• The fractional type KdV-BBM equation is derived as a pyhsical model. • The long time existence result for the Cauchy problem for the fractional KdV-BBM equation is established. • The maximal existence time is extended beyond hyperbolic time scale by using a modified energy technique. • A Fourier pseudospectral method is proposed for the numerical investigations of solutions to the fractional KdV-BBM equation. We consider a fractional Korteweg de Vries-Benjamin Bona Mahony (KdV-BBM) type equation including both fractional dispersive terms of fractional KdV and fractional BBM equations. We aim to enhance the existence time of solutions with small initial data ∥ u 0 ∥ H N + α / 2 = ϵ from 1 ϵ to 1 ϵ 2 . The proof relies on the combination of a modified energy method with Fourier techniques. In addition, the long time existence issues are investigated numerically. Numerical observations of the lifespan give an evidence of existence of solutions beyond the hyperbolic time scale. This study provides a detailed analysis from both analytical and numerical aspects for the existence of smooth solutions.

Controllability of fractional backward evolution equations with order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg" display="inline" id="d1e23"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo linebreak="goodbreak" linebreakstyle="after">∈</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math>

A generalized Caputo fractional jerk equation with Caputo antiperiodic boundary conditions: Existence of solutions, stability and numerical simulations

Analysis of semilinear fractional integro-differential equations governed by compact semigroups

On energy equality of the fractional magnetohydrodynamic equations

Existence and uniqueness results for non-linear fractional integral equation with reciprocal singularities

Variable-order fractional wave equation: Analysis, numerical approximation, and fast algorithm

Strong convergence analysis of a finite difference scheme for fractional Langevin equations with mixed fractional damping

We study pressure pulse generation and propagation in lipid monolayers by an experimental approach employing rapid photoisomerization of photoswitchable lipids (azoPC). This allows us to generate longitudinal surface pressure pulses by optical flash excitation in both free and constrained layer geometries. We compare the observed pulse shapes with a theoretical approach based on a nonlinear fractional wave equation for a surface displacement field, where a fractional time derivative term captures the hydrodynamics of the monolayer subphase. We explore channel geometries of different lengths and widths and find quantitative agreement between theory and experiment regarding pulse speed and pulse shapes. For narrow channels, we employ a one-dimensional version of the fractional wave equation to study pulse propagation without any fit parameters by using the pressure signal at a close pressure sensor as a boundary condition to predict the pressure signal at a second far sensor. A full two-dimensional description can capture all effects arising from the channel geometry for wider channels using one common set of fit parameters for the pulse excitation that can be applied to all geometries. The nonlinearity in the fractional wave equation plays no role in explaining the observed pulse shapes because pulse amplitudes generated by azoPC photoswitching remain very small.

Fractional models are essential for describing nonlinear, memory dependent wave phenomena in complex media, yet solving high order fractional nonlinear PDEs such as the Modified Caputo Fractional Kawahara Equation (MCFKE) remains challenging. This work introduces the Yang Residual Power Series Method (Yang RPSM), which integrates the Yang transform with a residual-based power series expansion to generate efficient semi-analytical solutions. Stability and convergence of the iterative scheme are established. Numerical comparisons show that the Yang RPSM outperforms natural transform decomposition technique (NTDT) and homotopy analysis technique (HAT) in accuracy and computational behavior. Applications to intracellular Ca[Formula: see text] propagation further demonstrate that the MCFKE effectively captures memory effects, nonlinear wave steepening, and dispersion-driven attenuation. The Yang RPSM provides a reliable computational tool for high order fractional PDEs and highlights the MCFKE as a biologically meaningful model for anomalous, memory driven wave processes.

Stabilization of fractional parabolic equations through designed robust Robin boundary controller

Periodic Solutions of the Discrete Fractional Relaxation Equation

Fast and accurate high-order discretization of nonlinear fractional integro-differential equations in multi-dimensional settings

This research introduces and investigates a novel class of generalized Itô-type complex Hermite polynomials, which generalize several well-known Hermite polynomials. We explore their fundamental properties, including generating functions, differential equations, recurrence relations, and connections and expansions with other special functions and polynomials. Explicit expressions for the product of two generalized Itô-complex Hermite polynomials, along with multiplication and addition formulas, are also presented. Additionally, we derive Burchnall-type and Rodrigues-type operational representations and establish the monomial properties. Finally, we have shown that the novel polynomials solve fractional and classical heat equations in one, two, and three dimensions, underscoring their versatility and potential for practical applications. Graphical representations and concluding remarks are also given.

In this paper, we analyze the stability of the fractional distributed delay models. We use the linear chain trick to convert these models into an incommensurate fractional order systems. We get the stability regions by studying the characteristic equation around equilibrium points. We investigate how the fractional order alpha _{1}, rho and a affect the stability of the models. We study the fractional order delay logistic equation and compare the influence of the distributed delay on the stability regions. Numerical simulations are exhibited to confirm the analytical results.

From Dirichlet to fractional Neumann: sensitivity and attractors of fractional reaction-diffusion equations on smoothly varying domains

In this paper, to investigate exact optical soliton solutions and their dynamical properties for the paraxial wave model involving the M-fractional derivative. To achieve this, employ the improved modified Sardar sub-equation method, which provides a systematic and efficient analytical framework for constructing closed-form solutions of nonlinear fractional wave equations. The resulting fractional complex paraxial wave dynamical (FPWD) model is of considerable importance due to its wide applicability in nonlinear optics, optical fiber communications, quantum electronics, and plasma physics. The mathematical analysis reveals the existence of new solitary wave solutions that demonstrate complex nonlinear patterns that govern the system. The research discovers a needle-type soliton structure that operates within an M-fractional paraxial wave framework, thus differentiating itself from existing research, which uses classical and other fractional operators. The research displays three-dimensional surface plots and contour maps to enhance understanding of the obtained wave structures. The study assesses how essential parameters affect solution stabilityAQ through a sensitivity analysis, which shows solitons maintain their strength and operational behavior across various physical conditions.

This paper presents a numerical approach for solving Fractional Volterra Integro-Differential Equations (FVIDEs) based on the Least Squares Method (LSM). Shifted Chebyshev polynomials are employed as basis functions to construct the approximate solution. The proposed scheme efficiently handles the combined challenges of fractional derivatives, integer-order derivatives, and a Volterra-type integral term. The accuracy and efficiency of the proposed scheme are demonstrated through several numerical examples, where the results are compared against known exact solutions. The findings indicate that the method achieves rapid convergence to the exact solution, confirming its effectiveness and potential as a powerful tool for handling this complex class of equations.

This study introduces a numerical strategy to address a new set of fractional optimal control problems governed by a singular third-kind Volterra-Fredholm fractional integro-differential equation. The governing system features a coefficient function that vanishes at the initial time, coupled with two nonlinear hereditary integral terms, representing a challenging category of fractional dynamical systems with memory. To address this problem, we develop a direct method using the generalised hat functions. The proposed methodology proceeds in three key stages. First, we derive a Hadamard fractional integral operational matrix for the chosen basis, enabling the efficient algebraic treatment of fractional operators. Second, the control variable and the Caputo-Hadamard derivative of the state function are expanded via the generalised hat functions, thereby reducing the primary infinite-dimensional problem to a finite-dimensional constrained optimisation problem. Finally, the fractional dynamics are enforced at the grid points, and the resulting constraints are incorporated via Lagrange multipliers, yielding a coupled system of algebraic equations. The effectiveness, accuracy and robustness of the method are demonstrated through two designed illustrative problems. Numerical results confirm the method's satisfactory convergence and consistent performance across different fractional orders. The approach effectively handles singular coefficients and hereditary integrals while achieving accurate approximations for the problem.

We develop a high-order space-time spectral method for nonlinear convection–diffusion equations with a Riemann–Liouville time-fractional derivative and a spectrally defined space-fractional Laplacian. The spatial discretization uses a Fourier spectral method that diagonalizes the fractional Laplacian under periodic boundary conditions. The temporal discretization employs a Petrov–Galerkin method based on generalized Jacobi functions which capture the initial singularity exactly. The nonlinear convection term is treated pseudo-spectrally, and the resulting algebraic system is solved with a damped Newton iteration. Rigorous error analysis proves exponential convergence in both space and time. Numerical experiments for various fractional orders confirm the spectral accuracy. Simulations of the fractional Burgers equation demonstrate that increasing the viscosity enhances diffusion and stabilizes the solution, while a nonlinear coefficient that significantly exceeds the viscosity leads to error growth over long time intervals. The method provides an efficient and accurate tool for simulating anomalous transport phenomena.