In this talk, we investigate a class of generalized time-variable fractional integro-differential equations involving multi-term Caputo derivatives of variable order and nonlocal initial conditions. The model consists of a second-order classical derivative coupled with several variable-order fractional operators and a nonlinear source term depending on a fractional derivative of the unknown function. The initial condition is given in a nonlocal integral form, reflecting global memory effects.

Variable-order fractional operators arise naturally in applications where the memory intensity evolves over time, but their analysis is significantly more challenging than in the constant-order case. In particular, fundamental properties such as the semigroup law and simple inverse relations between fractional integrals and derivatives no longer hold. Using the definition of variable-order Caputo derivatives and without relying on invalid composition rules, we derive an equivalent Volterra-type integral formulation of the problem. This formulation provides a suitable framework for analysis and allows us to establish the existence of mild solutions via fixed point techniques, such as the Leray–Schauder alternative, under natural continuity and growth assumptions.

Finally, a numerical example is presented to illustrate the theoretical results and to demonstrate the influence of the time-dependent fractional order and the nonlocal initial condition on the qualitative behavior of solutions.

The classical Hardy–Littlewood–Sobolev theory establishes fundamental mapping properties for integral operators of potential type. It provides precise boundedness conditions for Riesz potentials acting between Lebesgue spaces, based on the order of the potential, the space dimension, and the integrability exponent. The present research develops a methodological framework for extending this theory beyond its classical limits, as demonstrated through a series of recently obtained results on Riesz potential-type operators. In particular, boundedness is studied not only in classical Lebesgue spaces but also in their modern extensions, such as grand Lebesgue spaces. These spaces form a refined scale for describing integrability, especially for functions with borderline singularities. A key focus lies in the case where the classical Sobolev condition is violated. It is shown that under certain parameters, the Riesz potential-type operator with a power–logarithmic kernel remains bounded from an $L^p$ space to a generalized Hölder space, even when the order of the potential exceeds the classical critical exponent. In this setting, the image of an integrable function is proven to possess a quantified generalized Hölder smoothness. The fundamental methodological contribution of this research is the development of a mapping theory for generalized function spaces, namely, grand Lebesgue spaces and generalized Hölder spaces. The techniques used include spectral analysis via Fourier–Laplace multipliers, Zygmund-type estimates for the continuity modulus, and the properties of grand Lebesgue spaces. This talk aims to outline how such a methodology paves the way for new, more general forms of the Hardy–Littlewood–Sobolev theory, with potential implications for problems in mathematical physics and fractional calculus.

In this talk we analyze the existence of positive solutions for an h-Riemann–Liouville fractional differential equation (E) that involves a positive parameter and a singular nonlinearity that changes sign, subject to nonlocal boundary conditions (BC) incorporating Riemann–Stieltjes integrals and h-Riemann-Liouville fractional derivatives of various orders. Because the nonlinearity may take negative values, the problem is referred to as a semipositone fractional boundary value problem. The h-Riemann–Liouville fractional derivative extends several well-known fractional derivatives, including the classical Riemann–Liouville derivative when h(t)=t, the Hadamard derivative when h(t)=ln t, as well as other related fractional operators. We establish intervals for the parameter for which problem (E), (BC) admits at least one positive solution. We begin by deriving the Green function corresponding to the problem and examining several of its key properties. Afterwards, through a suitable change of variables, we reformulate the original problem into an equivalent one. An operator is then constructed in a suitable Banach space, and its fixed points correspond precisely to the solutions of the equivalent problem. Our main results are obtained by applying the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type. In the end, we present examples that highlight the usefulness of the established results.

Measles, a highly contagious disease, remains a significant public health challenge despite the availability of effective vaccines whose spread dynamics are influenced by population density, contact rates, and vaccination coverage. Mathematical modeling is essential for forecasting outbreaks and evaluating intervention strategies. Thus, we aimed to analyze the threshold dynamics of measles in Algeria.

This study used the SEIRV+D (Susceptible–Exposed–Infected-Recovered–Vaccinated+Deceased) model, which tracks six key groups, to analyze measles transmission. We performed equilibrium analysis, the Disease-Free Equilibrium (DFE) and derived Endemic Equilibrium (EE). The stability of these equilibria was analyzed through the basic reproduction number ($\mathcal{R}_0$). Stability analysis confirms that the DFE is locally asymptotically stable when $\mathcal{R}_0<1$, while the EE becomes locally asymptotically stable when $\mathcal{R}_0>1$, and the DFE is globally asymptotically stable when $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$.

Furthermore, the detailed bifurcation analysis reveals the occurrence of a forward bifurcation at $\mathcal{R}_0=1$, and sensitivity analysis helped identify the parameters that most influence the basic reproduction number $\mathcal{R}_0$, highlighting the critical role of vaccination-related factors. Numerical simulations for Algeria demonstrate that increasing the vaccination rate from suboptimal coverage to the WHO-recommended threshold of $95 \%$ significantly reduces both the cumulative number of infections and measles-attributable deaths. The model clearly shows the direct relationship between vaccination coverage, the magnitude of $\mathcal{R}_0$, and the long-term disease burden. The main results show that measles elimination in Algeria was achieved through modest vaccination coverage.

This research explores the construction of exact traveling wave solutions for significant nonlinear wave equations, specifically focusing on the coupled Korteweg–de Vries (KdV) equations and the coupled Hirota–Satsuma system. These systems are vital in mathematical physics for modeling complex wave interactions in fields such as fluid dynamics and plasma physics. To address the inherent complexity of these nonlinear partial differential equations (NLPDEs), the study employs two robust analytical techniques: the tanh--coth method and the generalized $\exp(-\phi(\xi))$-expansion (GEE) method. The core of the methodology involves applying a traveling wave transformation, $\xi = x - ct$, which reduces the original NLPDEs into more manageable nonlinear ordinary differential equations (ODEs). By assuming specific expansion forms for the solutions—hyperbolic functions for the tanh--coth approach and an exponential-based auxiliary equation for the GEE method—the researchers are able to convert the differential problems into systems of nonlinear algebraic equations. The study leverages the symbolic computation power of Maple 17 to solve these intricate algebraic systems, leading to the derivation of a diverse array of exact solutions. These results encompass various wave structures, including solitary waves, kink-type solutions, periodic waveforms involving trigonometric functions, and rational wave profiles. The comparative application of these two methods reveals that while the tanh--coth method is exceptionally efficient for capturing solitonic and kink behaviors, the GEE method offers greater flexibility by generating a broader class of solutions. Ultimately, the work demonstrates that the integration of these analytical frameworks with symbolic computing provides a powerful toolkit for understanding nonlinear phenomena. The derived solutions not only validate the effectiveness of the chosen methods but also provide deeper physical insights into how waves propagate and interact within complex nonlinear media, reinforcing the importance of exact solutions in the ongoing study of nonlinear dynamics.

Difference equations play a fundamental role in describing the evolution of various phenomena over discrete time intervals. Unlike differential equations, which model continuous processes, difference equations are particularly suitable for systems where changes occur at specific time steps. Over the last twenty-five years of the twentieth century, the theory of difference equations witnessed remarkable development, driven by both theoretical advances and the increasing demand for discrete-time models in applied sciences.

A difference equation is essentially a functional relation among the terms of an unknown sequence, derived from known physical, biological, or economic principles. Once such a relation is established, the equation can be analyzed and solved using various mathematical tools, including analytical methods, numerical approximations, and computational techniques. Of particular interest in modern research are nonlinear and rational difference equations, whose dynamics often exhibit complex behaviors such as oscillations, bifurcations, and chaos.

In this context, several researchers have focused on the qualitative analysis of systems of difference equations. In particular, in 2013, Y. Yazlık, D. T. Tollu, and N. Taşkara investigated the form of solutions of certain systems of rational difference equations. Their work contributed significantly to our understanding of the structural properties and long-term behavior of such systems, and it has motivated further research on stability, boundedness, and convergence properties, especially for systems involving special number sequences such as Fibonacci and Tribonacci numbers.

In this work, we explored the form of solutions, stability character and asymptotics of thenon-linear system of difference equations of the order p + 1.
8><>:
xn+1 = ±1
yn-(p-1)(xn-p±1)+1
x ±1
n-(p-1)(yn-p±1)+1yn+1 = ;
n 2 N0; p ≥ 1
where x
-p; x-(p-1); x-(p-2); : : : ; x0; y-p; y-(p-1); y-(p-2); : : : ; y0 are real initial values with
certain conditions.

There are several methods and techniques available for finding exact solutions to nonlinear differential equations, including the Darboux transformation, Ricatti method, Kudryashov method, Hirota bilinear transformation method, Lie symmetry method, extended tanh function method, G’/G expansion method, G’/G² expansion method, and the sine-Gordon approach, among others. Regarding the Drinfel’d–Sokolov–Wilson (DSW) equation, modified extended direct algebraic methods yielded solutions in the form of bell, anti-bell, periodic, and dark solitary waves in 2017, while series solutions were obtained using the Adomian decomposition method in 2022. In the field of fractional calculus, various types of fractional derivatives have been applied, such as the Beta derivative, Caputo fractional derivative, conformable fractional derivative, Riemann–Liouville derivative, and truncated M-fractional derivative. Many researchers are focused on constructing exact solutions for fractional differential equations. In 2023, singular bright, dark, periodic, bell, and lump-type water wave solutions to the coupled nonlinear fractional Drinfel’d–Sokolov–Wilson (FDSW) model with the Beta derivative were explored using the generalized rational exponential function method. However, the exact form of Weierstrass function-type solutions has not yet been established. Therefore, in this study, we aim to construct new travelling wave solutions for the FDSW model using the complex method and compare our results to those that are already known.

Introduction. The numerical range is a central tool in spectral theory,
providing geometric insight into stability, perturbations, and spectral
localization in associative Banach algebras and operator theory. In the
non-associative setting of Banach Jordan algebras, however, a complete
analogue has remained underdeveloped. This work addresses this gap by
presenting a systematic theory of numerical ranges adapted to Banach
Jordan algebras, with particular emphasis on their interaction with Jordan
spectra and pseudospectra.
Methods. We introduce the quadratic numerical range WJ (a), defined
via the quadratic operator Ua, which naturally replaces left–right
multiplication in the associative case. The analysis combines Jordan functional
calculus, duality arguments, and convexity methods to study structural
properties of WJ (a) in general unital Banach Jordan algebras, with
special attention to the case of special Jordan algebras embedded in associative
C∗-algebras.
Results. We prove that the quadratic numerical range is non-empty,
bounded, closed, and convex. A central result establishes spectral containment,
showing that the Jordan spectrum is contained in the quadratic
numerical range. In addition, we obtain pseudospectral inclusion results,
demonstrating that Jordan pseudospectra are contained in explicit neighborhoods
of WJ (a). In the special Jordan setting, we show that WJ (a) is
contained in the square of the classical numerical range, thereby extending
known results from the associative theory. Several illustrative examples
in finite-dimensional settings are provided.
Conclusions. This work places numerical range theory in Banach
Jordan algebras on a rigorous and complete foundation, opening new avenues
for the study of spectral stability, perturbation theory, and nonassociative
spectral analysis.

In this paper, we study composition operators on the Hilbert space of complex-valued harmonic functions in the unit disc, focusing on their boundedness and structural properties. We analyze how analytic self-maps of the disc determine the behavior of the induced operators and extend several classical results from spaces of analytic functions to Hilbert space of complex-valued harmonic setting. We identify classes of analytic self-maps that generate isometric composition operators and prove that the boundedness of a composition operator implies that its symbol is a self-map of the disc. Boundedness is further characterized using Poisson integral estimates and integral mean techniques, which yield norm inequalities capturing the interaction between the analytic and co-analytic components of complex-valued harmonic functions. These inequalities provide refined upper and lower bounds for the operator norm in terms of the symbol evaluated at the origin, highlighting differences between harmonic and purely analytic theories. Additionally, we examine the relationship between reproducing kernels and composition operators, demonstrating that the adjoints of composition operators map reproducing kernels into reproducing kernels. This property leads to a characterization of composition operators via their adjoints and clarifies the structure of bounded operators on this space. Together, these results establish a comprehensive framework for understanding bounded composition operators in the context of Hilbert space of complex-valued harmonic functions.

This research presents a comprehensive investigation into the convergence analysis of approximate solutions for the generalized one-dimensional linear telegraph equation, utilizing the Reproducing Kernel Hilbert Space (RKHS) method. The primary objective of this study is to establish a robust mathematical framework that bridges the gap between numerical approximations and exact analytical solutions. By employing a sophisticated iterative procedure within the RKHS environment, we rigorously demonstrate that the generated approximate results exhibit a strong and consistent convergence behavior toward the exact solutions derived from Fourier series expansions. The theoretical foundations of the RKHS technique allow for a systematic treatment of the telegraph equation, which is a fundamental model in various physical and engineering phenomena, such as signal propagation and vibration analysis. To evaluate the practical performance and reliability of the proposed methodology, several numerical experiments were conducted. The empirical results consistently highlight the method's exceptional precision, stability, and computational integrity. Furthermore, the study underscores a remarkably rapid convergence rate, signifying that high-fidelity results can be achieved with minimal computational effort compared to traditional numerical schemes. These findings validate the RKHS method as a highly efficient and potent alternative for solving linear partial differential equations, confirming its significant potential for broader applications in the field of applied mathematics and numerical analysis.