Numerical solution of fractional differential equations using the Bernstein matrix for fractional differentiation

Fractional differential equations (FDEs) represent a highly significant aspect of physical and engineering models, albeit the exact solutions are still a difficult issue. In this research, we suggest using Bernstein polynomials to come up with numerical sol utions to FDEs. The approach is based on the Bernstein operational matrix for fractional derivatives, which enables the FDE to be restated as a system of algebraic equations, and the system can be solved to find the unknown coefficients of approximation. \r\nIn addition to presenting the efficacy of the suggested method, we also utilize it for solving three different FDEs. As the first \r\ninstance, a problem of a Bagley-Torvik equation is studied, and the result of, which is the exact solution, is achieved by the numerical method. The second example is concerned with the investigation of a linear fractional differential equation with \r\nboundaries, where the error is considerably decreased by increasing the degree of the polynomial. Specifically, for, the \r\nerror is 6.5×10−4, and the error is 1.5×10−3, whereas for , these errors reduce to 4.6×10−7 and 6.7×10, respectively.\r\nThrough Example 3, you are faced with a scenario where the solution of the equation contains a square root. The results show that for, the error is 2.9×10−3 and error is 6.5×10−3, decreasing to 3.4×10−5 and 4.1×10−7, respectively, for .\r\nThe developed technique offers a methodological, practical, and correct way to tackle FDEs. The computational results ascertain\r\nthat the Bernstein polynomial method is a viable choice in solving FDEs very accurately.\r\n