Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes logtj=loga+logtNajNr,j=0,1,2,⋯,N with a≥1 and r≥1, where loga=logt0<logt1<⋯<logtN=logT is a partition of [logt0,logT]. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes logtj=loga+logtNaj(j+1)N(N+1),j=0,1,2,⋯,N. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., DCHa,tαy(t)∉C1[a,T] with α∈(0,2), the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio r≥1. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.

Marine gas-phase sulfur emissions during an induced phytoplankton bloom

The oxidation of dimethyl sulfide (DMS; CH3SCH3), emitted from the surface ocean, contributes to the formation of Aitken mode particles and their growth to cloud condensation nuclei (CCN) sizes in remote marine environments. It is not clear whether other, less commonly measured marine-derived, sulfur-containing gases share similar dynamics to DMS and contribute to secondary marine aerosol formation. Here, we present measurements of gas-phase volatile organosulfur molecules taken with a Vocus proton transfer reaction high resolution time-of-flight mass spectrometer during a mesocosm phytoplankton bloom experiment using coastal seawater. We show that DMS, methanethiol (MeSH; CH3SH), and benzothiazole (C7H5NS) account for on average over 90 % of total gas-phase sulfur emissions, with non-DMS sulfur sources representing 36.8 ± 7.7 % of sulfur emissions during the first nine days of the experiment in the pre-bloom phase prior to major biological growth, before declining to 14.5 ± 6.0 % in the latter half of the experiment when DMS dominates during the bloom and decay phases. The molar ratio of DMS to MeSH during the pre-bloom phase (DMS : MeSH = 4.60 ± 0.93) was consistent with the range of previously calculated ambient DMS to MeSH sea-to-air flux ratios. As the experiment progressed, the DMS to MeSH emission ratio increased significantly, reaching 31.8 ± 18.7 during the bloom and decay. Measurements of dimethylsulfoniopropionate (DMSP), heterotrophic bacteria, and enzyme activity in the seawater suggest the DMS : MeSH ratio is a sensitive indicator of the bacterial sulfur demand and the composition and magnitude of available sulfur sources in seawater. The evolving DMS : MeSH ratio and the emission of a new aerosol precursor gas, benzothiazole, have important implications for secondary sulfate formation pathways in coastal marine environments.

789 Split-thickness Skin Graft Meshing: The True Mesh Ratio

Introduction Skin graft meshing is frequently used to allow coverage of extensive wound areas, reduction of donor site size, and prevention of fluid accumulation underneath the skin graft. Meshers allow for various ratios to be used but no studies have confirmed the accuracy of the mesh ratio provided by the manufacturers. The objective of this study was to measure the true mesh ratio achieved using some of the most commonly used skin meshers. Methods A prospective cohort study was performed in 2019. The true expansion ratio was calculated for standardized human split-thickness skin grafts (4x4cm harvested 0.0012in depth using an air dermatome) of the most commonly used meshing ratios (1:1, 1:5:1, 2:1, 3:1, 4:1, and 6:1). Results We had a total of 18 patients resulting in 86 measurements. The study population consisted of 12 males and 6 females; 89% white, 11% non-white; average age of 43 years (SD 21.2). The true mesh ratios of 1:1, 1.5:1, 2:1, 3:1, 4:1, and 6:1 meshers were found to be 1.20 (SD 0.14), 1.55 (SD 0.32), 1.68 (SD 0.32), 2.62 (SD 0.51), 2.87 (SD 0.92), 4.94 (SD 1.35), respectively. The percent errors were 20.2%, 3.3%, -15.8%, -12.5%, -28.3%, -17.7% respectively. Conclusions Mesh ratios of 2:1 meshers and above established by manufacturers are inaccurate. Therefore, overestimation of 2:1 meshers and above is recommended. We recommend an overestimation of 10% or more for meshers 2:1 and above. Applicability of Research to Practice This has significant applicability to practice as it can affect surgical decisions related to estimating the extent of donor area needed to cover skin and soft tissue defects.

652 Topical Delivery of Fidgetin-Like 2 siRNA to Enhance Cell Migration for Burn Wound Healing in a Swine Model

Introduction Donor site skin is a precious commodity for burn wounds and graft-sparing procedures are critical for larger total body surface area burns. Currently, large ratio skin grafts have been found to preserve donor skin but at the expense of delayed reepithelialization and increased scar formation at the recipient site. In this study, we propose to utilize a topical gene therapy treatment as an adjunct to higher meshing ratio (4:1) split thickness skin grafts (mSTSG) to improve re-epithelialization and reduce donor skin requirements. Administration of small interfering RNA (siRNA) to suppress the expression of one microtubule regulatory protein, Fidgetin-Like 2 (FL2), has been shown to increase cell migration rate both in vitro and in vivo. We hypothesize that topical application of nanoparticle (NP) encapsulated FL2 siRNA will increase cell migration resulting in expedited reepithelialization in large mesh ratio mSTSG in a full-thickness porcine burn model. A pilot study was conducted to down-select siRNA dose by measuring the rate of reepithelialization of full-thickness grafted burns. Methods Ten full-thickness (FT) 5x5cm burn wounds were created on the dorsum of anesthetized Yorkshire pigs. Four days later, the wounds were surgically debrided to a bleeding wound bed and grafted using a 4:1 mSTSG. Treatments groups (n= 2 per pig) consisted of vehicle control (PBS), scrambled sequence NP siRNA (non-targeting siRNA; negative control), and either 5, 10, or 20uM concentration of NP FL2 siRNA were applied topically to the mSTSG. Rechecks were performed twice a week for four week during which time digital images, non-invasive measurements, punch biopsies were acquired, and treatment was reapplied. Quantitative measurements include rate of reepithelialization and laser speckle imaging. Histopathology was assessed by a blinded pathologist. Results All wounds reepithelialized within 28 days post-grafting without infection. There was a positive trend with increasing concentration of NP FL2 siRNA on the burn wound healing. LSI data showed no statistical differences to vehicle control. Pathology analysis is ongoing. Conclusions Topical application of NP FL2 siRNA did increase the rate of reepithelialization of large ratio mSTSG treated full-thickness injuries. Additional pilot animals are currently ongoing to down-select siRNA dose for future studies comparing the experimental treatment to the current clinical gold standard 1:1.5 mesh ratio. Applicability of Research to Practice A topical gene therapy treatment combined with a higher meshing ratio to improve current treatment modalities by increasing reepithelialization and decreasing multiple donor site harvesting.

Spectrally Consistent Approximations to the Matrix Exponent and Their Applications to Boundary Layer Problem

In an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞ uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

Resolvent Estimates in l_p for Discrete Laplacians on Irregular Meshes and Maximum-norm Stability of Parabolic Finite Difference Schemes

In an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞ uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

Diagonal scaling of stiffness matrices in the Galerkin boundary element method

A boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.