An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation

In this paper, we proposed a new approximate solution method of the Bagley- Torvik fractional order differential equation with damping term. The solution of this equation is of great practical interest and has been studied extensively. In general, the solution of the Bagley-Torvik equation is known to be computationally expensive and the process is complicated. However, the new proposed method significantly reduced the computational effort while ensuring the accuracy of the solution in a novel way. Since 1.5 order derivative of absolutely continuous a function on some interval in the Caputo’s sense is equal to the temperature gradient at the boundary of the one-dimensional heat conduction problem in the semi-infinite interval with 1 order derivative of the function as boundary conditions, we transformed the given fractional differential equation into a general differential equation. Then, according to the characteristics of the obtained equation, we used the variables separation and Fourier series approximation. So the proposed method transforms the Bagley-Torvik fractional order differential equation into an integer order differential equation. Prior to the main content, we have given and proved two definitions and two theorems as preliminaries. And the convergence analysis of this method is discussed. And then we solved two different example problems for the cases with and without damping by using various methods including proposed method and verified that the computational effort is significantly small and the accuracy is guaranteed through comparison of the results with other methods. So the effectiveness of the proposed method is analyzed.