Solutions of (1+1) and (m+1)-dimensional time-fractional delay PDEs with the Hilfer derivative: Separable and invariant subspace methods

Abstract

The main aim of this work is to systematically present two analytical approaches that are known as (i) the separable method and (ii) the invariant subspace method to solve the scalar and coupled time-delay linear and nonlinear time-fractional PDEs with the Hilfer arbitrary-order derivative. Also, this work investigates how to compute different possible types of exact solutions for the $k$-component coupled $(m+1)$-dimensional time-delay time-fractional PDEs with the Hilfer arbitrary-order derivative through the invariant subspace method together with and without the linear space variable transformation. More precisely, we show the effectiveness and usefulness of the separable and invariant subspace methods to obtain various types of variable separable forms of exact solutions for the scalar and k-component coupled $(1+1)$-dimensional time-delay linear and nonlinear time-fractional heat equations with the Hilfer arbitrary-order derivative. In addition, we explicitly illustrated the importance of the invariant subspace method together with and without the linear space variable transformation to compute the variable separable forms of exact solutions for the 2-component coupled $(2+1)$-dimensional time-delay nonlinear time-fractional diffusion convection reaction systems with the Hilfer arbitrary-order derivative subject to suitable initial and boundary conditions. From this study, we notice that the Euler-gamma, trigonometric, exponential, three-parameter Mittag-Leffler, and polynomial functions are involved in the derived exact solutions. Further, we provide the comparative study of the discussed methods along with illustrative examples in the appropriate places as well as with the existing literature wherever possible.