Efficient Numerical Solutions of Proportional Fractional Differential Equations

This article proposes a new family of basis functions named exp-Bernstein functions, which generalize classical Bernstein polynomials by adding an exponential term that is compatible with proportional fractional derivatives. These functions are tailored to efficiently treat generalized proportional fractional calculus operators. We propose a numerical method using exp-Bernstein functions and a collocation method for solving linear and nonlinear generalized proportional fractional Cauchy problems (GPFCPs). Our method uses an operational matrix approach to calculate proportional fractional derivatives exactly. Theoretical convergence results are also given, providing error estimates using fractional Gronwall inequalities. Numerical results show that the proposed method provides high accuracy. The exp-Bernstein basis function is particularly useful for problems with exponential kernels, which are typical of proportional fractional derivatives.