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.

This paper presents the development of reverse Minkowski and reverse Hölder's fractional integral inclusions. We propose a generic class of ${\eta }_{\hslash }$ interval-valued $(\mathrm{I}.\mathrm{V})$ convex functions, which unifies various existing classes. Additionally, we obtain a discrete Jensen-type inclusion within this convexity setup. By leveraging this advanced convexity structure together with tempered fractional integral operators, we derive new Hermite–Hadamard ( $\mathrm{H}$- $\mathrm{H}$)-type, Fejér- $\mathrm{H}$- $\mathrm{H}$-type, and other fractional inclusions. Moreover, we explore the broader significance of our results, supporting them with graphical visualizations. The applications of our results are demonstrated through average value computations.

This article established a comprehensive analytical framework for fractional Hermite functions using the Atangana-Baleanu Caputo (ABC) derivative. We derived a convergent power series solution (radius $|x|<1$ for $\alpha \in (0,1)$) with explicit recurrence relations for its coefficients. Even and odd fractional Hermite functions were constructed via novel termination conditions, and a generalized Rodrigues-type formula was presented. A central result was the proof of orthogonality with respect to the weight function $W_{\alpha }(x)=e^{-x^2}{E}_{\alpha }(-\frac{\alpha }{1-\alpha }|x{|}^{2/\alpha })$, accompanied by the derivation of exact normalization constants ${\mathrm{\Lambda }}_n(\alpha )$. Numerical validation confirmed theoretical predictions, with errors $<0.5\mathrm{\%}.$ The functions $H_{n,\alpha }^{ABC}(x)$ preserved key classical properties while exhibiting distinct fractional behavior, such as cusp-like formation at the origin. Quantitative analysis demonstrated convergence to classical Hermite polynomials as $\alpha \to 1^{-}$, with root errors $<1\mathrm{\%}$ for $\alpha =0.95$. This work extends Hermite theory into the fractional domain, providing essential tools for modeling systems with memory and non-local interactions.

In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.