HOMOTOPY ANALYSIS METHOD FOR SYSTEMS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

Abstract: In this article, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve systems of fractional integro-differential equations. Comparing with the exact solution, the HAM provides us with a simple way to adjust and control the convergence region of the series solution by introducing an auxiliary parameter h . Four examples are tested using the proposed technique. It is shown that the solutions obtained by the Adomian decomposition method (ADM) are only special cases of the HAM solutions. The present work shows the validity and great potential of the homotopy analysis method for solving linear and nonlinear systems of fractional integro-differential equations. The basic idea described in this article is expected to be further employed to solve other similar nonlinear problems in fractional calculus

Exact solution for linear and nonlinear systems of PDEs by homotopy-perturbation method

In this paper, the homotopy-perturbation method (HPM)proposed by J.-H. He is adopted for solving linear and nonlinear systems of partial differential equations (PDEs). In this method, a homotopy parameter p, which takes the values from 0 to 1, is introduced. When p = 0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p gradually increases to 1, the system goes through a sequence of ‘deformations’, the solution of each of which is ‘close’ to that at the previous stage of ‘deformation’. Eventually at p = 1,the system takes the original form of the equation and the final stage of ‘deformation’ gives the desired solution. Some examples are presented to demonstrate the efficiency and simplicity of the method

Mortality from gastrointestinal congenital anomalies at 264 hospitals in 74 low-income, middle-income, and high-income countries: a multicentre, international, prospective cohort study

Summary Background Congenital anomalies are the fifth leading cause of mortality in children younger than 5 years globally. Many gastrointestinal congenital anomalies are fatal without timely access to neonatal surgical care, but few studies have been done on these conditions in low-income and middle-income countries (LMICs). We compared outcomes of the seven most common gastrointestinal congenital anomalies in low-income, middle-income, and high-income countries globally, and identified factors associated with mortality. Methods We did a multicentre, international prospective cohort study of patients younger than 16 years, presenting to hospital for the first time with oesophageal atresia, congenital diaphragmatic hernia, intestinal atresia, gastroschisis, exomphalos, anorectal malformation, and Hirschsprung’s disease. Recruitment was of consecutive patients for a minimum of 1 month between October, 2018, and April, 2019. We collected data on patient demographics, clinical status, interventions, and outcomes using the REDCap platform. Patients were followed up for 30 days after primary intervention, or 30 days after admission if they did not receive an intervention. The primary outcome was all-cause, in-hospital mortality for all conditions combined and each condition individually, stratified by country income status. We did a complete case analysis. Findings We included 3849 patients with 3975 study conditions (560 with oesophageal atresia, 448 with congenital diaphragmatic hernia, 681 with intestinal atresia, 453 with gastroschisis, 325 with exomphalos, 991 with anorectal malformation, and 517 with Hirschsprung’s disease) from 264 hospitals (89 in high-income countries, 166 in middleincome countries, and nine in low-income countries) in 74 countries. Of the 3849 patients, 2231 (58·0%) were male. Median gestational age at birth was 38 weeks (IQR 36–39) and median bodyweight at presentation was 2·8 kg (2·3–3·3). Mortality among all patients was 37 (39·8%) of 93 in low-income countries, 583 (20·4%) of 2860 in middle-income countries, and 50 (5·6%) of 896 in high-income countries (p<0·0001 between all country income groups). Gastroschisis had the greatest difference in mortality between country income strata (nine [90·0%] of ten in lowincome countries, 97 [31·9%] of 304 in middle-income countries, and two [1·4%] of 139 in high-income countries; p≤0·0001 between all country income groups). Factors significantly associated with higher mortality for all patients combined included country income status (low-income vs high-income countries, risk ratio 2·78 [95% CI 1·88–4·11], p<0·0001; middle-income vs high-income countries, 2·11 [1·59–2·79], p<0·0001), sepsis at presentation (1·20 [1·04–1·40], p=0·016), higher American Society of Anesthesiologists (ASA) score at primary intervention (ASA 4–5 vs ASA 1–2, 1·82 [1·40–2·35], p<0·0001; ASA 3 vs ASA 1–2, 1·58, [1·30–1·92], p<0·0001]), surgical safety checklist not used (1·39 [1·02–1·90], p=0·035), and ventilation or parenteral nutrition unavailable when needed (ventilation 1·96, [1·41–2·71], p=0·0001; parenteral nutrition 1·35, [1·05–1·74], p=0·018). Administration of parenteral nutrition (0·61, [0·47–0·79], p=0·0002) and use of a peripherally inserted central catheter (0·65 [0·50–0·86], p=0·0024) or percutaneous central line (0·69 [0·48–1·00], p=0·049) were associated with lower mortality. Interpretation Unacceptable differences in mortality exist for gastrointestinal congenital anomalies between lowincome, middle-income, and high-income countries. Improving access to quality neonatal surgical care in LMICs will be vital to achieve Sustainable Development Goal 3.2 of ending preventable deaths in neonates and children younger than 5 years by 2030

Modeling and Analyzing Neural Networks Using Reproducing Kernel Hilbert Space Algorithm

In this paper, we present a new method for solving some certain differential systems in the artificial neural networks field. The analytic and approximate solutions are given with series form in the spaces W[a,b] and H[a,b]. The method used in this thesis has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, need less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this thesisshow potentiality, generality, and superiority of our method as compared with the Range Kutta method

Second Order Fuzzy Fractional Differential Equations Under Caputo’s H-Differentiability

The aim of this paper is to use the concept of the generalized H-derivative to define fuzzy Caputo’s H-derivative of order β ∈(1,2]. Our definition is an extension of fuzzy Caputo’s H-derivative of order β ∈(0,1] and higher order H-derivative of integerorder. After that, we study fuzzy fractional initial value p roblems of order β ∈(1,2] and give an algorithm to solve them based onthe characterization theorem. Finally, we apply the reprod ucing kernel Hilbert space method to obtain approximate solutions of second order fuzzy fractional initial value problems and give some numerical examples

The multistage homotopy-perturbation method: a powerful scheme for handling the Lorenz system

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy-perturbation method (HPM) is presented. The HPM is treated as an algorithm in a sequence of intervals (i.e. time step) for finding accurate approximate solutions to the famous Lorenz system. Numerical comparisons between the multistage homotopy-perturbation method (MHPM) and the classical fourth-order Runge–Kutta (RK4) method reveal that the new technique is a promising tool for the nonlinear systems of ODE

Direct solution of second-order BVPs by homotopy-perturbation method (Penyelesaian Secara Langsung MNS Berperingkat-Dua Melalui Kaedah Homotopi-usikan)

In this paper, systems of second-order boundary value problems (BVPs) are considered. The applicability of the homotopy-perturbation method (HPM) was extended to obtain exact solutions of the BVPs directly. Dalam makalah ini, sistem masalah nilai sempadan (MNS) berperingkat dua dipertimbangkan. Kegunaan kaedah homotopi-usikan (KHU) diperluaskan bagi memperoleh penyelesaian tepat MNS tersebut secara langsung

Generalizing the meaning of derivatives and integrals of any order differential equations by fuzzy-order derivatives and fuzzy-order integrals

This paper develops the correlation between fuzzy numbers and order of differential equations and overcomes the limitation in the existence of fractional order in the formulation of equation. In the view of fractional calculus, a new logic called fuzzy order by generalizing the meaning of derivatives and integrals of any order as fuzzy-order derivatives and fuzzy-order integral. We discuss Dα, where Dα is derivative of order α and α may be a triangular fuzzy number or trapezoidal fuzzy number, and propose to rewrite Dαy(x)=gx,y(x), when α=A,B,C and A,BandC∈N (where N is the set of natural numbers) and rewrite Riemann-Liouville integral, Riemann-Liouville derivative and Caputo fractional derivatives with respect to this new logic of fuzzy order. The proposed approach also covers multi cases, where the order is either integer or fractional. At the end, three numerical examples are presented to demonstrate the application of new logic, when the order of derivatives and integrals are given as triangular fuzzy numbers. These include time fractional heat equation represented as a time fuzzy-order heat equation and the time-fractional diffusion wave equation represented as a time-fuzzy-order diffusion wave equation. Keywords: Fractional calculus, Fuzzy numbers, Riemann-Liouville fuzzy definitions, Caputo fuzzy definition

Analytical Solutions of Fuzzy Fractional Boundary Value Problem of Order 2α by Using RKHS Algorithm

In this article, an effective numerical solution for fractional fuzzy differential equations of order 2α subject to appropriate fuzzy boundary conditions has been provided by using the Reproducing Kernel (RK) algorithm in Caputo sense. The reproducing kernel functions are built, in which the constraint conditions are satisfied, to yield a fast and accurate RK algorithm for handling these BVPs. The solution methodology is based on constructing the fractional series solution based on the reproducing-kernel theory in the form of a rapidly convergent series with a minimum size of calculations using symbolic computation software. The analytical solution is formulated in the form of a finite series, while, the n-term numerical solution is obtained and proved to converge uniformly to the analytical solution in the space of interest. Simulations, as well as the computational algorithm, are provided to guarantee the RK procedure, to show potentiality, generality, and superiority of RK algorithm and to illustrate the theoretical statements of the present algorithm