A spectral collocation scheme for solving nonlinear delay distributed-order fractional equations

The basic purpose of this paper is to implement a variant of the Jacobi-Gauss collocation scheme for solving nonlinear delay (in difference format) fractional differential equations of distributed order type. At the first stage of the solving procedure, the Legendre-Gauss quadrature rule implemented to approximate the main problem by a difference delay fractional differential equation of multi-term type. So, one can apply the Jacobi-Gauss collocation approach for localizing the resulting multi-term fractional differential equation with the difference delay factor. Moreover, some special cases of Jacobi-Gauss quadrature rules can be used for approximating the involved integrals in the aforementioned problem. By considering the Lagrange interpolating polynomials, as the numerical solutions, the solution of the main problem can be approximated by solving the associated system of nonlinear algebraic equations in terms of unknown Lagrange multipliers. Convergence analysis of the problem is investigated rigorously under some mild conditions at a spectral rate. Extensive test problems are considered to justify the theoretical analysis.