Open research questions in Fractional Differential Equations Solutions
232 gap statements mined from Fractional Differential Equations Solutions papers in our 7.9M-paper local library, which holds 2,499 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
The lack of positivity results for fractional perturbed problems. The inability to obtain the exact expression of the related Green's function for perturbed fractional problems. The need for a new approach to study singular perturbed nabla fractional problems.
The analysis is limited to a specific type of nonlinear Volterra integral equation. The method is sensitive to the choice of r and the grading of the mesh. The results are based on numerical experiments and may not be generalizable to all cases.
Nonlinear Volterra Integral Equation with Solution-Dependent Singular Kernels: Well-Posedness, Regularity, and Numerical Approximation · 2026 · DOIThe lack of a general framework for variable-order kernels. The need for a numerical method that can compensate for the initial singularity. The importance of deriving error estimates for the numerical method.
Nonlinear Volterra Integral Equation with Solution-Dependent Singular Kernels: Well-Posedness, Regularity, and Numerical Approximation · 2026 · DOITo the best of the authors’ knowledge, nonlinear third-order fractional heat conduction models combined with a Green’s function based fixed-point iterative formulation have not been investigated in the present setting.
Generalized third-order fractional heat conduction with nonlinear sources: a Green’s function fixed-point approach · 2026 · DOIThe paper does not mention any specific limitations of the study. The method used is limited to finding exact travelling wave solutions and may not be applicable to other types of solutions.
Analytical solutions of time-fractional non-linear model Clannish Random Walker’s Parabolic equation and its sensitivity · 2026 · DOIFuture research can apply the method used in the paper to other non-linear fractional partial differential equations. Future research can explore the applications of the novel soliton structures discovered in the paper. Future research can investigate the stability and robustness of the solutions obtained in the paper.
Analytical solutions of time-fractional non-linear model Clannish Random Walker’s Parabolic equation and its sensitivity · 2026 · DOIThe need for continued research on fuzzy fractional-order models and their advantages. The limitation of integer-order differential equations in representing complex systems.
A review on fuzzy fractional order modeling in health systems with application to cardiovascular disease · 2026 · DOIThe paper identifies the need for a posteriori estimates that can efficiently evaluate the quality of a particular numerical solution. The paper identifies the lack of fully computable error estimates for the problem.
The gap is that conventional numerical techniques may not provide accurate solutions.
The lack of a numerical scheme for solving fractional differential equations involving the Kayo-Kengne-Akgül derivative. The need for a method that can accurately model complex phenomena.
Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling · 2026 · DOIConventional methods are limited in their ability to handle strong nonlinearities efficiently and accurately.
A Novel Computational Framework for Nonlinear Differential Equations Employing the Modified Laplace Adomian Polynomial Method · 2026 · DOIThe need for more accurate and efficient methods to solve time-fractional convection–reaction–diffusion equations. The existing solutions have limitations in terms of accuracy and efficiency.
Mathematical analysis of fractional-order convection–reaction–diffusion equations under the Caputo fractional derivative · 2026 · DOIThe method is limited to solving the nonlinear Duffing equation with specific boundary conditions. The numerical results are based on a finite number of examples.
Numerical solution of the Duffing equation with three types of boundary conditions using shifted Legendre polynomials · 2026 · DOIThe lack of effective methods for solving the nonlinear Duffing equation with different boundary conditions. The need for a novel approach to handle the boundary conditions.
Numerical solution of the Duffing equation with three types of boundary conditions using shifted Legendre polynomials · 2026 · DOIThe Caputo-Fabrizio temporal fractional wave equation is a recent development in fractional calculus and needs to be solved. There is a lack of numerical methods for solving this type of equation.
Numerical Solution of a Wave Partial Differential Equation With the Caputo-Fabrizio Time-Fractional Derivative Using the Finite Element Method Under Non-Homogenous Dirichlet and Neumann Boundary Conditions · 2026 · DOIThe paper suggests that future research can focus on applying the proposed framework to study complex systems. The paper suggests that future research can focus on developing new numerical schemes for the equation.
Fractional Langevin Equation Driven by Multifractional Brownian Motion: Integral Equation Approach · 2026 · DOIThe lack of a theoretical framework for analyzing the stability of nonlinear systems with multi-term Hilfer fractional derivatives and variable coefficients. The need for a systematic approach to establish the existence and uniqueness of solutions for such systems.
STABILITY OF NONLINEAR SYSTEMS WITH MULTI-TERM HILFER FRACTIONAL DERIVATIVES AND VARIABLE COEFFICIENTS · 2026 · DOIThe lack of understanding of the Prabhakar integral operator's boundedness and continuity. The need for further study of the operator's properties and applications.
The paper suggests that more sophisticated methods can be used to estimate the integrals in the right hand side of (3.1). The paper suggests that the results can be applied to a wide range of problems involving fractional differential operators.
Existing methods may not efficiently address this singularity.
Fourth-order RBF-CFD scheme for solving the one-dimensional time-fractional convection–diffusion equation with initial weak singularity · 2026 · DOIExploring dynamics of multipeak and breathers-type solitary wave solutions in generalized higher-order nonlinear Schroedinger equation and their optical applications, - Study of generalized higher-order nonlinear Schroedinger…
To extend research on fuzzy fractional-order models and their applications. To develop more effective treatment strategies for cardiovascular diseases using the suggested technique.
A review on fuzzy fractional order modeling in health systems with application to cardiovascular disease · 2026 · DOIThe paper identifies a gap in the existing literature on numerical solutions of time-fractional Burgers equations. The paper proposes a novel method to fill this gap.
An efficient higher-order trigonometric cubic B-spline collocation method for timefractional Burgers equations · 2026 · DOIThe application of the proposed method to other complex systems. The development of new numerical schemes for solving fractional differential equations.
Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling · 2026 · DOITo extend the results to more general types of systems and equations. To apply the results to practical problems in various fields such as circuits, population dynamics, and control systems.
STABILITY OF NONLINEAR SYSTEMS WITH MULTI-TERM HILFER FRACTIONAL DERIVATIVES AND VARIABLE COEFFICIENTS · 2026 · DOI
Most-cited papers in Fractional Differential Equations Solutions
- Panel data methods for fractional response variables with an application to test pass rates · Journal of Econometrics · 2008 · 1,006 citations
- A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative · Chaos Solitons & Fractals · 2020 · 740 citations
- On the power of Dickey-Fuller tests against fractional alternatives · Economics Letters · 1991 · 370 citations
- A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control · Chaos Solitons & Fractals · 2021 · 355 citations
- Sumudu transform: a new integral transform to solve differential equations and control engineering problems · International Journal of Mathematical Education in Science and Technology · 1993 · 314 citations
- Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency · The Journal of the Acoustical Society of America · 2004 · 306 citations
- Multivariable regression model building by using fractional polynomials: Description of SAS, STATA and R programs · Computational Statistics & Data Analysis · 2005 · 295 citations
- A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods · Mathematical Methods in the Applied Sciences · 2020 · 295 citations
- Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian · The Journal of the Acoustical Society of America · 2010 · 286 citations
- Discrete fractional calculus with the nabla operator · Electronic journal of qualitative theory of differential equations · 2009 · 276 citations
Most recent work
- Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking · Reports on Progress in Physics · 2026
- NUMERICAL ANALYSIS OF A FRACTIONAL ROTATING PENDULUM OSCILLATOR · Fractals · 2026
- Fractional MHD analysis of Oldroyd–B fluid with variable thermal conductivity and thermoelectric coupling · Ain Shams Engineering Journal · 2026
- Fast and accurate high-order discretization of nonlinear fractional integro-differential equations in multi-dimensional settings · International Communications in Heat and Mass Transfer · 2026
- Fractional order mem-elements: from modeling to applications : a review · Nonlinear Dynamics · 2026
- Analytical solutions of fractional Burgers’ equation with exponential extension kernel derivatives · Journal of Computational and Applied Mathematics · 2026
- Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Network · SIAM Journal on Scientific Computing · 2026
- A study on fractional logistic growth models in the frame of singular kernel derivatives · Nonlinear Dynamics · 2026
- Computational analysis of fractional heat conduction with fading memory · Fractional Calculus and Applied Analysis · 2026
- A new endpoint-adaptive spectral method for nonlinear tempered fractional terminal value problems with non-smooth solutions · Applied Numerical Mathematics · 2026
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