An L1 type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion–reaction equations with fixed delay

A linearized spectral Galerkin/finite difference approach is developed for variable fractional-order nonlinear diffusion–reaction equations with a fixed time delay. The temporal discretization the variable-order fractional derivative performed by L1-approximation. An appropriate basis function in terms of Legendre polynomials used to construct Galerkin method spatial second-order operator. main advantage proposed that implementation iterative process avoided term problem. Convergence and stability estimates constructed scheme are proved theoretically discrete energy estimates. Some numerical experiments finally provided demonstrate efficiency accuracy theoretical findings.