Baboucarr Ceesay

The purpose of this work is to examine new exact soliton solutions for the doubly dispersive equation (DDE) by utilizing a finite series in terms of Jacobi elliptic functions (JEFs). For the exact analytical solution of nonlinear partial differential equations (NLPDEs), the JEF expansion approach is frequently employed. The suggested approach yields several solutions, including hyperbolic-type solutions, singular periodic wave solutions, JEF solutions, exponential solutions, bright, dark and bright-dark combo soliton solutions and Weierstrass elliptic double periodic solutions. Experts in engineering models will be interested in the findings, which will help them comprehend waves. To this end, we use the Galilean transformation to derive a dynamical system closely related to the equation. The bifurcation behaviours observed in this derived system were then investigated using concepts from the theory of planar dynamical systems. To examine the possible existence of chaotic behaviours, we carefully evaluate the DDE and add a perturbed term to the dynamical system. Comprehensive two- and three-dimensional (2D and 3D) phase portraits are presented to further enhance this inquiry. Some solutions are selected for the graphical behaviour, and these plots are explained according to real-life applications.

Dengue fever is a mosquito-borne viral disease posing a major health challenge in tropical and subtropical regions, including Malaysia, where the climate supports year-round transmission. It affects millions annually, leading to severe illness and hospitalization, while the lack of specific antiviral treatment and the expanding range of Aedes mosquitoes make dengue a persistent global threat. To overcome this issue, this study develops a time delay SEITR-SEI model incorporating two biologically motivated delays representing the human intrinsic and vector extrinsic incubation periods. Firstly, time delay of intrinsic incubation period (IIP) in system captures the dynamics of dengue with time delay in human. The model's dynamics are examined by identifying dengue free and existing equilibria, evaluating the basic reproduction number, and analyzing local and global stability. The dengue free state is stable when the reproduction number is below one; otherwise, a stable endemic state arises. The second delay in the vector compartment represents a delay in the extrinsic incubation period (EIP), which suppresses dengue transmission while allowing the mosquito population to remain sustainable. The model was fitted using real hospital data from Penang General Hospital. These records correspond to patients admitted to Hospital between 2022 and 2023. The model parameters related to the dengue outbreak in this specific area were estimated using the least squares method. Furthermore, using these estimated parameter values, the equilibrium points of the model were also examined through the nonstandard finite difference (NSFD) scheme to ensure numerical stability and dynamic consistency of the system. In addition, a sensitivity analysis was performed to identify the most influential parameters affecting the transmission dynamics of dengue in Penang. Unlike previous studies on simple dengue fever models, this work applies optimal control to a more realistic double-delay model, where graphical comparison showed NSFD outperforming Euler. Simulation results emphasize the importance of biological control strategies, particularly eco-friendly and sustainable approach to limit dengue transmission without eliminating mosquito populations. All over Malaysia, the spread of dengue is nearly same, so the findings provide theoretical insights and practical guidance for dengue control policies in Malaysia.

In this study, the nonlinear time fractional equation is taken and made into a space and time nonlinear fractional equation. After that, fuzzy theory is utilized to study our simple modified space and time nonlinear fractional equation under fuzziness. Our aim is to extract analytical solutions for the proposed model under fuzziness. A fuzzified fractional traveling wave transformation and conformable time fractional derivative are applied to reduce the governing fuzzy space and time fractional nonlinear equation into a fuzzy ordinary differential equation (FODE), which is solved using the extended hyperbolic function method (EHFM). By considering different cases and fuzzy lower and upper bounds, multiple types of soliton solutions are derived, including bright solitons, periodic solutions, and exponential‐type wave solutions. These solutions describe various nonlinear wave structures that can propagate in dispersive media. To illustrate the physical behavior of the fuzzy space and time fractional nonlinear model, two‐dimensional, three‐dimensional, and contour plots are presented using Mathematica software to show the effects of fractional derivatives and fuzzy parameters on soliton amplitude, localization, and propagation. The final three graphs in every figure further quantify uncertainty by displaying the divergence between lower and upper fuzzy solutions, highlighting enhanced sensitivity near the soliton core.

This work addresses the main inducements: the modified extended direct algebraic approach and residual neural network (MEDARNN) structures to obtain the exact analytical solutions for the (2+1)-dimensional modified Veronese web (MVW) equation. This is an integrable nonlinear partial differential equation (NLPDE) and is used to admit a differential covering with non-removable physical parameters. We present for the first time a novel analytical technique called the MEDARNN method, which combines models of residual neural networks (RNNs) with the auxiliary equation to provide precise solutions for NLPDEs. In addition, by establishing particular activation functions within the ‘2-2-2-1’ and ‘2-2-3-1’ NNs models, the introduction of a novel activation function derived from the auxiliary equations solutions is a significant innovation of this approach, creating a new mathematical connection between deep learning and differential equation theory. The different forms of solutions, such as shock, complex solitary-shock, shock-singular and periodic-singular forms, exhibit both single and coupled wave structures. For the physical interpretation of some of the constructed solutions are draw in the form of three-dimensional (3D), two-dimensional (2D) and corresponding contours by choosing the different values of parameters. This paper offers a new methodological approach for dealing with NLPDEs that is widely applicable in engineering and science domains.

This study primarily aims to apply the Riccati equation rational expansion method (REREM) to the nonlinear coupled Konno–Oono system (NCKOS) in order to construct novel and flexible optical soliton solutions. In addition, the modulation instability of the NCKOS is examined. The NCKOS represents current–field strings that interact with an external magnetic field. By employing REREM, we obtain various new types of solutions, including hyperbolic, bright, dark, periodic, kink and anti-kink, peakon, bell–shaped, and many other solitary wave structures. Several physical illustrations of some of the derived solutions are provided to enhance clarity. The findings demonstrate that the proposed technique is more powerful, efficient, and effective for investigating the solutions of other complex nonlinear models. Furthermore, the newly obtained solutions have potential applications in hydrodynamics, solid-state physics, cosmology, ecology, and quantum electronics.

In this study, we present the neural networks generalized Kudryashov (NNGK) method for the first time to explore exact solutions of the generalized doubly dispersive equation. This is a novel analytical technique that combines neural networks (NNs) models with the generalized Kudryashov approach. The NNs are multilayer computational models composed of activation functions and weights that connect neurons in the input, hidden, and output layers. The generalized Kudryashov solutions are assigned to each neuron in the first hidden layer of the NNGK method. This is how the new trial functions are produced. A key novelty of this study is the construction of the novel activation functions from the Kudryashov approach solutions, which creates a new mathematical connection between differential equations theory and deep learning, which is a major innovation of this approach. There are different types of soliton solutions constructed, such as lump interaction, lump-singular and lump-dark, lump-bright, breather solitons, breather-kink, breather-antikink, and other interaction solitons. Some of these solutions are drawn in the form of 3D, 2D, and planar waves and corresponding contours for the physical interpretations of these solutions. This study offers a new methodological approach for dealing with NLPDEs that is widely applicable in engineering and science domains.

In this work, we present a complete dynamical analysis of lump, breather, M-shaped, and other waveforms propagating in a nonlinear PDE governing nonlinear low-pass electrical transmission lines. We utilize the Hirota bilinear transformation approach with the help of Mathematica to report a number of wave solutions, including bright and dark lumps, solitons, breathers, and kink waves, along with their periodic and aperiodic forms. Energy distribution, wave interactions, and changes are presented in the form of 3D, contour, and 2D plots, which demonstrate the nonlinear characteristics that govern the dynamics. These results provide a better understanding of the propagation, stability, and interaction of waveforms which are useful in signal and energy transport and also in the construction of complex nonlinear electric circuits.

In this paper, we investigate exact analytical solutions of the (2+1)-dimensional complex modified Korteweg–de Vries (CmKdV) system using the truncated M-fractional derivative together with the Jacobi elliptic function expansion method. The CmKdV system plays a central role in modeling nonlinear wave propagation in optics, plasma physics, and fluid dynamics. By applying the truncated M-fractional derivative, the system is transformed into a more tractable form, enabling the effective use of the Jacobi elliptic function expansion method to construct exact soliton and periodic wave solutions. The results offer deeper insight into the system’s nonlinear dynamics and highlight the robustness of the proposed method. Graphical simulations generated in Mathematica illustrate the physical behavior of the obtained solutions across multiple dimensions, such as two-dimensional, three-dimensional, and contour, and the influence of time on the wave propagations. Overall, combining the Jacobi elliptic function expansion method and truncated M-fractional derivatives creates a strong framework for solving complex differential equations, leading to new possibilities. Opportunities for research and development exist. This research adds to our understanding of the (2+1)-dimensional complex modified Korteweg-de Vries (CmKdV) system and demonstrates how theoretical mathematics can be applied to real concerns. Mathematical modeling and computational visualization can significantly impact engineering and science, and our findings promote a multidisciplinary approach to research.