The aim of this paper is to develop the inverse scattering transform (IST) for multi-component generalisations of nonlocal reductions of the nonlinear Schrödinger (NLS) equation with PTsymmetry related to symmetric spaces. This includes: the spectral properties of the associated Lax operator, Jost function, the scattering matrix and the minimal set of scattering data, the fundamental analytic solutions. As main examples we use the Manakov vector Schrödinger equation (related to A.III-symmetric spaces) and the multi-component NLS (MNLS) equations of Kullish-Sklyanin type (related to BD.I-symmetric spaces). Furthermore, the 1and 2-soliton solutions are obtained by using an appropriate modification of the Zakharov-Shabat dressing method. It is shown, that the MNLS equations of these types allow both regular and singular soliton configurations. Finally, we present here different examples of 1-and 2-soliton solutions for both types of models, subject to different reductions.Here, S ± J and T ± J are upper-and lower-block-triangular matrices, which can be written in the form:

Background: Acute upper gastrointestinal bleeding is a serious complication in cirrhotic patients. Without recommended management, recurrent bleeding happens in 30–40% within the next 2–3 days, and up to 60% within 1 week. Aim was to determine predictors of re-bleeding after oesophageal variceal banding in cirrhotic patients for 4 weeks. It was a descriptive study, conducted at the Department of Medicine, Sheikh Zayed Hospital, Rahim Yar Khan. Six months from June 21 to December 21, 2021. Methods: A total of 93 patients with active oesophageal variceal bleeding were included in this study. Upper gastrointestinal (UGI) endoscopy was performed to look for bendable varices (grades 1–4) and band ligation was applied. Patients were followed for 4 weeks for the history of hematemesis or Malena, fall in haemoglobin of 2 grams per decilitre or more and endoscopic rebleeding findings. Results: Out of 93 patients, 67(72.0%) were males, while 26(28.0%) were females. The Mean age of the patients was 45.66±16.61 years. According to Child-Pugh Classification, the majority of the patients 45(48.4%) had Child-Pugh Class-A, while 33 (35.5%) were Child B and 15 (16.1%) patients belonged to Child-Pugh Class C. Red wale sign was noted in 22 patients (23.7%). Among 93 cirrhotic patients who presented with variceal bleeding, 9 (9.7%) had re-bleeding within 4 weeks. Amongst 9 patients, 8 patients (88.9%) had red wale sign, grade II or above oesophageal varices and belonged to severe liver disease with child class B or C. Conclusion: Endoscopic variceal band Ligation is an effective treatment modality for the control of oesophageal variceal bleeding. Re-bleeding after band ligation was 9.7%. The major contributing factors to re-bleeding were the severity of cirrhosis, grades and columns of oesophageal varices, number of bands ligation and findings of red wale sign. Increasing age and duration of cirrhosis were contributing predictors of increased re-bleeding risk

Our goal in this paper is to generalize the integral transforms and use it with He’s polynomial method to find the solution of the nonlinear partial differential equations. All results of theoretical studies regarding the generalization and its properties are presented. For the He’s polynomial method, it is used to solve the nonlinear part of the partial differential equation. It is shown that the importance of my research is the combination of generalization of integral transforms with He’s polynomial method allows for exact and approximate solutions configurations to be determined. Furthermore, the generalization of integral transforms has been shown to include most, or even all, of the integral transforms and be applicable to a variety of equations, making it a crucial tool in solving them. Finally, the capability of solutions to be obtained quickly and easily through this combined technique provides an invaluable tool for solving problems.

‫الملخص‬ : ‫اق‬ ‫ة‬ ‫مباشر‬ ‫جديدة‬ ‫يقة‬ ‫طر‬ ‫ترحت‬ ‫و‬ ‫الخطية‬ ‫غير‬ ‫التكاممية‬ ‫لممعادلة‬ ‫عددي‬ ‫حل‬ ‫لحساب‬ ‫مقبولة‬ ‫يدهولم‬ ‫فر‬ ‫نوع‬ ‫من‬ -‫يدهولم‬ ‫فر‬ . ‫أساس‬ ‫هي‬ ‫يقة‬ ‫الطر‬ ‫متسمسمة‬ ‫حل‬ ‫معرفة‬ ‫غير‬ ‫لكمية‬ ‫ادم‬ ‫يقة‬ ‫بطر‬ ‫و‬ ‫ن‬ . ‫نتائج‬ ‫عمى‬ ‫حصمنا‬ ‫أفضل‬ ‫ادمون‬ ‫يقة‬ ‫بطر‬ ‫نة‬ ‫مقار‬ . ‫بعض‬ ‫األمثمة‬ ‫يقة‬ ‫الطر‬ ‫لهذه‬ ‫الفعالية‬ ‫وحساب‬ ‫دقة‬ ‫لتوضيح‬ ‫قدمت‬ . Abstract:A new and effective direct method to determine the numerical solution of non-linear Fredholm-Fredholm integral equations is proposed.The method is based on a series solution for the unknown quantity by Adomian's method. We obtained very good results compared with classic Adomian's method Some numerical examples are provided to illustrate accuracy and computational efficiency of the method.

In this paper a new technique for Aitken extrapolation method is used to solve F.I.E. st 1 k. This method requires many reiterative steps.