A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in $2D$ and $3D$ graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations.

In this paper, we deal with topology optimization attributed to the non stationary Navier-Stokes equations. We propose an approach where we analyze the sensitivity of a shape function relating to a perturbation of the flow domain. A numerical optimization algorithm based on topological gradient method is built and applied to the 2D Tesla micro valve reconstruction. Some numerical results confirm the efficiency of the proposed approach.

In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence and conditional stability. The accuracy of the method is demonstrated with 3D mesh plots, while the effects of time and fractional order are shown in 2D plots. Comparative evaluations with the cubic B-spline collocation method are provided. To illustrate the suitability and effectiveness of the proposed method, two examples are tested, with the results are evaluated using $L_2$ and $L_{\infty }$ norms.

The structural changes along the c axis, of the Ba-exchanged montmorillonite (Swy-2-Ba), under variable relative humidity (% RH), is investigated. In this regard, the arrangement, amount and position of both exchangeable cation and the water molecules in the interlamellar space (IS), are evaluated. This aim is achieved using the X-ray diffraction (XRD) profile modeling approach that consists of comparing experimental and theoretical patterns calculated from structural models. The contributions of the hydration states and the interlayer water amounts, as a function of the % RH, are registered by quantitative XRD investigation. The validated structural models are heterogeneous, suggesting various proportions of layer types at different RH ranges, which means the coexistence of different mixed layer structure MLS packages, exhibiting different proportions of layers with contrasting hydration states. This result is attributed to the orientation of the applied hydration sequence. Indeed, the interlayer water molecule amounts, which led to the appearance of a logic hydration hysteresis, are strongly affected by hydrous perturbation.

The time-fractional coupled Korteweg–De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable Hyperbolic Non-Polynomial Spline Method (CHNPSM), for solving TFCKdVEs. The method leverages the inherent symmetry in the structure of TFCKdVEs, exploiting conformable derivatives and hyperbolic non-polynomial spline functions to preserve the equations’ symmetry properties during computation. Additionally, first-derivative finite differences are incorporated to enhance the method’s computational accuracy. The convergence order, determined by studying truncation errors, illustrates the method’s conditional stability. To validate its performance, the CHNPSM is applied to two illustrative examples and compared with existing methods such as the meshless spectral method and Petrov–Galerkin method using error norms. The results underscore the CHNPSM’s superior accuracy, showcasing its potential for advancing numerical computations in the domain of TFCKdVEs and preserving essential symmetries in these physical systems.

This paper presents a new general subfamily ${\mathcal{N}}_{{\Sigma }_m}^{u,v}(\eta ,\mu ,\gamma ,\ell )$ of the family ${\Sigma }_m$ that contains holomorphic normalized m-fold symmetric bi-univalent functions in the open unit disk $\mathbb{D}$ associated with the Ruscheweyh derivative operator. For functions belonging to the family introduced here, we find estimates of the Taylor–Maclaurin coefficients $\left|a_{m+1}\right|$ and $\left|a_{2m+1}\right|$, and the consequences of the results are discussed. The current findings both extend and enhance certain recent studies in this field, and in specific scenarios, they also establish several connections with known results.