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 work, an epidemic model of a susceptible, exposed, infected and recovered SEIR-type is established for the distinctive dynamic compartments and epidemic characteristics of COVID-19 as it spreads across a population with a heterogeneous rate. The proposed model is investigated using a novel approach of fractional calculus known as piecewise derivatives. The existence theory is demonstrated through the establishment of sufficient conditions. In addition, result related to Hyers-Ulam stability is also derived for the considered model. A numerical method based on modified Euler procedure is also constructed to simulate the approximate solutions of the proposed model by employing various values of fractional orders. We testified the numerical results by using real available data of Japan. In addition, some results for the SEIR-type model are also presented graphically using the stochastic process, and the obtained results are discussed.

We address a classification problem for critical one-dimensional Hardy forms perturbed by a logarithmic remainder. On $(0,1)$, with the gauge $\log \frac{e}{x}=\log (e/x)$, we construct an explicit one-parameter family of Riccati weights that yields an identity-level ground-state representation. This produces a continuum of logarithmic improvements of the critical Hardy inequality with a computable remainder coefficient and an explicit positive ground state. We then derive a quantitative interior stability estimate: The Hardy deficit controls the distance to the associated ground state on every interior subinterval. We further classify the constant-coefficient logarithmic remainder class by reducing the associated ground-state ordinary differential equation (ODE) to a Euler equation in the logarithmic variable, and we obtain an interior compactness statement for sequences with vanishing deficit. As an application, we prove positivity and a priori bounds for a class of Dirichlet Schrödinger problems with critical singular potentials.

The basic objective of this paper is to utilize the factorization technique method to derive several properties such as, shift operators, recurrence relation, differential, integro-differential, partial differential expressions for Gould-Hopper-Frobenius-Genocchi polynomials, which can be utilized to tackle some new issues in different areas of science and innovation.

The primary objective of this paper is to investigate and establish existence and uniqueness results for solutions of nonlinear Volterra-Fredholm integro-differential equations (VFIDEs) of fractional order, specifically for $1<\eta <2$. By leveraging fixed-point theorems and contraction mapping principles within Banach spaces, we derive comprehensive results for both one-dimensional and two-dimensional nonlinear fractional-order equations. By presenting sufficient conditions, we ensure the existence and uniqueness of a fixed point associated with the operator form of the VFIDEs. Our analysis provides a rigorous framework for understanding the behavior of such equations, and the results obtained in this study enhance our knowledge of fractional integro-differential equations (FIDEs). To illustrate the practical application of these theoretical results, two examples are provided that demonstrate the uniqueness of solutions.

This study examines the influence of drought on predator–prey systems under the variable-order (VO) fractional derivative. It is applied to the wildebeest–lion system of the Serengeti. First, the well-posedness of the system is ensured by the existence, uniqueness, and Ulam–Hyers (UH) stability of the solution. A finite difference method is presented, coupled with a neural network (NN) approach for numerical validation. The numerical results show the effect of the VO fractional derivative and the intensity of the drought. The results demonstrate that a critical drought threshold exists for the drought impact parameter $\gamma$, beyond which the healthy prey populations decline by over $90\mathrm{\%}$ from 6643 when $\gamma$ is 0.20 to 407 when $\gamma$ is 0.40, and the risk of extinction is very high. As the fractional order decreases from 0.5, the ecological memory is increased, resulting in increased predator populations (from 4898 to 8974 when $\gamma$ is 0.1) and the long-term effects of the drought. The VO framework produces qualitatively different dynamics than constant-order models, featuring time-dependent stability and attractor morphing, which makes it more suitable for modelling real-world ecological systems under climate stress. The NN approach also demonstrates excellent predictive capabilities, achieving $R^2=1.0$ and RMSE $<12$ for all populations. These metrics validate our numerical scheme and provide a computationally efficient quick scenario analysis. The novelty of our analysis is the combination of a VO operator, finite difference method, and neural computing in a unified framework for analyzing nonlinear fractional ecological systems. This study provides a mathematically sound framework for understanding drought-induced population shifts and offers practical computational tools for ecological forecasting under climate change.