This paper analyzes the stability of a fully discrete finite element approximation for a class of nonlinear parabolic variational inequalities of obstacle type. The temporal discretization is based on a θ-scheme. We derive a stability condition for the scheme that depends critically on the parameter θ. We prove that the method is unconditionally stable in the L2-norm for θ in [1/2,1]. For θ in [0,1/2), we establish a precise Courant-Friedrichs-Lewy (CFL)-type condition, Delta t<=2gγ/(L2(1-2θ)), where γ is the coercivity constant and L is the Lipschitz constant of the nonlinear source term. The analysis is based on a careful choice of test functions in the variational inequality and by deriving sharp estimates of the associated bilinear form.
This paper investigates coincidence point results for self-mappings in partial-metric spaces via simulation functions. By introducing a generalized contraction condition involving a simulation function and an auxiliary mapping $H$, we establish sufficient conditions for the existence and uniqueness of coincidence points and common fixed points. Our approach not only unifies several existing fixed point theorems in the literature but also provides a genuine extension by weakening conventional contraction assumptions. The theoretical findings are illustrated by a concrete example in a nonstandard partial-metric space setting, confirming the applicability and effectiveness of the proposed framework. As a special case, our results recover and generalize recent fixed point theorems in both metric and partial-metric spaces.
This paper proposes a Galerkin method based on Boubaker wavelets (BWGM) for the numerical solution of a class of differential equations. The method employs Boubaker wavelets as both weight functions and basis elements to construct approximate solutions. The accuracy of the proposed method is evaluated by comparing numerical results with exact solutions and with existing schemes such as the Galerkin method using Fibonacci and Gegenbauer wavelets. Several examples are provided to demonstrate the validity and applicability of the method. The results indicate that BWGM yields high accuracy with minimal absolute error, making it an efficient tool for solving linear, singular, and nonlinear boundary value problems.
This paper presents a novel numerical scheme for solving linear Volterra-Fredholm integral equations (V-FIEs) of the second kind, utilizing exponential spline functions (ESFs) in combination with fractional derivatives. The method simplifies computational implementation by converting the original integral equation into a matrix system. To prove the precision and stability of the suggested approach, a thorough convergence analysis is carried out. Numerical experiments, backed by graphical representations, validate the method's high accuracy and computational efficiency, even with a limited number of subintervals. All simulations and visualizations are implemented using Python. The results indicate that the suggested ESF approach performs noticeably better than traditional methods.
This paper presents a detailed comparison of three relatively recent methods for the numerical solution of systems of Nonlinear Volterra Integral Equations of the second kind (NVIEs-II): the Modified Adomian Decomposition Method (MADM), the Hussein-Jassim Method (H-JM), and the Cubic Non-Polynomial Spline Function Method (CNPSFM). The objective of this study is to evaluate the performance of these methods in terms of accuracy, convergence, and numerical stability. To achieve this, all three methods are applied to standard benchmark problems with known exact solutions, enabling quantitative assessment. The numerical results reveal distinct performance characteristics for each method. Both MADM and H-JM demonstrate excellent performance, yielding solutions with high accuracy and very low errors, occasionally approaching machine precision. MADM exhibits rapid convergence, while H-JM provides robust numerical stability and ease of implementation. CNPSFM displays good numerical stability and accurately captures the overall solution behavior; however, it produces relatively larger errors, particularly as the integration interval lengthens. This comparison concludes that the optimal choice among these methods is highly problem-specific. MADM and H-JM are best suited for high-precision applications requiring analytical insight (e.g., quantum mechanics or population dynamics), whereas CNPSFM remains viable for applications prioritizing solution smoothness over absolute accuracy. This study provides practical, evidence-based recommendations to assist researchers and engineers in selecting appropriate solvers for real-world systems modeled by NVIEs-II. Future research should extend these methods to systems with singularities and/or delays, which have been underexplored in the current literature. Another promising direction involves developing hybrid approaches that integrate artificial neural networks with traditional computational solvers.
