The objective of the present paper is to solve a one-dimensional quasilinear parabolic singularly perturbed problem with a discontinuous source term. Due to the presence of such a discontinuity, an interior layer exists at the location of the discontinuity. The problem is solved by discretizing the spatial variable on a piecewise uniform Shishkin mesh using the standard upwind approach, while the backward Euler scheme is employed on a uniform mesh to discretize the time variable. The method is $\epsilon$-uniformly convergent, providing first-order convergence in the time domain and almost first-order convergence in the spatial variable. To validate the theoretical findings, the scheme was tested by numerically solving two examples.

This paper establishes new criteria for the oscillation of solutions to a specific class of third-order delay differential equations. These equations, which have numerous applications in the physical and biological sciences, pose intriguing analytical challenges. By employing novel ordering properties, the comparison principle, Riccati transformations, and other analytical techniques, we derive effective oscillation criteria. This approach addresses and overcomes several restrictions previously imposed on the coefficients of such equations. To demonstrate the novelty and practical relevance of our results, we include illustrative examples.

This research introduced a physics informed neural network (PINN) framework designed to effectively solve the highly nonlinear Bratu equation, which arises in various physical contexts such as chemical reaction theory and combustion processes. PINNs provide a mesh-free solution by embedding physical laws directly into the loss function of the neural network and utilizing automatic differentiation for accurate derivative calculations. However, standard PINNs often face challenges in strictly enforcing boundary conditions (BCs), resulting in numerical inaccuracies and slow convergence. To overcome this, we proposed an innovative method that precisely enforces BCs through a transformation, thereby eliminating residual errors and significantly improving the reliability and performance of the PINN framework. Numerical experiments validated the effectiveness of the proposed approach, showing improved accuracy, faster convergence, and more stable training dynamics. Detailed analyses were conducted to investigate the influence of key hyperparameters, such as activation functions, network architecture, and learning rates, on the model's performance.

This study examines the dynamic response of a fibrillar adhesive floating breakwater positioned near a porous structure at a finite distance from Gaussian undulating seabed. The problem is addressed using linearized water wave theory, with numerical simulations based on the multi-domain boundary element method. The study primarily focuses on the analysis of crucial elements such as the added mass and damping coefficients associated with heave, surge, and pitch motions, considering the influence of both wave and structural parameters. Validation against existing literature confirms the accuracy and reliability of the proposed method. The study reveals that an increase in the number of seabed ripples leads to higher added mass and damping coefficients, particularly at larger wave incidence angles. Further, the frictional interaction between the water and the porous structure modifies the added mass coefficient, resulting in a shift in the resonance peak and significantly affecting the dynamic response of the breakwater. Moreover, surge and pitch motions are highly damped in intermediate waves as the porosity of the structure decreases.

We investigated a nonlinear singularly perturbed elliptic reaction-diffusion coupled system having non-smooth data networked by a $k$-star graph. We considered all possible boundary conditions at the free boundary located at the tail of the edge and imposed the continuity condition with Kirchhoff's junction law at the junction point of the $k$-star graph to obtain a continuous solution for this coupled system. First, we showed the existence and uniqueness of the solution using the variational formulation approach. Then, we reformulated it into a minimization problem over a function space to conclude the uniqueness of the solution. For the approximation of the continuous problem, note that the upwind scheme for the flux condition at the free boundary leads to a parameter uniform first-order approximation. To obtain a higher-order uniform accuracy, we utilized a three-point scheme for first-order derivatives and a five-point approximation at the point of discontinuity. These approximations typically did not yield an M-matrix or strict diagonally dominant structure of the stiffness matrix. Hence, we provided a suitable transformation that could lead to a sufficient condition for preserving the strict diagonally dominant structure of the stiffness matrix. We performed a comprehensive convergence analysis to demonstrate the almost second-order uniform accuracy on each edge of the $k$-star graph. Numerical experiments highly validate the theory on the $k$-star graph.