In this work, we present a detailed study of the interplay between even- and odd-indexed symmetric functions and classical number sequences, approached from both algebraic and combinatorial perspectives. Symmetric functions, which occupy a central role in algebraic combinatorics, provide a natural framework for exploring underlying structures in sequences of numbers and polynomials. By focusing on the families S_{2n-1}​ and ϕ2n​, we systematically examine how the parity of indices affects the algebraic behavior and combinatorial interpretations of these functions. This investigation is facilitated by constructing exponential generating functions associated with these families, enabling a precise encoding of their structure and revealing the intricate relationships among their terms.

Through this approach, we derive new convolution identities connecting classical sequences, such as Bernoulli, Euler, and Genocchi numbers, with important polynomial families, including bivariate Fibonacci, Lucas, Mersenne, and Balancing polynomials. These formulas not only generalize existing results but also provide closed-form generating functions and unified symmetric representations that encompass multiple polynomial structures within a single coherent framework. The distinction between even and odd indices emerges as a crucial factor, uncovering fundamental algebraic properties that govern the structure of these identities and illuminating the subtle effects of parity on recurrence relations and summation patterns.

Moreover, our methodology extends naturally to Gaussian (p,q)-Fibonacci and (p,q)-Lucas numbers, where the introduction of complex initial conditions enriches the symmetric relations and leads to novel convolution formulas. This extension highlights the versatility of the approach and establishes a systematic bridge between symmetric function theory and generalized number sequences. The results offer a unifying methodology for studying polynomial families, generating functions, and their combinatorial properties, opening new avenues for research in combinatorics, number theory, and the theory of special functions.

This study presents a generalized integral transform (GIT) approach for analyzing and modeling a wide class of fractional-order mathematical systems arising in physical, engineering, and biological sciences. The investigated models include Newton’s law of cooling, the logistic population growth equation, and the blood alcohol concentration model, each formulated using distinct fractional derivatives such as the Caputo, Caputo–Fabrizio (CF), modified Atangana–Baleanu–Caputo (mABC), and constant proportional Caputo (CPC) derivatives. These fractional operators effectively describe memory-dependent and non-local characteristics inherent in many natural and engineered processes. Analytical solutions of the proposed models are derived through the generalized transform method, and graphical illustrations are provided to demonstrate the influence of various fractional orders on system dynamics. The results reveal that the GIT technique offers a unified, powerful, and efficient framework for solving a broad spectrum of fractional differential equations with diverse kernels. Furthermore, it integrates several classical and modern transforms—including the Laplace, Sumudu, Elzaki, and Formable transforms—as special cases, thereby simplifying computation and enhancing both generality and adaptability. This unified formulation provides researchers with a flexible analytical tool capable of addressing diverse problems without redefining operators for each model. The comparative analysis validates the stability, accuracy, and consistency of the proposed technique. Overall, this work highlights the efficacy, robustness, and broad applicability of the generalized integral transform, establishing a firm foundation for future explorations of hybrid fractional models and their interdisciplinary applications.

This paper investigates an approximate analytical solution of the imbibition phenomenon arising in multiphase flow during secondary oil recovery. Imbibition is a capillarity-driven displacement process in which a wetting phase penetrates a porous medium and displaces a non-wetting phase, significantly influencing recovery efficiency in fractured and water-wet reservoirs. The structural characteristics of the porous matrix play a decisive role in this mechanism. In heterogeneous porous media, pore geometry and permeability vary spatially, resulting in non-uniform flow behavior, whereas homogeneous media exhibit uniform pore distribution and consistent transport properties. A comparative analysis of counter-current imbibition in heterogeneous and homogeneous porous formations is presented to assess the influence of medium variability on saturation evolution and displacement dynamics. The governing nonlinear partial differential equation describing the imbibition process is formulated using fundamental principles of mass conservation and multiphase flow theory. Owing to the inherent nonlinearity of the model, obtaining closed-form exact solutions is generally intractable. To address this challenge, the Optimal Homotopy Analysis Method (OHAM) is employed to construct a convergent homotopy series solution. The method provides flexibility through an auxiliary convergence-control parameter, enabling improved accuracy and stability without requiring small perturbation parameters. The approximate analytical results are validated numerically, and graphical representations are presented to illustrate the effects of key physical parameters on fluid saturation profiles. The study demonstrates that OHAM serves as an efficient and reliable semi-analytical technique for modeling nonlinear imbibition processes and offers valuable insights into fluid transport behavior in complex porous systems relevant to enhanced oil recovery applications.

In this study, exact analytical solutions of the nonlinear Tzitzéica–Dodd–Bullough (TDB) equation are obtained by employing the φ⁶-expansion method. The TDB equation is an important nonlinear evolution model that arises in several areas of applied mathematics and physics, including differential geometry, plasma physics, nonlinear field theory, and wave propagation phenomena. By introducing an appropriate traveling wave transformation, the governing nonlinear partial differential equation is reduced to an ordinary differential equation, which is subsequently solved using a systematic algebraic procedure associated with the φ⁶-expansion framework. The resulting solutions are expressed in logarithmic forms, leading to diverse classes of nonlinear wave structures such as solitary waves, periodic waves, and singular wave profiles. These analytical expressions reveal the rich dynamical behavior of the model and provide insight into the influence of system parameters on wave amplitude, localization, and propagation characteristics. Furthermore, the obtained results demonstrate that the proposed method offers an efficient and reliable mathematical tool for constructing exact solutions of nonlinear equations involving exponential nonlinearities. The derived solutions may also serve as useful benchmarks for validating numerical simulations and for investigating nonlinear wave interactions in related physical systems. Overall, this work contributes to the theoretical understanding of nonlinear wave dynamics and highlights the applicability of the φ⁶-expansion method to a broad class of nonlinear evolution equations arising in applied science and engineering contexts.

This study presents a rigorous stability analysis of Interior Penalty Discontinuous Galerkin (IP-DG) spatial discretizations for the complex Ginzburg–Landau equation (CGLE), a pivotal model governing nonlinear wave dynamics in optics, superconductivity, and fluid mechanics. The CGLE’s capacity to describe the evolution of slowly varying wave envelopes—characterized by intense nonlinearity and dissipative effects—is essential for modeling chaotic regimes and vortex-dominated systems. However, these dynamics pose significant numerical challenges; preserving physical properties such as amplitude boundedness and nonlinear stability is paramount for reliable long-term simulations.

We investigate a unified framework of IP-DG formulations, specifically comparing the Symmetric (SIPG), Non-symmetric (NIPG), and Incomplete (IIPG) methods. Our analysis establishes semi-discrete stability estimates and validates them through systematic numerical experiments. A central focus is placed on the role of the penalty parameter, where we demonstrate that its magnitude is the primary driver of numerical stability. We show that while the SIPG and IIPG schemes require a threshold penalty value to ensure coercivity and bound the solution norm, the NIPG scheme remains stable for a broader range of parameters, albeit with different convergence characteristics. Notably, the SIPG method exhibits the highest robustness in highly nonlinear settings, provided the penalty parameter is sufficiently tuned to suppress unphysical oscillations.

By clarifying the interplay between the penalty parameter, numerical dissipation, and nonlinear growth, this work offers a definitive guide for selecting DG discretizations tailored to specific physical regimes. These findings facilitate the development of high-fidelity computational tools for laser dynamics, turbulence modeling, and Bose–Einstein condensation research.

McKean-Vlasov stochastic differential equations (MVSDEs), also known as mean-field
SDEs, are essential for describing systems where particle coefficients depend on both the
individual state and the marginal distribution of the population. While classical theory is
well-established, this study explores these equations within the G-expectation framework
to address model uncertainty and volatility ambiguity.
We investigate the G-MVSDE under the assumption of Lipschitz continuity for the
drift and diffusion coefficients with respect to the state variable and the 2-Wasserstein
distance. To establish the existence and uniqueness of solutions, the Picard successive
approximation scheme is employed. The convergence of this iterative process is rigorously
analyzed using G-stochastic calculus, specifically leveraging the subadditive properties of
G-expectation and Burkholder-Davis-Gundy (BDG) type inequalities.
The research demonstrates that the Picard iteration sequence forms a Cauchy se-
quence in the complete space L2(Ω, C([0, T ], R^d)), ensuring convergence to a unique strong
solution. Furthermore, we establish the stability of these solutions by deriving quanti-
tative bounds. By applying Gronwall’s inequality, we prove that the solution depends
continuously on small perturbations in initial conditions and coefficients.
This study advances the theoretical foundation of nonlinear diffusions under G-Brownian
motion. By confirming that G-MVSDEs are well-posed and stable, this work provides a
robust framework for applications in mean-field games, financial risk management, and
large-scale interacting particle systems where uncertainty is a primary factor.

This paper develops an innovative derivative-free optimization method for solving large-scale nonlinear monotone systems on Hadamard manifolds. The proposed algorithm is built as a convex combination of two classical conjugate gradient schemes, Fletcher–Reeves and Polak–Ribiere–Polyak, thereby inheriting the good global convergence properties of the former and the practical efficiency of the latter. Unlike conventional Riemannian optimization methods, which typically require gradients or Jacobians that may be unavailable or prohibitively expensive to compute on curved spaces, the new approach is genuinely derivative-free. It integrates hybrid conjugate gradient strategies with hypersurface projection techniques defined via retractions and vector transport, ensuring that all iterates remain on the manifold while preserving suitable conjugacy and descent properties. One feature of the algorithm is its function-based line search, which employs Armijo- and Wolfe-type conditions to guarantee sufficient descent without using derivatives of the underlying operator, thereby mitigating stagnation issues commonly observed in derivative-free schemes. Under standard assumptions of monotonicity and Lipschitz continuity, a rigorous convergence analysis establishes global convergence of the iterates to a solution of the monotone system. Extensive numerical experiments on test problems with dimensions up to 50,000 illustrate the method’s efficiency, robustness, and competitiveness relative to existing state-of-the-art algorithms for large-scale nonlinear systems. Finally, when applied to an image restoration problem, the method successfully reconstructs severely degraded images, achieving high-quality reconstructions as quantified by standard performance metrics such as Peak Signal-to-Noise Ratio (PSNR) and Signal-to-Noise Ratio (SNR), demonstrating its practical relevance in imaging applications.

Introduction:
Abdominal aortic aneurysm (AAA) rupture risk is primarily assessed by maximum diameter, although biomechanical and hemodynamic mechanisms play a central role in wall degeneration and thrombus formation. From a fluid-mechanics standpoint, pathological remodeling is associated with low shear magnitude and oscillatory near-wall flow. This study formulates AAA evaluation as a patient-specific computational hemodynamics problem with spatial quantification of adverse shear environments.

Methods:
Fifteen AAA and ten healthy infrarenal aortic geometries were reconstructed and discretized using unstructured tetrahedral meshes with boundary layer refinement. Blood flow was modeled as an incompressible Newtonian fluid governed by the unsteady three-dimensional Navier–Stokes equations. Pulsatile inlet conditions were derived from a validated multiscale (0D–1D) cardiovascular model, and three-element Windkessel (RCR) boundary conditions were imposed at the iliac outlets. Hemodynamic indices were computed over the cardiac cycle: Time-Averaged Wall Shear Stress (TAWSS), Oscillatory Shear Index (OSI), and Relative Residence Time (RRT). The percentage of vessel surface exceeding pathological thresholds was quantified regionally.

Results:
AAA geometries exhibited significantly greater surface exposure to low TAWSS and elevated RRT compared with healthy controls (p < 0.001), indicating expanded regions of flow deceleration and prolonged near-wall residence. OSI showed weaker discriminatory capability in the infrarenal region. Disturbed flow patterns extended beyond the aneurysm sac into adjacent segments.

Conclusions:
Surface-based mathematical characterization of shear stress distributions enhances discrimination between healthy and aneurysmal aortas. The combined metrics TAWSS and RRT provide stronger predictive insight than oscillatory measures alone, supporting their integration into quantitative AAA risk modeling.

The numerical solution of linear systems is fundamental in engineering fields, such as structural analysis, heat transfer, fluid mechanics, and electrical circuit modeling. These applications typically generate systems of the form Tx = u, which may be large, sparse, and moderately ill-conditioned. Refinement-based iterative techniques have been developed to improve convergence characteristics of stationary schemes. This study focuses on the application of existing third refinement iterative methods to engineering linear system problems, with particular emphasis on evaluating their relative computational performance. Engineering-based linear systems were formulated from representative application models and expressed in standard matrix form. Selected third refinement methods, specifically the Third Refinement Jacobi (TRJ) and Third Refinement Gauss-Seidel (TRGS) schemes, were implemented and applied to the test problems. Convergence analysis was conducted using spectral radius criteria derived from the associated iteration matrices. Performance assessment was based on spectral radius, iteration number required to meet a prescribed tolerance, CPU time, and convergence rate. Results were systematically organized in comparative tables. Numerical finding indicates clear performance variation between the refinement schemes. Methods with smaller spectral radii required fewer iterations and reduced computational time. In particular, TRGS demonstrated faster convergence and improved computational efficiency compared to TRJ for the tested engineering problem. The study confirms that third refinement iterative methods are effective for solving engineering linear systems, with performance strongly influenced by spectral properties. These methods provide reliable and computationally efficient solvers for practical engineering applications.

Nonlinear conjugate gradient (NCG) methods are among the most effective techniques for solving large-scale unconstrained optimization problems. Their efficiency arises from low memory requirements, straightforward implementation, and strong performance in high-dimensional settings. A key component of NCG methods is the parameter βk\beta_kβk​, which governs how the new search direction is generated from the current gradient and the previous direction. The selection of βk\beta_kβk​ plays a crucial role in determining both the convergence behavior and computational efficiency of the algorithm.

Several well-known formulas have been proposed for βk\beta_kβk​, including the Fletcher–Reeves (FR), Polak–Ribiere (PRP), and Hestenes–Stiefel (HS) methods. The FR method is recognized for its solid theoretical convergence properties under standard line search conditions, yet it may exhibit slow progress in practice. On the other hand, PRP and HS often achieve faster numerical performance and better practical behavior, particularly for nonconvex problems. However, these methods may fail to guarantee global convergence unless additional safeguards are introduced to preserve the descent property.

To address these challenges, this work proposes a new adaptive hybrid formulation for βk\beta_kβk​ that combines the complementary strengths of FR, PRP, and HS. The proposed strategy dynamically balances robustness and efficiency by adjusting the influence of each formula based on the local characteristics of the objective function. A safeguarding mechanism is incorporated to ensure that the generated search direction maintains the descent condition at every iteration. Under standard assumptions and appropriate line search strategies, global convergence of the proposed method is established. Numerical experiments on benchmark problems demonstrate improved stability and competitive performance compared to classical NCG methods, without sacrificing theoretical guarantees.