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

• Published in: Chaos, solitons and fractals Vol. 199; p. 116738
• Main Authors: Priyendhu, K.S., Prakash, P., Victor, Stéphane
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2025
• Subjects: Exact solutions; Fractional diffusion delay systems; Hilfer fractional derivative; Initial–boundary value problems; Invariant subspace method; Separable method; Invariant subspace method; Fractional diffusion delay systems; Separable method; Hilfer fractional derivative; Initial–boundary value problems; Exact solutions
• ISSN: 0960-0779
• DOI: 10.1016/j.chaos.2025.116738

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. •The Hilfer fractional derivative is considered.•The theory of separation and invariant subspace methods are discussed.•Scalar and coupled systems of fractional time-delay PDEs are investigated.•Multiplicative, additive, and generalized separable solutions are derived.•The Hilfer fractional nonlinear diffusion convection reaction time-delay systems are considered.