Numerical Solution of Linear Ordinary Differential Equations and Differential-Algebraic Equations by Spectral Methods

This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudo-spectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriate choice of Gauss-Chebyshev-Radau points, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear

Juxta Cortical Tibia Metastatic Deposition in Gastric Cancer: A Case Report

We report a 41 years old man with rapidly growing and tender lump on the anteromedial surface of tibia. The patient had the history of gastrectomy and gastrojejunostomy due to gastric carcinoma. On admission, the Simple X-ray of lower extremity disclosed a slight thinning of the anterior cortex of tibia without cortical destruction. The whole-body bone scan with 99mTC MDP revealed activity of lesion in all 3 phases. The histopathological evaluation showed an infiltration of bone with tumor cells. Review No the literature revealed in previous cases of skeletal metastasis from gastric cancer in the tibia like this

Sprengel's Deformity Associated with Musculoskeletal Dysfunctions and Renal Anomalies: A Case Report

Background. Sprengel's deformity is a rare congenital anomaly of the shoulder girdle. The deformity is due to failure of descent of the scapula in intrauterine life. Case Presentation. We report a case of unilateral Sprengel's deformity associated with several other musculoskeletal and renal disorders consisting of absence of pectoralis major, weakness of trapezius and serratus anterior muscles, one kidney agenesis, and severe hydronephrosis of the other kidney in a 7-year-old boy. Conclusion. Sprengel's deformity can be associated with other musculoskeletal abnormalities and it is much more than a cosmetic problem

A first course in ordinary differential equations: analytical and numerical methods

This book presents a modern introduction to analytical and numerical techniques for solving ordinary differential equations (ODEs). Contrary to the traditional format—the theorem-and-proof format—the book is focusing on analytical and numerical methods. The book supplies a variety of problems and examples, ranging from the elementary to the advanced level, to introduce and study the mathematics of ODEs. The analytical part of the book deals with solution techniques for scalar first-order and second-order linear ODEs, and systems of linear ODEs—with a special focus on the Laplace transform, operator techniques and power series solutions. In the numerical part, theoretical and practical aspects of Runge-Kutta methods for solving initial-value problems and shooting methods for linear two-point boundary-value problems are considered. The book is intended as a primary text for courses on the theory of ODEs and numerical treatment of ODEs for advanced undergraduate and early graduate students. It is assumed that the reader has a basic grasp of elementary calculus, in particular methods of integration, and of numerical analysis. Physicists, chemists, biologists, computer scientists and engineers whose work involves solving ODEs will also find the book useful as a reference work and tool for independent study. The book has been prepared within the framework of a German–Iranian research project on mathematical methods for ODEs, which was started in early 2012

Nonlinear ordinary differential equations: analytical approximation and numerical methods

The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march method. This book comprehensively investigates various new analytical and numerical approximation techniques that are used in solving nonlinear-oscillator and structural-system problems. Students often rely on the finite element method to such an extent that on graduation they have little or no knowledge of alternative methods of solving problems. To rectify this, the book introduces several new approximation techniques

A First Course in Ordinary Differential EquationsAnalytical and Numerical Methods /

XIV, 288 p. 10 illus.online resource

Design of Forward/reverse Logistics with Environmental Consideration

Growth of environmental issues has caused to consider various factors that influence on condition of environment. Green supply chain has absorbed care of researchers because of its considerable impacts on environment. In this regard, this study designs the forward/revers logistics network by putting emphasis on environmental aspects in its model such as quantity of CO2 emission. In this logistics network, three objective functions such as minimizing the total cost and quantity of CO2 emission as well as maximizing the satisfaction of customers have been considered, simultaneously. Because of considering of three objective functions in this model, multi objective optimization methods persuade the researchers to implement them. Non-dominated sorting genetic algorithms (NSGA-ӀӀ) and Multi-objective particle swarm optimization (MOPSO) are proposed to cope with this problem. The results acquired from experiments on several test problems are verified by GAMS software. Finally, the results obtained through experiments on different problems verify the superiority of NSGA-ӀӀ over MOPSO in terms of all comparison metrics

A stochastic time-dependent green capacitated vehicle routing and scheduling problem with time window, resiliency and reliability: a case study

This paper presents a new multi-objective model for a vehicle routing problem under a stochastic uncertainty. It considers traffic point as an inflection point to deal with the arrival time of vehicles. It aims to minimize the total transportation cost, traffic pollution, customer dissatisfaction and maximizes the reliability of vehicles. Moreover, resiliency factors are included in the model to increase the flexibility of the system and decrease the possible losses that may impose on the system. Due to the NP-hardness of the presented model, a meta-heuristic algorithm, namely Simulated Annealing (SA) is developed. Furthermore, a number of sensitivity analyses are provided to validate the effectiveness of the proposed model. Lastly, the foregoing meta-heuristic is compared with GAMS, in which the computational results demonstrate an acceptable performance of the proposed SA

Anterior Shoulder Dislocation and Ipsilateral Humeral Shaft Fracture

Simultaneous dislocation of shoulder and humeral shaft fracture is a rare injury, and there is no clear protocol for its treatment. Herein we present a case of a 15-year-old boy, who suffered from a job-related accident and sustained fracture of humeral shaft associated with ipsilateral anterior shoulder dislocation and fracture of greater tuberosity 15 years ago. He received closed reduction of both injuries and coaptation plaster splint for four weeks, followed by Sarmiento splint at that time. Fifteen years after the injury, he has no problem related to the previous injury, and does not experience any episode of shoulder instability