Open research questions in Fractional Differential Equations Solutions

What the literature leaves open

• The paper identifies a gap in the understanding of the properties and dependence on parameters of hyperbolic Ateb-functions. It also identifies a need for a more efficient method for calculating Ateb-functions, which is addressed by the proposed Taylor series approach.

  Investigating Impact of Parameters on Hyperbolic Function Generalization · 2026 · DOI

• Investigate the same problem for other types of processes - Determine the precise limit value for other parameters - Study the small ball probabilities for other types of norms

  Chung-type laws of the iterated logarithm for m-fold weighted integrated fractional processes · 2026 · DOI

• The unknown exact value of the Chung-type law of the iterated logarithm for fractional Brownian motion. The lack of small ball probabilities for weighted integrated fractional processes. The need for a precise limit value of the Chung-type law of the iterated logarithm for fractional Brownian motion.

  Chung-type laws of the iterated logarithm for m-fold weighted integrated fractional processes · 2026 · DOI

• The angular domain remains a genuine restriction - The method supplies a closed representation through the Wright function, whereas the complex arguments generated by inversion require stronger assumptions than weak-solution or numerical approaches - The front is prescribed, linear, and of constant speed

  Diffusion Equation with Riemann–Liouville Derivative in the Angular Domain · 2026 · DOI

• The paper identifies a gap in the existing literature by considering a boundary value problem for a fractional-order diffusion equation with a variable lower limit of integration. The gap is related to the lack of explicit solutions for such problems.

  Diffusion Equation with Riemann–Liouville Derivative in the Angular Domain · 2026 · DOI

• The 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 · DOI

• Future 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 · DOI

• 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 · DOI

• The 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 · DOI

• The 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 · DOI

• 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.

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

• The 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.

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

• Future research can focus on applying the proposed method to other types of differential equations. Future research can focus on improving the accuracy of the proposed method.

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

• The paper identifies a gap in the numerical solution of Fredholm integral equations. The gap is that conventional numerical techniques may not provide accurate solutions.

  An Effective Numerical Approach for Solving Second-Kind Fredholm Integral Equations · 2026 · DOI

• The 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 · DOI

• 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 · DOI

• The traditional Laplace transform faces challenges when dealing with nonlinear differential equations. Conventional 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 · DOI

• The 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 · DOI

• The lack of a comprehensive framework for analyzing coupled fractional differential inclusions with non-convex set-valued nonlinearities. The need for a multivalued fixed-point approach that can accommodate both convex and non-convex set-valued nonlinearities.

  On Sequential Coupled Caputo-Type Fractional Differential Inclusions with Coupled Boundary Conditions: A Multivalued Fixed-Point Approach · 2026 · DOI

• The 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 · DOI

• The 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 · DOI

• The 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 · DOI

• The 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 · DOI

• A critical research gap in analyzing GIFDSs for both commensurate and incommensurate weight functions. The need for a mathematical framework that captures non-uniform multicomponent system dynamics.

  Generalized Incommensurate Fractional Differential Systems: Commensurate and Incommensurate Weight Analyses, Existence-Uniqueness, HU Stability, and Neural Network Applications · 2026 · DOI

• Traditional models cannot fully explain the complexity of brain tumor growth. There is a need for a model that incorporates past effects and memory-based behavior.

  Fractional Analysis of Brain Tumor–Immune System Interaction · 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
• On the power of Dickey-Fuller tests against fractional alternatives · Economics Letters · 1991 · 370 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
• 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
• On the power of unit root tests against fractional alternatives · Economics Letters · 1994 · 251 citations
• Causal theories and data for acoustic attenuation obeying a frequency power law · The Journal of the Acoustical Society of America · 1995 · 199 citations
• APPLICATION OF THE CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL TO KORTEWEG-DE VRIES-BURGERS EQUATION∗ · Mathematical Modelling and Analysis · 2016 · 195 citations

Most recent work

• Solutions of Volterra–Fredholm type fractional integro-differential equations in terms of shifted Gegenbauer wavelets compared with the solutions by Genocchi polynomial method · Journal of Computational and Applied Mathematics · 2026
• Fractional order modeling of hepatitis C transmission dynamics with physics-informed neural network solutions · BMC Infectious Diseases · 2026
• Functional a Posteriori Estimates for the Fractional Laplacian Problem · Computational Methods in Applied Mathematics · 2026
• An RBF-based method with optimal point selection for solving two-dimensional multi-term time-fractional PIDEs with weakly singular kernels · Mathematics and Computers in Simulation · 2026
• Fractional Logarithmic Double Phase Problems: Qualitative Analysis in the Anisotropic Case · SIAM Journal on Mathematical Analysis · 2026
• The generalized Duhamel principle for fully coupled systems of fractional order · Fractional Calculus and Applied Analysis · 2026
• A Crank-Nicolson ADI compact difference scheme for the two-dimensional tempered space-fractional diffusion equation · Computers & Mathematics with Applications · 2026
• Globally minimizing a class of fractional multiplicative problems using the separability of relaxation problem · Journal of Computational and Applied Mathematics · 2026
• Dual memory effects and epidemic thresholds in a fractional-order SEIR model · Applied Mathematics and Computation · 2026
• A review on fuzzy fractional order modeling in health systems with application to cardiovascular disease · International Journal of Mathematics and Computer in Engineering · 2026

Related topics in Mathematics