We develop a generating function framework for a fractional Hermite family $\{H_n^{(\alpha )}(x)\}$ defined through an explicit gamma function representation, within the setting of formal power series. The gamma normalized generating function ${\mathcal{G}}_{\alpha }(x,t)=\sum _{n=0}^{\mathrm{\infty }}{H}_n^{(\alpha )}(x)\frac{t^n}{\mathrm{\Gamma }(\alpha n+1)}.$ encodes the entire family in a single algebraic object. Working in the formal power series ring $\mathbb{C}[x^{\alpha }][[t]]$, we establish several structural properties of ${\mathcal{G}}_{\alpha }(x,t)$. We derive coefficient extraction and reconstruction formulas, obtain explicit double-sum representations, and establish an even–odd decomposition of the generating series. These results show that the generating function completely determines the underlying fractional Hermite family. We further introduce the Euler operator ${\mathcal{E}}_t=t{\mathrm{\partial }}_t$ and develop a formal functional calculus in the index variable. This leads to the dilation identity $\exp \left(s{\mathcal{E}}_t\right){\mathcal{G}}_{\alpha }(x,t)={\mathcal{G}}_{\alpha }(x,e^{s}t),$ together with the associated semigroup and infinitesimal identities. As additional algebraic consequences, we establish a homogeneous decomposition of the formal power series space, characterize the corresponding homogeneous components as eigenspaces of the Euler operator, construct the associated projection operators, derive commutation relations, and prove the invariance of the even and odd subspaces under the Euler action. All results are obtained within a purely formal coefficientwise framework and require no additional analytic assumptions. The resulting theory provides a consistent algebraic setting for studying this fractional Hermite family through generating function methods and suggests possible extensions to other classes of fractional special functions.

This article presents an extensive study of Gelfand–Shilov spaces of Roumieu type defined via sequences of positive real numbers satisfying natural growth and convexity conditions. We introduce generalized anisotropic spaces $S_{\{N\}}^{\{M\}}\left({\mathbb{R}}^n\right)$ and explore their fundamental properties, including stability under differentiation, multiplication, translation, and dilatation operators. We establish boundedness results that demonstrate these spaces form differential subalgebras of the space of continuous functions. These results enrich the functional–analytic framework of ultradifferentiable spaces and provide a foundation for further studies in microlocal analysis and partial differential equations involving ultradistributions.

The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $\varsigma =1$, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.

This paper presented a stable numerical method for solving convection dominated singularly perturbed nonlinear Volterra integro-differential equations. The equation involved a small perturbation parameter $\epsilon \in (0,1]$, which leads to a boundary layer in the solution, causing sharp gradients that classical numerical methods struggle to resolve without excessive computational effort. To handle this challenge, an exponentially fitted difference method was proposed for the differential part, which incorporated $\epsilon$ into the discrete operator to accurately capture the boundary layer on coarse meshes. The Volterra integral term was approximated using the composite Trapezoidal rule. The stability and convergence analysis confirmed that the proposed method was uniformly convergent with respect to $\epsilon$, preserving accuracy even for small values of $\epsilon$. Numerical experiments were implemented on four test problems to validate the theoretical results, demonstrating the accuracy, uniform convergence, and the method's ability to handle a boundary layer. We extended the method to handle a nonlinear problem by incorporating Newton's linearization technique. The computed result showed that the proposed method was accurate and stable under strong boundary layer condition.

This paper studies a class of fractional stochastic integro-differential systems with memory effects and control inputs. The model involves a Caputo fractional derivative of order $\alpha \in (1/2,1)$, a Volterra-type memory kernel, and stochastic perturbations driven by a Wiener process. Under standard Lipschitz and boundedness assumptions, we establish the existence and uniqueness of mild solutions in the space of mean-square continuous processes via the Banach contraction principle, together with an explicit contraction condition. We further prove Ulam–Hyers stability, providing quantitative bounds that characterize the sensitivity of solutions to perturbations. In addition, we investigate local approximate controllability through a scaled bounded-control approximation framework. We show that, for sufficiently small terminal times and for targets approaching the initial state at the fractional scaling rate $\Vert x_T-x_0\Vert \le \rho T^{\alpha -1/2}$ the controllability of the nonlinear stochastic system can be inferred from that of an associated reduced linear system using controls whose $L^2$-norms remain uniformly bounded as $T\to 0^{+}$. A finite-dimensional example is provided to demonstrate that the scaled bounded-control approximation hypothesis can be verified explicitly in a concrete setting and to illustrate the applicability of the controllability framework. The results provide a unified analytical framework for studying well-posedness, stability, and controllability in fractional stochastic systems with memory.

In this article, we discuss the existence and uniqueness results for mix derivative involving fractional operators of order $\beta \in (1,2)$ and $\gamma \in (0,1)$. We prove some important results by using integro-differential equation of pantograph type. We establish the existence and uniqueness of the solutions using fixed point theorem. Furthermore, one application is likewise given to represent our fundamental results.

In this paper, we study the existence of the solutions for a tripled system of Caputo sequential fractional differential equations. The main results are established with the aid of Mönch's fixed point theorem. The stability of the tripled system is also investigated via the Ulam-Hyer technique. In addition, an applied example with graphs of the behaviour of the system solutions with different fractional orders are provided to support the theoretical results obtained in this study.

In this article, the existence of a solution to a system of fractional equations of sequential type was investigated via Mönch's fixed point theorem. In addition, the stability of this solutions was verified by the Ulam-Hyers method. Finally, an applied example is presented to illustrate the theoretical results obtained from the existence results.

In the present paper, we consider the linear and nonlinear relaxation equation involving $\psi$-Riemann-Liouville fractional derivatives. By the generalized Laplace transform approach, the guarantee of the existence of solutions for the linear version is shown by Ulam-Hyer's stability. Then by establishing the method of lower and upper solutions along with Banach contraction mapping, we investigate the existence and uniqueness of iterative solutions for the nonlinear version with the non-monotone term. A new condition on the nonlinear term is formulated to ensure the equivalence between the solution of the nonlinear problem and the corresponding fixed point. Moreover, we discuss the maximal and minimal solutions to the nonlinear problem at hand. Finally, we provide two examples to illustrate the obtained results.

Piecewise polynomial functions serve as powerful tools for function approximation and the numerical solution of differential equations. In this study, we presented a robust numerical method for solving the time-fractional diffusion-wave equation involving the Caputo-Fabrizio fractional derivative. The proposed scheme combines the uniform hyperbolic polynomial B-spline basis for spatial discretization with a $\theta$-weighted finite difference approach for temporal integration. The uniform hyperbolic polynomial B-spline, an advanced generalization of B-splines, integrates hyperbolic functions to enhance smoothness and flexibility, making it especially well-suited for problems exhibiting hyperbolic behavior. Rigorous stability and convergence analyses were carried out to ensure the reliability of the method. To demonstrate its effectiveness, the scheme was applied to several benchmark problems. Numerical results reveal that the proposed approach is highly accurate and computationally efficient.