In this article, we design and analyze a hybrid high-order method for a semilinear Sobolev model on polygonal meshes. The method offers distinct advantages over traditional approaches, demonstrating its capability to achieve higher-order accuracy while reducing the number of unknown coefficients. We derive error estimates for the semi-discrete formulation of the method. Subsequently, these convergence rates are employed in full discretization with the Crank-Nicolson scheme. The method is demonstrated to converge optimally with orders of O(tau(2) + h(k+1)) in the energy-type norm and O(tau(2) + h(k+2)) in the L-2 norm. The reported method is supported by a series of computational tests encompassing linear, semilinear and Allen-Cahn models.

In this study, we proposed a weak Galerkin finite element method (WG-FEM) for solving two-parameter singularly perturbed parabolic problems (TP-SPPPs) of convection-diffusion-reaction type on nonuniform mesh. The WG-FEM approach incorporates an interpolation operator I-N and achieves optimal order convergence in an energy-like norm for continuous and fully discrete schemes. In the fully-discrete analysis, we combined WG-FEM for spatial space on a Shishkin-type mesh with the Crank-Nicolson scheme for temporal discretization on an equidistant mesh. Since the energy norm is insufficient to capture the behavior of the boundary layer functions accurately, we have derived the optimal order of convergence in a strong balanced norm. This norm exhibits an order of O(N-1 ln N)(k) in spatial convergence and second-order convergence in time when using an L-2-projection Q(N) as an intermediate operator. We conducted numerical examples to validate the proposed method's theoretical findings. The results of these experiments confirmed the theoretical conclusions and demonstrated the robustness of the proposed method.

In this paper, we design and develop a weak Galerkin finite-element numerical method for solving singularly perturbed convection–diffusion–reaction equations with a non-conservation convection term. Many advantages of the proposed method include support for the higher order of convergence and general polygonal meshes. The convergence study for the weak Galerkin algorithm is performed in both the triple-bar norm and the L^2 norm. We achieve an optimal order of convergence of 𝒪(h^k) in the triple-bar norm and 𝒪(h^k+1) in the L^2 norm. Several numerical experiments in a two-dimensional setting are carried out to demonstrate the convergence of our theories.

The article presents the development of the weak Galerkin finite element method (WG-FEM) for semilinear hyperbolic problems. Semidiscrete error estimate in L^2 -norm as well as H^1 -norm have been executed for the weak Galerkin space (P_k (𝒦), P_k (∂𝒦), [P_k-1 (𝒦)]^2), where k ≥ 1 is an integer. For a fully discrete scheme, we employ the Newmark scheme for temporal discretization. Finally, a few numerical results are provided to validate theoretical results.

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

Competent treatment techniques were explored to curb the environmental pollution of dye-laden wastewater. In the current study, eucalyptus biomass contemplated as agricultural waste is translated into eucalyptus graphitic activated carbon (EPGAC) using ZnCl 2 at 600 °C in the N 2 atmosphere. The present investigation illustrated awareness about the nature of EPGAC’s dye elimination by employing Direct Yellow 12 dye (DY12) as a model dye. EPGAC was characterized using multiple characterization tools such as Fourier transform infrared spectroscopy (FTIR), Boehm titrations, pH zpc , X-ray diffraction (XRD), Raman, field emission scanning electron microscopy (FESEM), energy dispersive X-ray analysis (EDX), high-resolution transmission electron microscopy (HRTEM), and Brunauer-Emmett-Teller (BET) surface area analysis techniques. Electron micrographs disclosed the availability of high pore density for the adsorption of DY12 dyes. BJH analysis reported the distribution of mesopores having a 3 nm diameter on the EPGAC surface. Further, the surface area available for adsorption per gram of the adsorbent is estimated as 178.35 m 2 employing BET analysis. XRD and Raman’s data revealed the graphitic nature of EPGAC. Influences of adsorbent parameters such as EPGAC mass, initial dye concentration, contact time, solution pH, and temperature on the eviction of DY12 by EPGAC were examined to achieve a deeper insight into the adsorption mechanism. The optimum EPGAC adsorbent dose was found to be 0.15 g. The equilibrium was attained at 120 min for DY12 dye. Pseudo-second-order kinetics entirely relates to the perfect fit associated with the investigational results. The aptness of the equilibrium data relevant to the Langmuir adsorption isotherm eventually recommends a maximum unilayer adsorption capacity of 42.01 mg/g for EPGAC. Thermodynamic studies further reveal the spontaneous, endothermic, and chemisorption nature of adsorption. Adsorbent viability was established through stability and recyclability studies carried out up to 5 run cycles with 0.15 g of EPGAC. Adsorption mechanisms were explained considering hydrogen bonding, π-π interactions, and electrostatic interactions, ultimately confirming the adsorption tendency displayed by EPGAC for the eviction of DY12 dye present in industrial wastewater.

Due to the importance and advancements of the dye adsorption capacity for waste water treatments, it becomes really important to understand the efficient adsorption process and mechanism. In this study, (NH4)2V6O16.1.5H2O consisting of porous 3D nest-like nanostructures were prepared using the sol-gel method for the removal of methylene blue dye from contaminated wastewater. The prepared material was systematically characterized to determine its structural and morphological properties using various techniques, including X-ray diffraction, field emission scanning electron microscopy, and high resolution transmission electron microscopy. The impact of various agitation techniques on the dye adsorbent material was analyzed using different mixing techniques such as turbulent stirring and orbitally shaking. These mechanical processes enhance the interaction between dye molecules and adsorbent nanostructures by promoting mass transfer and increasing the collision frequency. The kinetic models of dye adsorption on the (NH4)2V6O16.1.5H2O nanostructures were conducted to investigate the dye adsorption mechanism. In the intraparticle diffusion model, the rate constant (kd) offers insights into the diffusion rate within the adsorbent nanostructures. The higher value of kd for turbulent stirring in comparison to orbital shaking can be dedicated to the creation of more active adsorption sites by disruption of consecutive liquid layers by stirring.

In this paper, a Galerkin finite element method is designed and analyzed to simulate the nonlinear Korteweg-de Vries-Rosenau-regularized long-wave (KdV-RRLW) model. We establish the existence and uniqueness results in H02(Ω) Sobolev space by applying the Banach–Alaoglu theorem. Using appropriate projection, we derive the error estimates of a semidiscrete scheme for the finite element solution of the KdV-RRLW model. Furthermore, a second-order Crank-Nicolson scheme is employed for the temporal discretization and obtain the optimal order of convergence in the maximum norm. Finally, several numerical examples are provided to visualize the nature of the wave phenomena in one and two dimensional spaces and demonstrate the robustness of the developed finite element algorithm.

Tungsten oxide/molybdenum oxide nanocomposites have been synthesized using a cost-effective internal combustion method by incorporating two different phases of tungsten oxide into hexagonal and orthorhombic molybdenum oxide nanostructures. The synthesized materials were subjected to different analytical techniques, including X-ray diffraction, Fourier transform infrared spectroscopy, field emission scanning electron microscopy, and energy-dispersive X-ray spectroscopy, to investigate their structural and morphological properties. The synthesized materials were employed for the adsorption of methylene blue by varying different parameters, such as the initial dye concentration, contact time, adsorbent dose, pH, and temperature of the solution. The thermodynamical parameters were calculated to validate the adsorption phenomenon in terms of its spontaneity and feasibility. Various kinetic studies, such as pseudo-first-order, pseudo-second-order, and intra-particle diffusion models, were used to determine kinetic parameters and understand the adsorbate-adsorbent interactions. The detailed characteristics of adsorbent surface and adsorption behavior were studied using the Langmuir, Freundlich, and Temkin isotherms. Among these, monoclinic tungsten oxide/orthorhombic molybdenum oxide exhibits the best adsorption efficiency and higher kinetic rate constant, followed by the orthorhombic tungsten oxide/orthorhombic molybdenum oxide nanocomposite. Dye adsorption over the adsorbent surface was confirmed by investigating the materials characteristics before and after the process using Fourier infrared transform spectroscopy. The enhanced adsorption efficiency and higher rate constant can be attributed to the improved surface properties of these materials.

Background The global burden of HIV remains significant, particularly in India. Antiretroviral therapy (ART) has improved outcomes for children with HIV, yet understanding the virus's impact on respiratory health is essential. Pulmonary complications, common in HIV-infected adults, are poorly understood in children. Despite India's high HIV prevalence, data on pediatric lung function are lacking. This study aims to evaluate spirometry-based pulmonary function in perinatally HIV-infected children, exploring associations with disease severity, immune status, and other factors. Methods This prospective cross-sectional study conducted in a North Indian tertiary care hospital aimed to assess pulmonary function using spirometry in children (6-18 years) with HIV infection. Ethical approval and informed consent were secured. Data on demographics, clinical history, CD4+ T-cell counts, and viral load were collected. Certified respiratory therapists performed spirometry using standardized protocols. Descriptive statistics were computed, and differences in pulmonary function based on CD4+ T-cell counts, viral load, and opportunistic infection were analyzed. The study adhered to ethical guidelines and maintained participants' confidentiality. Results This cross-sectional study enrolled 57 children (mean age 13.6±3.2 years) with HIV infection. Age distribution was <9 years (24.6%), 9-11 years (28.1%), and >11 years (47.4%). Males constituted 56.1%. The mean BMI was 15.92±2.78 kg/m². HIV viral load (87.23±56.28 copies/μL) and CD4 count (1146.32±103.98 cells/mm³) were recorded. ART duration averaged 6.21±1.36 years. Viral load groups were <1 (52.6%), 1-1000 (26.3%), and >1000 copies/μL (21.1%). CD4 categories were >500 cells/mm³ (47.4%), 200-499 (42.1%), and <200 cells/mm³ (10.5%). Spirometry showed 71.9% normal and 28.1% abnormal (mild/moderate obstruction: 18.8%, mild/moderate restriction: 81.3%). No significant spirometric differences were observed among CD4 or viral load groups (p>0.05), nor with opportunistic infections (p>0.05). Conclusion This study reveals complex associations between spirometric parameters and CD4 count, viral load, and opportunistic infections in children with HIV. Further research, including longitudinal studies, is needed to unravel the intricate interplay and improve management strategies for this population.

Metal-organic frameworks (MOFs) are an innovative class of porous materials, permit exceptional structural and compositional diversity beyond the conventional solid-state materials. MOFs are generally produced by joining the metal-containing units with desired organic linkers by applying the reticular synthesis that constructs the strong bonds between the two subunits. Circumspect selection of these two subunits may yield the MOFs with extraordinary porosity and tunability and these characteristics allow them for various applications such as high density energy storage, separation, catalysis, drug delivery as well as ease the host-guest interactions for the precise control of electronic structure. They also control the functions of a material by the integration of inorganic and organic components. It is also known that MOFs with their chemical properties and shapes of their building units can be varied within a structure and could then lead to materials that offer a combination of synergistic properties. Thus, these kinds of materials are the need of the hour. Herein, we have explored the relevant challenges and possible opportunities toward the MOFs for the betterment of human beings.

Guided wave based techniques are among the promising techniques for structural health monitoring due to their ability to detect damage with high precision, ability to scan large area and low power consumption. However, practical implementation of these techniques to structures with relatively complex geometries such as stiffened plates demand further research due to several challenges. Some of such challenges include the presence of stiffeners and edges in close vicinity causing multiple reflections of waves, possibility of multiple damages and lack of baseline data. To address all these issues, a modified hyperbola based approach using mode converted signal and a data based matching scheme has been proposed in this paper. A stiffened plate is taken as the example structure. Corrosion like defects have been considered as the damages. Each panel of the stiffened plate has been scanned separately through phased array actuation. The effectiveness and robustness of the proposed approach have been shown through a case study using 3D finite element simulation data.

The effect of varying methylene blue dye concentrations on the adsorption activity of molybdenum oxide nanostructures has been investigated. The removal percentage and maximum adsorption efficiency were determined by varying the dye concentration from 20 to 150 mg/L. The adsorption rate constants, nature and type of adsorption were determined using pseudo-first-order, pseudo-second-order, and intraparticle diffusion kinetics models. Four different isotherm, that is, Langmuir, Freundlich, Temkin and Dubinin-Radushkevich models were used to understand the interaction of the adsorbate-adsorbent at interface and to calculate maximum adsorption efficiency. The mixed phase shows adsorption efficiency of 633.1 mg/g while orthorhombic phase gives much lesser efficiency, that is, 425 mg/g. The removal percentage decreases from 99% to 25.8% (orthorhombic phase) and 99.4% to 67.8% (mixed phase) with increasing dye concentrations.

In this paper, we describe weak Galerkin finite element methods for solving hyperbolic problems on polygonal meshes. We propose both semidiscrete and fully discrete schemes to numerically solve the second-order linear wave equation. For the time discretization, we have used implicit second order Newmark scheme. For sufficiently smooth solutions, optimal order error estimate in the L 2 norm is shown to hold as O ( h k + 1 + τ 2 ), where h is the mesh size and τ the time step. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the proposed method.

In this paper, we present a convergence analysis of a weak Galerkin finite element method (WG-FEM) using polygonal meshes for the semilinear singularly perturbed time-dependent convection-diffusion-reaction equations. Piecewise polynomials of degree k≥1 are used in the interior of each element, and polynomials of degree k≥0 are employed on the edge of each element in this finite element technique. The convergence analysis is explored in two phases: first in space and later in time. The proposed method employs a Crank-Nicolson scheme for temporal discretization and a WG-FEM for spatial discretization. The key finding shows that the error bound of the weak Galerkin solution is O(τ2+hk) in the energy-like norm and O(τ2+hk+1) in the L2 norm. Several numerical experiments are carried out to validate the theoretical results.

In this study, we design and analyze weak Galerkin finite element methods to approximate diffusive viscus wave equations with variable coefficients on polygonal meshes. The proposed method has numerous assets, including supporting a higher order of convergence and general polygonal meshes. We investigated the convergence analysis using a two-step technique that discretizes first in space and then in time. A second-order Newmark scheme is employed to develop the temporal discretization and obtain the optimal order of convergence rate in L^∞(L^2) and L^∞(H^1) norms. In other words, we attain 𝒪(h^k+1+τ ^2) in L^∞(L^2) norm and 𝒪(h^k+τ ^2) in L^∞(H^1) norm. We performed several numerical experiments in a two-dimensional setting, illustrating our theoretical convergence findings.

We analyze the weak Galerkin finite element methods for second-order linear parabolic problems with L2 initial data, both in a spatially semidiscrete case and in a fully discrete case based on the backward Euler method. We have established optimal L2 error estimates of order O(h2/t) for semisdiscrete scheme. Subsequently, the results are extended for fully discrete scheme. The error analysis has been carried out on polygonal meshes for discontinuous piecewise polynomials in finite element partitions. Finally, numerical experiments confirm our theoretical convergence results and efficiency of the scheme.

In this paper, we describe weak Galerkin finite element methods for solving hyperbolic problems on polygonal meshes. We propose both semidiscrete and fully discrete schemes to numerically solve the second-order linear wave equation. For the time discretization, we have used implicit second order Newmark scheme. For sufficiently smooth solutions, optimal order error estimate in the L2 norm is shown to hold as O(hk+1+τ2), where h is the mesh size and τ the time step. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the proposed method.

This paper considers a stabilizer-free weak Galerkin (SFWG) finite element method for the time-dependent convection diffusion reaction equation. We describe error estimate for both semidiscrete and fully discrete schemes and achieve the supercloseness convergence rate, which is two orders higher than the optimal order associated with SFWG finite element space (Pk(K),Pk+1(∂K),[Pk+1(K)]2). More precisely, we obtain O(hk+2+τ2) in L∞(H1) norm and O(hk+3+τ2) in L∞(L2) norm. Numerous numerical examples are provided to confirm the theoretical findings and efficiency of the proposed method.

Sinus of valsalva aneurysms are uncommon and can be congenital or acquired. They may have variable clinical presentation ranging from asymptomatic cases to congestive heart failure and, in extreme situations, cardiac arrest. Both ruptured and nonruptured valsalva sinus aneurysms can cause deadly consequences, though the prognosis is excellent after treatment. As a result, timely and precise diagnosis is essential. Rupture of sinus of valsalva (RSOV) aneurysm in pregnancy during antepartum is a dreaded complication and can be life threatening for both mother and fetus. However, it is infrequent with only few cases reported during pregnancy. We report a case of ruptured sinus of valsalva aneurysm in postpartum period with acute right heart failure following uneventful normal vaginal delivery and favorable outcome postpercutaneous intervention. present with palpitations or acute heart failure. We present a case of rupture of SVA in postpartum period with uneventful normal vaginal delivery.