Approximate Solution of Fractional Integro-Differential Equations by Least Squares Method

In this paper, least squares approximation method is developed for solving a class of linear fractional integro-differential equations comprising Volterra and Fredhlom cases. This method is based on a polynomial of degree n to compute an approximate solution of these equations. The convergence analysis of the proposed method is proved. In addition, to show the accuracy and the efficiency of the proposed method, some examples are presented

Pseudo-Chilblains in Adult Patients with Confirmed COVID-19: A Systematic Review

Background: Pseudo-chilblains have been associated with COVID-19. Many reports, however, lack confirmation of COVID-19 infection. While likely associated, all chilblains/chilblain-like lesions during this time should not be assumed to be COVID-19 related. This study examines the characteristics of adults with pseudo-chilblains and confirmed COVID-19. Methods: A systematic review of PubMed/MEDLINE database was performed using the PRISMA guidelines. Adults (>18 years) with confirmed COVID-19 were included. De-identified registries were excluded to avoid duplication. We extracted study design, age, sex, race, geographic location, relationship of COVID-19 diagnosis to chilblains onset, confirmatory testing, hospitalization status, anatomical location, cold/damp exposure, presence/absence/description of pseudo-chilblains symptoms, presence/absence of biopsies/histopathologic findings, tissue IHC/PCR, presence/absence/details of extracutaneous COVID-19 disease, pre-existing chilblains, treatment and resolution timeline. The search was completed in July 2022. Results: We identified 13 studies (29 patients). In COVID-19-infected adults, pseudo-chilblains were reported primarily from North America and Europe, occurring in both sexes over a wide age-range, affected well and ill patients, favored the hands and feet and could be symptomatic or asymptomatic. Most patients had extracutaneous symptoms. Resolution time ranged from 50 days. There was marked variation in treatment strategies and appearance of pseudo-chilblains relative to entire disease course. Biopsies were infrequently performed but findings similar to classical chilblains were described. Conclusions: Many patients reported as pseudo-chilblains of COVID-19 lack confirmed infection. Infection confirmation, photographic documentation and histopathology are critical to establish homogeneity in reported pseudo-chilblains during this global pandemic. Further work clarifying the relationship of acral eruptions and COVID-19 is necessary

Approximate Solution of Fractional Integro-Differential Equations by Least Squares Method

In this paper, least squares approximation method is developed for solving a class of linear fractional integro-differential equations comprising Volterra and Fredhlom cases. This method is based on a polynomial of degree n to compute an approximate solution of these equations. The convergence analysis of the proposed method is proved. In addition, to show the accuracy and the efficiency of the proposed method, some examples are presented

A new operational matrix for solving two-dimensional nonlinear integral equations of fractional order

In this paper, first, we derive the operational matrix of two-dimensional orthogonal triangular functions (2D-TFs) for two-dimensional fractional integrals. Then, we apply this operational matrix and properties of Two-dimensional orthogonal triangular functions to reduce two-dimensional fractional integral equations to a system of algebraic equations. Finally, in order to show the validity and efficiency, we present some numerical examples

Evaluation of second order and parallel second order approaches to model temperature variation in chlorine decay modelling

All drinking water receives some form of disinfection and a minimum residual should remain at the customer’s tap. Most popular disinfectant of all is chlorine. Chlorine reacts with compounds in water and hence leads to decay. Temperature is one of the important factors that control the rate of decay. Annual water temperature variations of more than 20°C are common in distribution systems, so that dosing needs to be adjusted substantially between seasons to maintain residuals within desired limits. Arrhenius equation has been successfully used to estimate the temperature effects on chlorine decay reactions, especially when temperature is below 30°C. The temperature dependence parameter estimated is activation energy (E)/universal gas constant (R). A number of chlorine decay tests were conducted, by varying temperature from 15–50°C. Resulting chlorine measurements were input into AQUASIM, data fitting was performed using the parallel second order model (PSOM) proposed by Kastl et al. [1] and second order model (SOM) proposed by Clark [2]. The model parameters for all modelling approaches were estimated using AQUASIM. PSOM has two reactants and two respective decay coefficients. Results showed that PSOM fitted the data very well when either single or two E/Rs were used. On the contrary, the SOM did not show a good fit to the experimental chlorine decay profile for the same data sets. The results, therefore, indicated PSOM is more convenient to describe chlorine decay profile over a wide range of temperature