutions. This is further reinforced on the basis of absolute errors, which are found to be small, reaching as low as the order of 10−13. The convergence order of the method is obtained to be two, confirming its high precision, and the stability is analyzed using the Von Neumann method, and the proposed method is found to be unconditionally stable.