Solution Techniques Based on Adomian and Modified Adomian Decomposition for Nonlinear Integro-Fractional Differential Equations of the Volterra-Hammerstein Type

هذا البحث يطبق بفعاليه طريقه التحليل الادوميانى وطريقه التحليل الادوميانى المعدله كتقنيات عددية لتعيين الحل شبه التحليلى او الحل شبه التقريبى للمعادلات التفاضليه التكامليه اللاخطيه للرتب الكسريه (IFDE) من نوع فولتيرا-هاميرشتين (V-H) والتى توصف فيها المشتقه الكسريه المتعدده العليا بنمط كابوتو. فى هذا النهج سنغير بشكل جذرى ال (IFDE) لنوع  (V-H)  الى بعض معادلات جبريه تكراريه وان الحل لهذه المعادلات هو بمثابه مجموع من المتتابعات اللاعدديه (Countless) لمركبات متقاربه نوعيا للحل المستند (المعتمد)  على الحدود الضوضائيه وذلك فى حاله عدم حصولنا على حل من النوع المغلق وان الحدود المقطوعه (المحذوفه) يستخدم للاغراض العدديه. واخيرا تم اعطاء امثله لتوضيح هذه الافكار والاعتباراتThis paper efficiently applies the Adomian Decomposition Method and Modified Adomian Decomposition Method as computational techniques to locate the semi-analytical solution or semi-approximate solution for the considered nonlinear Integro Differential Equations for the fractional-order (IFDE) of the Volterra-Hammerstein (V-H) type, in which the higher-multi fractional derivative is described in the Caputo sense.In this procedure, we radically change the IFDE’s of V-H type into some iterative algebraic equations and the solution of this equations is considered as the sum of the countless sequence of components typically converging to the solution based on the noise terms where a closed-form solution is not obtainable, a truncated number of terms is usually used for numerical purposes.Finally, examples are prepared to illustrate these considerations

Laplace Adomian Decomposition and Modify Laplace Adomian Decomposition Methods for Solving Linear Volterra Integro-Fractional Differential Equations with Constant Multi-Time Retarded Delay

     في هذا العمل نقدم تحويلات لابلاس مع طريقة أدوميان التحليلية المتسلسلة و كما اننا نعدل طريقة أدوميان التحليلية للمرة الاولى لحل معادلات فولتيرا التفاضلية-التكاملية الخطيه للرتب الكسرية كما في مفهوم كابوتو مع التأخير الحدي المتضاعف الثابت. هذه الطريقة تعتمد على مزيج ممتاز من طريقة تحويلات لابلاس، طريقة تحديد المتسلسلات، طريقة متعددات الحدود لادوميان مع التعديلات. أن التقنية المستخدمة تحول التأخير الحدي للمعادلات التفاضلية ذات التكاملات الكسرية الى معادلات جبرية متكررة عندما تكون نواة الفروق من نوع المنحل البسيط. و أخيراَ أعطيت أمثلة لتوضيح فعالية و ديقة الطرق المقترحة.In this work, we present Laplace transform with series Adomian decomposition and modify Adomian decomposition methods for the first time to solve linear Volterra integro-differential equations of the fractional order in Caputo sense with constant multi-time Retarded delay. This method is primarily based on the elegant mixture of Laplace transform method, series expansion method and Adomian polynomial with modifications. The proposed technique will transform the multi-term delay integro-fractional differential equations into some iterative algebraic equations, and it is capable of reducing computational analytical works where the kernel of difference and simple degenerate types. Analytical examples are presented to illustrate the efficiency and accuracy of the proposed methods

Solving a System of Fractional-Order Volterra Integro-Differential Equations Based on the Explicit Finite Difference Approximation via the Trapezoid Method with Error Analysis

The well-known central finite difference approximation was combined with the trapezoid quadrature method in this study to provide a numerical solution of the linear system of Volterra integro-fractional differential equations (LSVI-FDEs) of arbitrary orders, where the fractional derivative is described in the Caputo sense and the orders are between zero and one. The method works by first using the central finite difference approximation to approximate the Caputo derivative at any fixed point and then using the trapezoidal rule to obtain a finite difference expression for our fractional equation, while recalling the linear spline approximation for the first steps. This new, more efficient method involves converting sets of equations and conditions into matrix relationships, from which symmetry matrices can be created in some cases. We also present a new approach for error analysis of the discrete numerical scheme and the explicit numerical technique for LSVI-FDEs. The multi-level explicit finite difference approximation’s stability and convergence were explored, and a MatLab application was created to explain the results. Finally, several numerical examples are offered to demonstrate the technique’s application

Bessel Collocation Method for Solving Fredholm–Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021)

Numerical Computation of Mixed Volterra–Fredholm Integro-Fractional Differential Equations by Using Newton-Cotes Methods

In this article, the numerical solution of the mixed Volterra–Fredholm integro-differential equations of multi-fractional order less than or equal to one in the Caputo sense (V-FIFDEs) under the initial conditions is presented with powerful algorithms. The method is based upon the quadrature rule with the aid of finite difference approximation to Caputo derivative usage collocation points. For treatments, our technique converts the V-FIFDEs into algebraic equations with operational matrices, some of which have the symmetry property, which is simple for evaluating. Furthermore, numerical examples are presented to show the technique’s validity and usefulness as well comparisons with previous results. The majority of programs are performed using MATLAB v. 9.7

Bessel Collocation Method for Solving Fredholm–Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021)

Prospective observational cohort study on grading the severity of postoperative complications in global surgery research

Background The Clavien–Dindo classification is perhaps the most widely used approach for reporting postoperative complications in clinical trials. This system classifies complication severity by the treatment provided. However, it is unclear whether the Clavien–Dindo system can be used internationally in studies across differing healthcare systems in high- (HICs) and low- and middle-income countries (LMICs). Methods This was a secondary analysis of the International Surgical Outcomes Study (ISOS), a prospective observational cohort study of elective surgery in adults. Data collection occurred over a 7-day period. Severity of complications was graded using Clavien–Dindo and the simpler ISOS grading (mild, moderate or severe, based on guided investigator judgement). Severity grading was compared using the intraclass correlation coefficient (ICC). Data are presented as frequencies and ICC values (with 95 per cent c.i.). The analysis was stratified by income status of the country, comparing HICs with LMICs. Results A total of 44 814 patients were recruited from 474 hospitals in 27 countries (19 HICs and 8 LMICs). Some 7508 patients (16·8 per cent) experienced at least one postoperative complication, equivalent to 11 664 complications in total. Using the ISOS classification, 5504 of 11 664 complications (47·2 per cent) were graded as mild, 4244 (36·4 per cent) as moderate and 1916 (16·4 per cent) as severe. Using Clavien–Dindo, 6781 of 11 664 complications (58·1 per cent) were graded as I or II, 1740 (14·9 per cent) as III, 2408 (20·6 per cent) as IV and 735 (6·3 per cent) as V. Agreement between classification systems was poor overall (ICC 0·41, 95 per cent c.i. 0·20 to 0·55), and in LMICs (ICC 0·23, 0·05 to 0·38) and HICs (ICC 0·46, 0·25 to 0·59). Conclusion Caution is recommended when using a treatment approach to grade complications in global surgery studies, as this may introduce bias unintentionally

The surgical safety checklist and patient outcomes after surgery: a prospective observational cohort study, systematic review and meta-analysis

© 2017 British Journal of Anaesthesia Background: The surgical safety checklist is widely used to improve the quality of perioperative care. However, clinicians continue to debate the clinical effectiveness of this tool. Methods: Prospective analysis of data from the International Surgical Outcomes Study (ISOS), an international observational study of elective in-patient surgery, accompanied by a systematic review and meta-analysis of published literature. The exposure was surgical safety checklist use. The primary outcome was in-hospital mortality and the secondary outcome was postoperative complications. In the ISOS cohort, a multivariable multi-level generalized linear model was used to test associations. To further contextualise these findings, we included the results from the ISOS cohort in a meta-analysis. Results are reported as odds ratios (OR) with 95% confidence intervals. Results: We included 44 814 patients from 497 hospitals in 27 countries in the ISOS analysis. There were 40 245 (89.8%) patients exposed to the checklist, whilst 7508 (16.8%) sustained ≥1 postoperative complications and 207 (0.5%) died before hospital discharge. Checklist exposure was associated with reduced mortality [odds ratio (OR) 0.49 (0.32–0.77); P\u3c0.01], but no difference in complication rates [OR 1.02 (0.88–1.19); P=0.75]. In a systematic review, we screened 3732 records and identified 11 eligible studies of 453 292 patients including the ISOS cohort. Checklist exposure was associated with both reduced postoperative mortality [OR 0.75 (0.62–0.92); P\u3c0.01; I2=87%] and reduced complication rates [OR 0.73 (0.61–0.88); P\u3c0.01; I2=89%). Conclusions: Patients exposed to a surgical safety checklist experience better postoperative outcomes, but this could simply reflect wider quality of care in hospitals where checklist use is routine

Critical care admission following elective surgery was not associated with survival benefit: prospective analysis of data from 27 countries

This was an investigator initiated study funded by Nestle Health Sciences through an unrestricted research grant, and by a National Institute for Health Research (UK) Professorship held by RP. The study was sponsored by Queen Mary University of London