In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function $\Psi .$ The authors prove several inequalities for newly defined GPF-integral with respect to another function $\Psi .$ Our consequences will give noted outcomes for a suitable variation to the GPF-integral in the sense of another function $\Psi$ and the proportionality index $\varsigma$ . Furthermore, we present the application of the novel operator with several integral inequalities. A few new properties are exhibited, and the numerical approximation of these new operators is introduced with certain utilities to real-world problems.

We investigate a time scale version of two auxiliary functions for the class of convex functions. We derive several novel dynamic inequalities for these classes of convex functions. Applications of these consequences are taken into consideration in special means. Furthermore, illustrative examples are introduced to help our outcomes. Meanwhile, we communicate a few particular cases which may be deduced from our main outcomes.

The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics and genetics, specifically in chemistry. The Caputo and Antagana-Baleanu fractional derivatives in the Caputo sense are used to test the intricacies of this mechanism numerically. In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. Moreover, the stability of the approach was demonstrated using fixed point theory. Several illustrative cases are described that strongly agree with the precise solutions. Moreover, tables and graphs are included in order to conceptualize the influence of the fractional order and on the previous findings. The projected technique illustrates that only a few terms are sufficient for finding an approximate outcome, which is computationally appealing and accurate to analyze. Additionally, the offered procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, but it could be generalized to other complex physical phenomena.

In this work, a new strategy to derive inequalities by employing newly proposed fractional operators, known as a Hilfer generalized proportional fractional integral operator ($ \widehat{\mathcal{GPFIO}} $). The presented work establishes a relationship between weighted extended Čebyšev version and Pólya-Szegö type inequalities, which can be directly used in fractional differential equations and statistical theory. In addition, the proposed technique is also compared with the existing results. This work is important and timely for evaluating fractional operators and predicting the production of numerous real-world problems in varying nature.

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

In Hilbert space, we develop a novel framework to study for two new classes of convex function depending on arbitrary non-negative function, which is called a predominating ℏ-convex function and predominating quasiconvex function, with respect to $\eta$ , are presented. To ensure the symmetry of data segmentation and with the discussion of special cases, it is shown that these classes capture other classes of $\eta$ -convex functions, $\eta$ -quasiconvex functions, strongly ℏ-convex functions of higher-order and strongly quasiconvex functions of a higher order, etc. Meanwhile, an auxiliary result is proved in the sense of $\kappa$ -fractional integral operator to generate novel variants related to the Hermite–Hadamard type for $pth$ -order differentiability. It is hoped that this research study will open new doors for in-depth investigation in convexity theory frameworks of a varying nature.