The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM) for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.

Background Syria has witnessed more than a decade of armed conflict through which healthcare workers and facilities have not only been affected, but targeted. Amidst this targeting of healthcare workers, subsequent displacement, and ‘weaponization’ of healthcare, the medical education and health professional training (MEHPT) of those that remain has split into at least two distinctive contexts: government controlled, and non-government controlled. Efforts to rebuild MEHPT in light of this polarisation and fragmentation have led to a new MEHPT system in non-government controlled northwest Syria, that functions through what we describe as a ‘hybrid kinetic model’. This mixed-methods study provides an in-depth analysis of this MEHPT system as a case study to inform future policy planning and interventions in the context of future post-conflict health workforce development. Methods We used mixed methods to investigate the state of MEHPT in northwest Syria during September 2021 and May 2022. This included a) Stakeholder analysis, b) 15 preparatory experts consultations c) 8 Focus group discussions d) 13 Semi-structured interviews e) 2 Questionnaires and f) Validation workshops. Results We identified three main categories of key stakeholders working on MEHPT in northwest Syria: 12 newly established academic institutions, 7 local governance authorities involved in MEHPT, and 12 non-governmental organisations (NGOs). The MEHPT system operated through these stakeholders in a three-layer system to provide undergraduate and postgraduate MEHPT. In the first, top, layer, external NGOs and donors hold the strongest capacity at the expense of relatively under resourced internal governance in the second, middle, level. In the third, bottom, level, local academic bodies operate. We uncovered several levels of challenges facing these stakeholders including governance challenges, institutional challenges, individual challenges, and political challenges. Despite these challenges, participants in our study highlighted significant opportunities within the MEHPT system and that MEHPT can be a peace building pillar for the community. Discussion To our knowledge, this is the first paper that provides an in-depth situational analysis of the MEHPT system in a conflict setting while engaging the voice of local key stakeholders. We found that local actors in MEHPT in non-government controlled northwest Syria have made efforts towards (re)building a new, hybrid and kinetic MEHPT system, through a bottom-up approach. Despite these efforts, the MEHPT system remains fragile and polarised, suffering from several levels of challenges with limited involvement of internal governance. Building on our findings, to improve this approach and build bridges of trust among stakeholders and the MEHPT community, further studies are needed to determine feasible approaches to increasing the role of internal governance structures in the MEHPT system through: 1-Formalisation of efforts through establishing a MEPHT technical coordination unit. 2-Further shifting of power from external supporting NGOs and funders to internal governance structures. 3- Working towards achieving sustainable long-term partnerships.

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations. Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems. Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields. The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs). In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively. We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind. We introduced efficient method, constructed on modified Bernstein polynomials. The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind. This method is tested for some examples from nonlinear integral equations. Maple software was used to obtain the solutions of these examples. The results demonstrate reliability of the proposed method. Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

Abstract. The aim of this work is to present the optimal homotopy asymptotic method (OHAM) and homotopy perturbation method (HPM) for solving Fredholm-Hammerstein integral equations. Several examples are discussed to show the ability of the methods. The results indicated that the methods are very effective and simple.

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