In the present research, we establish an effective method for determining the time-fractional coupled Korteweg–de Vries (KdV) equation’s approximate solution employing the fractional derivatives of Caputo–Fabrizio and Atangana–Baleanu. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. Some theoretical physical features of quantum mechanics are also explained by the KdV model. Many investigations have been conducted on this precisely solvable model. Numerous academics have proposed new applications for the generation of acoustic waves in plasma from ions and crystal lattices. Adomian decomposition and natural transform decomposition techniques are combined in the natural decomposition method (NDM). We first apply the natural transform to examine the fractional order and obtain a recurrence relation. Second, we use the Adomian decomposition approach to the recurrence relation, and then, using successive iterations and the initial conditions, we can establish the series solution. We note that the proposed fractional model is highly accurate and valid when using this technique. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Two examples are given to illustrate how the technique performs. Tables and 3D graphs display the best current numerical and analytical results. The suggested method provides a series form solution, which makes it quite easy to understand the behavior of the fractional models.

Fractional calculus extends conventional differentiation and integration to non-integer orders, facilitating a more suitable modeling framework for complicated dynamical processes characterized by memory and long-range dependence. The fractional and variable-order fractional Genesio-Tesi systems have recently attracted significant interest owing to their rich nonlinear dynamics and the added flexibility introduced by variable fractional orders. This work comparatively studies the stability and chaotic behavior of constant-order versus variable-order formulations of the fractional Genesio–Tesi system. The study of the system dynamics is carried out by numerical simulations, including time series, bifurcation diagrams, Lyapunov exponents, and phase portraits. We identify further stability boundaries and chaotic regimes through analytical investigations based on the Jacobian eigenvalue spectrum. It is found that variable-order derivatives intensify sensitivity and transient responses, disclosing chaotic patterns that contribute to a more profound understanding of fractional nonlinear dynamics.

In this research, we employ a dual-approach that combines the Laplace residual power series method and the novel iteration method in conjunction with the Caputo operator. Our primary objective is to address the solution of two distinct, yet intricate partial differential equations: the Foam Drainage Equation and the nonlinear time-fractional Fisher’s equation. These equations, essential for modeling intricate processes, present analytical challenges due to their fractional derivatives and nonlinear characteristics. By amalgamating these distinctive methodologies, we derive precise and efficient solutions substantiated by comprehensive figures and tables showcasing the accuracy and reliability of our approach. Our study not only elucidates solutions to these equations, but also underscores the effectiveness of the Laplace Residual Power Series Method and the New Iteration Method as potent tools for grappling with intricate mathematical and physical models, thereby making significant contributions to advancements in diverse scientific domains.

Edge detection is an essential problem in computer vision and is used in applications such as object recognition, scene analysis, and medical imaging. Conventional edge detectors based on integer-order derivatives are computationally efficient but sensitive to noise, whereas modern deep learning approaches generally achieve higher accuracy at the cost of increased model complexity. This paper presents GLF-ResFormer a lightweight hybrid CNN–Transformer architecture incorporating Grünwald–Letnikov (GL) fractional preprocessing. The discrete GL operator is approximated using a finite-difference convolution with a truncation level of $N=15$, where the fractional order $\alpha \in (0,1]$ controls the spatial memory of the operator. We establish an upper bound for the truncation error of the discrete GL approximation, $\mathcal{O}\left(h^{N-\alpha }\right)$ (Theorem 1), and present a gradient-sensitivity analysis (Lemma 1) that provides theoretical support for the proposed preprocessing strategy. Extensive experiments using 10 independent random seeds on the MNIST dataset show that, at the optimal fractional order of $\alpha =0.01$, GLF-ResFormer achieves a pixel-wise F1 score of $0.9967\pm 0.0002$ compared with $0.9891\pm 0.0033$ for a CNN baseline, while reducing the validation loss to $0.0043\pm 0.0003$. Additional experiments on the CIFAR-10 dataset and comparisons with the BSDS500 benchmark further demonstrate the effectiveness of the proposed framework across multiple edge detection evaluation settings while maintaining a lightweight architecture.