In this work, we consider a generalized form of the classical $(2+1)$-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results.

In this paper, we propose an efficient inertial iterative algorithm for solving a system of generalized quasi-variational inequalities (SGQVI) in Hilbert spaces. Using the projection operator technique, we establish an equivalence between SGQVI and fixed-point problems, thus developing a novel inertial method. The algorithm introduces an inertial term to accelerate convergence, and its performance is rigorously analyzed under some mild conditions, including relaxed co-coercivity and Lipschitz continuity of the involved mappings. Our framework unifies and extends several existing models, such as classical variational inequalities, quasi-variational inequalities, and related optimization problems. Some experiments demonstrate the effectiveness of the inertial method, which shows an improvement in convergence speed compared to noninertial methods. Our results generalize and enhance previous research results in the literature, making it more widely applicable in computational mathematics, engineering, and economics.

We study exponentially convex nonlinearities and exponentially monotone operators in Hilbert spaces for exponential variational inequalities (EVIs). We show that the exponential nonlinearity is the gradient of a convex functional, so it is monotone. We use this fact to prove existence, uniqueness, and contractivity of the associated fixed-point mapping for every strong monotonicity constant $\alpha >0$ of the underlying operator. This task is performed using an explicit, computable step-size range and linear convergence rate. We present a two-step inertial predictor–corrector technique and prove its strong convergence. Applying the framework to the Bratu obstacle problem (which models thermal ignition in combustion systems) yields the explicit, sharp requirement ${(b-a)}^2{e}^M<1$ for contractivity of the associated fixed-point map and, hence, for existence, uniqueness, and convergence in this application. When this requirement is satisfied, numerical experiments demonstrate a noticeable acceleration of up to $5\%$ from the inertial approach in the contractive regime’s mid-range. Neither the non-contractive regime nor the smallest tested domain length showed any obvious improvement.