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Open research questions in Fractional Differential Equations Solutions

46 unresolved questions extracted from the limitations and future-work sections of 493 Fractional Differential Equations Solutions papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • First, the model lacks validation against real-world lon- gitudinal ASD data; future research should calibrate predictions using multi-omics datasets (metagenomics, cytokine profiles, behavioral assessments).

    Analysis of a caputo fractional-order modeling approach to dietary-gut-brain interactions in autism spectrum disorder · 2026 · DOI

  • Future research should focus on integrating the HWOMM with advanced swarm intelligence or evolutionary optimization algorithms to tackle more challenging models, including those involving variable-order fractional derivatives, time-delay effects, and nonlocal boundary conditions. The method developed in this study is not limited to the problems considered and can be extended to more complex partial FDEs arising in multidimensional physical and engineering systems in the future.

    A Haar wavelet operational matrix method for fractional Riccati differential equations with Atangana’s beta derivative · 2026 · DOI

  • The paper demonstrates that parameter selection (n and k values) significantly affects accuracy and computational cost (Tables 9, 12), but systematic guidelines or adaptive strategies for automatically determining optimal n and k values for arbitrary Duffing equations with varying nonlinearity strength and boundary condition types are not provided.

    Numerical solution of the Duffing equation with three types of boundary conditions using shifted Legendre polynomials · 2026 · DOI

  • The method handles non-integral forcing terms in Example 6 (x²(τ) nonlinearity), but the behavior of the shifted Legendre polynomial approach for higher-order nonlinearities (e.g., x³, x⁴) or multiplicative nonlinear terms in Duffing-type equations has not been examined.

    Numerical solution of the Duffing equation with three types of boundary conditions using shifted Legendre polynomials · 2026 · DOI

  • The proposed L-RKM method using shifted Legendre polynomials outperforms reproducing kernel methods (Yao 2009, n=100), variational iteration methods (Geng 2011, n=50), and sinc-collocation methods (Saadatmandi & Yeganeh 2017, N=20) for standard Duffing problems, but comparative analysis against recent machine learning approaches (neural networks, physics-informed neural networks) for nonlinear Duffing equations with non-integral forcing terms is absent.

    Numerical solution of the Duffing equation with three types of boundary conditions using shifted Legendre polynomials · 2026 · DOI

  • The shifted Legendre polynomial method has been demonstrated only on Duffing equations with Dirichlet, Neumann, and Bitsadze–Samarskii boundary conditions; extension to other nonlinear differential equations with these three boundary condition types (e.g., nonlinear wave equations, nonlinear diffusion equations) remains unexplored.

    Numerical solution of the Duffing equation with three types of boundary conditions using shifted Legendre polynomials · 2026 · DOI

  • The error analysis in Theorem 3.2 depends on N = max_{t∈[t_n,t_{n+1}]} |u″(t, f(t))|, but the paper does not establish methods to estimate or bound this maximum for the chaotic flow system or provide guidance on detecting when this bound may be violated during numerical integration of fractional-order chaotic systems.

    Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling · 2026 · DOI

  • The paper demonstrates phase portraits and temporal evolution for α = 1.0, 0.99, 0.97, and 0.03, but does not provide quantitative measures of chaos (Lyapunov exponents, bifurcation analysis) to rigorously characterize whether chaotic properties are preserved across the fractional-order spectrum when using the Kayo-Kengne-Akgül derivative with the Adams-Bashforth scheme.

    Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling · 2026 · DOI

  • The chaotic flow system application uses only fixed parameter values (a=15, b=1.0) and a single initial condition (0, 0.5, 0.5). The paper lacks systematic investigation of how the Adams-Bashforth scheme's accuracy and stability vary across the parameter space of the chaotic system (different a, b values) and multiple initial conditions, particularly for α values near the critical regions (α ≈ 0.03 where self-stabilization occurs).

    Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling · 2026 · DOI

  • The Adams-Bashforth scheme convergence proof demonstrates that |Err| → 0 as h → 0, but the paper does not provide explicit convergence rates or bounds on the error reduction as a function of step size h for different values of the fractional order α in the range [0, 1], which is critical for comparing computational efficiency against classical schemes.

    Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling · 2026 · DOI

  • The stability analysis in Theorem 3.4 requires the condition that ‖u″(ε, f(ε))‖∞ → 0 as n → ∞, but the paper does not investigate which classes of chaotic systems or fractional-order differential equations satisfy this boundedness requirement, nor does it characterize the parameter ranges (α, a, b) where this condition holds for the Kayo-Kengne-Akgül derivative framework.

    Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling · 2026 · DOI

  • The Riesz-Hartree-Fock method is presented as reconciling the mathematical ambiguity of fractional correlation energy definitions by specifying fractional operators for the α=2 limit, but validation of its capacity to recover classical HF and correlation energies across the range 1<α<2 with various basis sets has not been demonstrated.

    Fractional paradigms in quantum chemistry · 2021 · DOI

  • The stability, convergence rate, and computational scaling properties of constant-order and variable-order fractional numerical methods remain largely unknown within quantum chemistry. Systematic benchmarking studies comparing fractional quantum chemical methods against classical approaches on test systems are essential to establish practical viability.

    Fractional paradigms in quantum chemistry · 2021 · DOI

  • To date, no computational study has applied variable-order fractional calculus to the electron correlation problem in many-body quantum systems. Development of variable-order fractional Schrödinger equations with α(t) or α(t,τ) dependence for describing dynamically evolving nonlocal electron correlation effects requires both theoretical formulation and numerical implementation.

    Fractional paradigms in quantum chemistry · 2021 · DOI

  • The mathematical ambiguity in Equation (18) for fractional electron correlation energy stems from the unspecified type of fractional operator used for generalizing the Hartree-Fock model. A systematic classification and comparison of different fractional operator definitions (Riemann-Liouville, Caputo, Riesz) applied to correlation energy calculations is needed.

    Fractional paradigms in quantum chemistry · 2021 · DOI

  • The detailed mathematical formulation of fractional tensor operators to replace the (anti-)Coulomb operator in fractional quantum chemistry requires development but is explicitly deferred from this manuscript. This core methodological component needs rigorous derivation and integration into fractional Hartree-Fock and Kohn-Sham DFT frameworks.

    Fractional paradigms in quantum chemistry · 2021 · DOI

  • Future work will focus on nonlinear extensions, numerical implementation and adaptive truncation of the generalized Peano–Baker series, optimal control applications, higher-order Caputo systems, and extensions to systems with delays, impulses, and more general input spaces such as Lp classes.

    Caputo fractional systems with variable coefficients: Existence and stability results via Peano–Baker series · 2026 · DOI

  • This work presents a Caputo fractional-order typhoid fever model that integrates screen- ing, sanitation, and treatment interventions—a combination rarely studied within a frac- tional calculus framework.

    Fractional-order modeling of typhoid fever dynamics with screening, sanitation, and treatment interventions · 2026 · DOI

  • These findings can be used in further studies of fractional models and in trans- forming problems into more applicable forms, such as the Caputo formulation.

    EQUIVALENCE OF INITIAL CONDITIONS FOR A FRACTIONAL DIFFERENTIAL EQUATION WITH THE RIEMANN–LIOUVILLE DERIVATIVE · 2026 · DOI

  • This spectral theory unifies and extends existing results in viscoelasticity, signal processing, and analysis, and makes progress on an open question of Abel regarding the solution of integral equations of the first kind.

    A spectral theory of scalar Volterra equations · 2026 · DOI

  • Despite the remarkable progress achieved in recent years, several open questions persist regarding the boundedness and continuity of the Prabhakar operator in classical function spaces such as C[a, b], C 1 (a, b).

    Some Boundedness Results for the Prabhakar Integral Operator · 2026 · DOI

  • Future research may focus on extending the comparative analysis to other numerical techniques, such as wavelet-based, collocation, and Newton-type methods, to further assess the strengths and limitations of PINNs.

    On the simulation of fractional Riccati equations with physics-informed neural networks · 2026 · DOI

  • Convergence analysis is not rigorously provided; formal theoretical proof of convergence for the modified Laplace-Adomian polynomial method is absent.

    A Novel Computational Framework for Nonlinear Differential Equations Employing the Modified Laplace Adomian Polynomial Method · 2026 · DOI

  • In practice, infinite series must be replaced by finite sums, and if the approximation contains an insufficient number of terms, then the solution may contain significant error even if eigenvalues and eigenfunctions are defined exactly.

    Functional a Posteriori Estimates for the Fractional Laplacian Problem · 2026 · DOI

  • Eigenvalues φj and λj are known exactly for only a limited amount of special domains Ω, so in general we must use certain approximations ψj and θj instead.

    Functional a Posteriori Estimates for the Fractional Laplacian Problem · 2026 · DOI

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46 open questions have been extracted from the limitations and future-work passages of 493 Fractional Differential Equations Solutions papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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