A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

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

Effective Computational Methods for Solving the Jeffery-Hamel Flow Problem

في هذا البحث، تم تنفيذ الطريقة الحسابية الفعالة (ECM) المستندة إلى متعددة الحدود القياسية الأحادية لحل مشكلة تدفق جيفري-هامل غير الخطية. علاوة على ذلك، تم تطوير واقتراح الطرق الحسابية الفعالة الجديدة في هذه الدراسة من خلال وظائف أساسية مناسبة وهي متعددات الحدود تشيبشيف، بيرنشتاين، ليجندر، هيرمت. يؤدي استخدام الدوال الأساسية إلى تحويل المسألة غير الخطية إلى نظام جبري غير خطي من المعادلات، والذي يتم حله بعد ذلك باستخدام برنامج ماثماتيكا®١٢. تم تطبيق تطوير طرق حسابية فعالة (D-ECM) لحل مشكلة تدفق جيفري-هامل غير الخطية، ثم تم عرض مقارنة بين الطرق. علاوة على ذلك، تم حساب الحد الأقصى للخطأ المتبقي ( )، لإظهار موثوقية الطرق المقترحة. تثبت النتائج بشكل مقنع أن ECM و D-ECM دقيقة وفعالة وموثوقة للحصول على حلول تقريبية للمشكلة.In this paper, the effective computational method (ECM) based on the standard monomial polynomial has been implemented to solve the nonlinear Jeffery-Hamel flow problem. Moreover, novel effective computational methods have been developed and suggested in this study by suitable base functions, namely Chebyshev, Bernstein, Legendre, and Hermite polynomials. The utilization of the base functions converts the nonlinear problem to a nonlinear algebraic system of equations, which is then resolved using the Mathematica®12 program. The development of effective computational methods (D-ECM) has been applied to solve the nonlinear Jeffery-Hamel flow problem, then a comparison between the methods has been shown. Furthermore, the maximum error remainder ( ) has been calculated to exhibit the reliability of the suggested methods. The results persuasively prove that ECM and D-ECM are accurate, effective, and reliable in getting approximate solutions to the problem

Collocation Method using Compactly Supported Radial Basis Function for Solving Volterra's Population Model

In this paper, indirect collocation approach based on compactly supported radial basis function is applied for solving Volterras population model. The method reduces the solution of this problem to the solution of a system of algebraic equations. Volterras model is a non-linear integro-differential equation where the integral term represents the effect of toxin. To solve the problem, we use the well-known CSRBF: Wendland3,5. Numerical results and residual norm 2 show good accuracy and rate of convergence.Comment: 8 pages , 1 figure. arXiv admin note: text overlap with arXiv:1008.233

Optimal control of systems with memory

The “Optimal Control of Systems with memory” is a PhD project that is borne from the collaboration between the Department of Mechanical and Aerospace Engineering of Sapienza University of Rome and CNR-INM the Institute for Marine Engineering of the National Research Council of Italy (ex INSEAN). This project is part of a larger EDA (European Defence Agency) project called ETLAT: Evaluation of State of the Art Thin Line Array Technology. ETLAT is aimed at improving the scientific and technical knowledge of potential performance of current Thin Line Towed Array (TLA) technologies (element sensors and arrays) in view of Underwater Surveillance applications. A towed sonar array has been widely employed as an important tool for naval defence, ocean exploitation and ocean research. Two main operative limitations costrain the TLA design such as: a fixed immersion depth and the stabilization of its horizontal trim. The system is composed by a towed vehicle and a towed line sonar array (TLA). The two subsystems are towed by a towing cable attached to the moving boat. The role of the vehicle is to guarantee a TLA’s constant depth of navigation and the reduction of the entire system oscillations. The vehicle is also called "depressor" and its motion generates memory effects that influence the proper operation of the TLA. The dynamic of underwater towed system is affected by memory effects induced by the fluid-structure interaction, namely: vortex shedding and added damping due to the presence of a free surface in the fluid. In time domain, memory effects are represented by convolution integral between special kernel functions and the state of the system. The mathematical formulation of the underwater system, implies the use of integral-differential equations in the time domain, that requires a nonstandard optimal control strategy. The goal of this PhD work is to developed a new optimal control strategy for mechanical systems affected by memory effects and described by integral-differential equations. The innovative control method presented in this thesis, is an extension of the Pontryagin optimal solution which is normally applied to differential equations. The control is based on the variational control theory implying a feedback formulation, via model predictive control. This work introduces a novel formulation for the control of the vehicle and cable oscillations that can include in the optimal control integral terms besides the more conventional differential ones. The innovative method produces very interesting results, that show how even widely applied control methods (LQR) fail, while the present formulation exhibits the advantage of the optimal control theory based on integral-differential equations of motion