Ordinary differential equations and calculus of variations

2020 ◽  
Author(s):  
Efim Kogan
Keyword(s):  
Mathematical Modeling ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Calculus Of Variations ◽  
Applied Mechanics ◽  
Technological Systems ◽  
Technological Means ◽  
Independent Work

The textbook contains theoretical information in a volume of the lecture course are discussed in detail and examples of typical tasks and test tasks and tasks for independent work. Designed for students enrolled in directions of preparation 15.03.03 "Applied mechanics" 01.03.02 "mathematics" (specialization "Mathematical modeling"), major 23.05.01 "Land transport and technological means" (specialization "Dynamics and strength of transport and technological systems"). Can be used by teachers for conducting practical classes.

A Steepest-Ascent Method for Solving Optimum Programming Problems

1962 ◽  
Vol 29 (2) ◽  
pp. 247-257 ◽  
Author(s):  
A. E. Bryson ◽  
W. F. Denham
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Calculus Of Variations ◽  
Numerical Procedure ◽  
Nonlinear Ordinary Differential Equations ◽  
Point Boundary ◽  
Numerical Examples ◽  
Steepest Ascent ◽  
Maximum Range

A systematic and rapid steepest-ascent numerical procedure is described for solving two-point boundary-value problems in the calculus of variations for systems governed by a set of nonlinear ordinary differential equations. Numerical examples are presented for minimum time-to-climb and maximum altitude paths for a supersonic interceptor and maximum-range paths for an orbital glider.

scholarly journals Using Ordinary Differential Equations to Explore Cancer-Immune Dynamics and Tumor Dormancy

2016 ◽  
Author(s):  
Kathleen P. Wilkie ◽  
Philip Hahnfeldt ◽  
Lynn Hlatky
Keyword(s):  
Mathematical Modeling ◽  
Immune Response ◽  
Differential Equations ◽  
Tumor Growth ◽  
Ordinary Differential Equations ◽  
Systemic Disease ◽  
Growth Dynamics ◽  
Tumor Dormancy ◽  
Cytotoxic Action ◽  
General Method

AbstractCancer is not solely a disease of the genome, but is a systemic disease that affects the host on many functional levels, including, and perhaps most notably, the function of the immune response, resulting in both tumor-promoting inflammation and tumor-inhibiting cytotoxic action. The dichotomous actions of the immune response induce significant variations in tumor growth dynamics that mathematical modeling can help to understand. Here we present a general method using ordinary differential equations (ODEs) to model and analyze cancer-immune interactions, and in particular, immune-induced tumor dormancy.

scholarly journals Simulation of several flying situations of paragliders

2021 ◽  
Vol 15 (1) ◽  
pp. 79-89
Author(s):  
Thanh Xuan Nguyen ◽  
Phuong Thi-Thu Phan ◽  
Tien Van Pham
Keyword(s):  
Mathematical Modeling ◽  
Differential Equations ◽  
Modeling And Simulation ◽  
Ordinary Differential Equations ◽  
State Space ◽  
Simulation Modeling ◽  
Time Varying ◽  
Gravity Forces

Paragliding is an adventure and fascinating sport of flying paragliders. Paragliders can be launched by running from a slope or by a winch force from towing vehicles, using gravity forces as the motor for the motion of flying. This motion is governed by the gravity forces as well as time-varying aerodynamic ones which depend on the states of the motion of paraglider at each instant of time. There are few published articles considering mechanical problems of paragliders in their various flying situations. This article represents the mathematical modeling and simulation of several common flying situations of a paraglider through establishing and solving the governing differential equations in state-space. Those flying situations include the ones with constant headwind/tailwind with or without constant upwind; the ones with different scenario for the variations of headwind and tailwind combined with the upwind; the ones with varying pilot mass; and the ones whose several parameters are in the form of interval quantities. The simulations were conducted using a powerful Julia toolkit called DifferentialEquations.jl. The obtained results in each situation are discussed, and some recommendations are presented. Keywords: paraglider; simulation; modeling; state-space; ordinary differential equations; Julia; DifferentialEquations.jl

scholarly journals Mathematical Modeling and Simulation of SIR Model for COVID−2019 Epidemic Outbreak: A Case Study of India

2020 ◽  
Author(s):  
Dr. Ramjeet Singh Yadav
Keyword(s):  
Mathematical Modeling ◽  
Infectious Diseases ◽  
Differential Equations ◽  
Modeling And Simulation ◽  
Ordinary Differential Equations ◽  
Initial Level ◽  
Sir Model ◽  
Euler Method ◽  
Epidemic Outbreak

The present study discusses the spread of COVID−2019 epidemic of India and its end by using SIR model. Here we have discussed about the spread of COVID−2019 epidemic in great detail using Euler method. The Euler method is a method for solving the ordinary differential equations. The SIR model has the combination of three ordinary differential equations. In this study, we have used the data of COVID−2019 Outbreak of India on 8 May, 2020. In this data, we have used 135710 susceptible cases, 54340 infectious cases and 1830 reward/removed cases for the initial level of experimental purpose. Data about a wide variety of infectious diseases has been analyzed with the help of SIR model. Therefore, this model has been already well tested for infectious diseases by various scientists and researchers. Using the data to the number of COVID−2019 outbreak cases in India the results obtained from the analysis and simulation of this proposed SIR model showing that the COVID−2019 epidemic cases increase for some time and there after this outbreak decrease. The results obtained from the SIR model also suggest that the Euler method can be used to predict transmission and prevent the COVID−2019 epidemic in India. Finally, from this study, we have found that the outbreak of COVID−2019 epidemic in India will be at its peak on 25 May 2020 and after that it will work slowly and on the verge of ending in the first or second week of August 2020.

scholarly journals A GEOMETRIC SETTING FOR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 08 (06) ◽  
pp. 1291-1327 ◽  
Author(s):  
IOAN BUCATARU ◽  
OANA CONSTANTINESCU ◽  
MATIAS F. DAHL
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Calculus Of Variations ◽  
Fourth Order ◽  
Jacobi Equation ◽  
Equivalence Problem ◽  
Nonlinear Connection ◽  
Geometric Setting ◽  
Jacobi Endomorphism

To a system of second-order ordinary differential equations one can assign a canonical nonlinear connection that describes the geometry of the system. In this paper, we develop a geometric setting that also allows us to assign a canonical nonlinear connection to a system of higher-order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third- or fourth-order ordinary differential equations.

Numerical solution technique for solving isoperimetric variational problems

2020 ◽  
pp. 2150002
Author(s):  
A. M. S. Mahdy ◽  
E. S. M. Youssef
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Calculus Of Variations ◽  
Variational Problems ◽  
Solution Technique ◽  
Transform Method ◽  
Change Technique ◽  
Main Target ◽  
Sumudu Transform

In this paper, we have a zeal for fulfilling the estimated scientific answers for the calculus of variations by using the Sumudu transform method (STM). The main target is to search the numerical arrangement of ordinary differential equations (ODEs) which emerge from the variational problems where first the fundamental condition for the arrangement of the issue is to fulfill the Euler–Lagrange condition and then solve the equations using STM. The valuable properties of the Sumudu change technique are used to downsize the calculation of the issue to a gathering of straight arithmetical conditions. We introduce four variational problems and discover the numerical solution of those problems using STM and plot the curves of those solutions. These models are picked such that there exist systematic answers for them to offer a reasonable diagram and show the effectiveness of the proposed strategy. Numerical outcomes are registered utilizing Maple programming.

Mathematical modeling of adaptive immune response after transplantation

2020 ◽  
Vol 13 (08) ◽  
pp. 2050167
Author(s):  
Anka Markovska
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Immune Response ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Ordinary Differential Equations ◽  
Cellular Immunity ◽  
Adaptive Immune Response ◽  
Nonlinear Ordinary Differential Equations ◽  
Adaptive Immune

A mathematical model of adaptive immune response after transplantation is formulated by five nonlinear ordinary differential equations. Theorems of existence, uniqueness and nonnegativity of solution are proven. Numerical simulations of immune response after transplantation without suppression of acquired cellular immunity and after suppression were performed.

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