System of Differential Equations

8 Key excerpts on "System of Differential Equations"

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  Stability Theory, Instability & Key Concepts of Physics and Mathematics

  • (Author)
  • 2014(Publication Date)
  • Academic Studio

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Types of Differential Equations 1. Differential equation Visualization of heat transfer in a pump casing, by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines. ________________________ WORLD TECHNOLOGIES ________________________ Differential equations arise in many areas of science and technology, specifically when-ever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly. An example of modelling a real world problem using differential equations is deter-mination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance may be modelled as proportional to the ball's velocity. This means the ball's acceleration, which is the derivative of its velocity, depends on the velocity.

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  Lectures, Problems And Solutions For Ordinary Differential Equations

  • Yuefan Deng(Author)
  • 2014(Publication Date)
  • World Scientific

    (Publisher)

  1 Chapter 1 First-Order Differential Equations 1.1 Definition of Differential Equations A differential equation (DE) is a mathematical equation that relates some functions of one or more variables with its derivatives. A DE is used to describe changing quantities and it plays a major role in quantitative studies in many disciplines such as all areas of engineering, physical sciences, life sciences, and economics. Examples Are they DEs or not?
    +  +  = 0 No Chapter 1 First-Order Differential Equations 2
    +  ′ +  = 0 Yes Here  ′ =  
    +  ′ +  ′ = 0 Yes Here  ′ =   and ′ =   ′′ =   Yes Here ′ =   To solve a DE is to express the solution of the unknown function (the dependent variable) in mathematical terms without the derivatives. Example
    +  = 0  ′ = − 
  is not a solution  = −  
   is a solution In general, there are two common ways in solving DEs, analytic and numerical. Most DEs, difficult to solve by analytical methods, must be “solved” by numerical methods although many DEs are too stiff to solve using numerical techniques. Solving DEs by numerical methods is a different subject requiring basic knowledge of computer programming and numerical analysis; this book focuses on introducing analytical methods for solving very small families of DEs that are truly solvable. Although the DEs are quite simple, the solution methods are not and the essential solution steps and terminologies involved are fully applicable to much more complicated DEs. Classification of DEs Classification of DEs is itself another subject in studying DEs. We will introduce classifications and terminologies for flowing the contents of the book but one may still need to lookup terms undefined here.

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  eBook - ePub

  Advanced Problem Solving with Maple

  A First Course

  • William P. Fox, William C. Bauldry(Authors)
  • 2019(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  3

  Introduction, Basic Concepts, and Techniques in Problem Solving with Systems of Ordinary Differential Equations

  Objectives

  (1)  Define and build a System of Differential Equations for a real problem. (2)  Find a solution. (3)  Graph a solution. (4)  Understand equilibrium values and stability. (5)  Solve both linear and nonlinear systems of differential equations. (6)  Solve systems of differential equations. (7)  Use numerical approaches to obtain good approximate solutions to differential equations and systems of differential equations.

  3.1  Systems of Differential Equations

  Interactive situations occur in the study of economics, ecology, electrical engineering, mechanical systems, control systems, systems engineering, and so forth. For example, the dynamics of population growth of various species is an important ecological application of applied mathematics.

  Your uncle recently retired and bought farm land in South Carolina. His desire is to have a fishing pond; his favorite fish to catch are rainbow trout and brown trout. He finds he has a fair size fresh water pond on his land, but it contains no fish. He takes a water sample to the local fish and game authority. They analyze his water and conclude that the pond can sustain a fish population. He visits the local fish hatchers where they provide him the growth rates of rainbow trout and brown trout in isolation, call these values r and b, respectively. The experts tell him that rainbow trout and brown trout have the same food sources and will compete for the oxygen in the water as well as for food for survival. Data from the American Fisheries Society estimates the interaction rates between rainbow trout and brown trout for survival; call these rates m and n respectively. We desire to build a mathematical model to help your uncle determine if the pond can sustain both species of fish. This leads to a competitive hunter

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  eBook - ePub

  Essential Math Skills for Engineers

  • Clayton R. Paul(Author)
  • 2011(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  4

  Solution of Linear, Constant-Coefficient, Ordinary Differential Equations

  Engineering systems have inputs that are usually functions of time, denoted as t. Static engineering systems are generally governed by algebraic equations. Dynamic engineering systems are generally governed by differential equations. These differential equations contain derivatives with respect to time t and are somewhat more difficult to solve than are algebraic equations. In this chapter we examine the most common type of differential equation that is frequently found in engineering systems: the linear, constant-coefficient, ordinary differential equation.

  4.1 HOW TO IDENTIFY LINEAR, CONSTANT-COEFFICIENT, ORDINARY DIFFERENTIAL EQUATIONS

  What do they look like? In these equations there is only one independent variable, which we denote as t (perhaps symbolizing time). The function to be solved for is denoted as x(t). Since the unknown is a function of t, any derivatives will be with respect to

  t: dx(t)/dt, d2 x(t)/dt2 ,

  and so on. These are said to be ordinary derivatives, as opposed to partial derivatives. A derivative

  dn x(t)/dtn

  is said to be of order n.

  The general form of a first-order ordinary differential equation is

  (4.1a)

  The general form of a second-order ordinary differential equation is

  (4.1b)

  The coefficients a and b are constants and are known (given). The right-hand-side function of t,f(t), is also known (given). The right-hand side,f(t), is called the forcing function since it represents the source that is driving the engineering system. The task will be to solve for a function x(t) which when substituted into the equation, satisfies it. It is always a good idea to make the leading coefficient unity I All of our results apply, of course, in identical fashion to differential equations where the independent and dependent variables might have different symbols and meanings, such as y(x).

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  Available until 9 Apr |Learn more

  A First Course in Differential Equations, Modeling, and Simulation

  • Carlos A. Smith, Scott W. Campbell(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Thus, the solution of the differential equation describes how the level in the tank changes as a function of time, including its final steady value . The differential equation provides more information than the algebraic equation. 5 Introduction 1.2 Differential Equations Up to this moment in your study of engineering and science you have become quite famil-iar with algebraic equations , for example, x 3 + 3 x 2 + 4 x = 3 t 2 + 4 t (1.15) From this equation we can solve for x at any t . We could repeat this for different values of t and make a graph of x versus t . Many natural and man-made phenomena and systems cannot be described by algebraic equations, or the description provided by algebraic equations is not complete (as in the 0 5 10 15 20 25 30 35 40 50 100 150 200 250 0 5 10 15 20 25 30 35 40 3 3.5 4 4.5 5 5.5 6 Time, min Time, min Flow 2 , kg/min Level h , m FIGURE 1.2 Response of level in tank. 6 A First Course in Differential Equations, Modeling, and Simulation example in Section 1.1). In these cases, differential equations (DEs) may provide the required description; an example of a differential equation is d x t dt d x t dt dx t dt x t F t 3 3 2 2 3 4 ( ) ( ) ( ) ( ) ( ) + + + = (1.16) Note that the variable x that is differentiated must be a function of t ; otherwise, the differ-ential would be zero. That is the reason for writing x ( t ). What often happens is that because obviously x is a function of t , we drop the t term and simply write d x dt d x dt dx dt x F t 3 3 2 2 3 4 + + + = ( ) (1.17) A differential equation is an equation containing one or more derivatives of an unknown function and perhaps the function itself . In Equation 1.16, x ( t ), or x in Equation 1.17, is the unknown function; in Equation 1.13, h is the unknown function.

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  eBook - ePub

  Mathematical Modeling

  Branching Beyond Calculus

  • Crista Arangala, Nicolas S. Luke, Karen A. Yokley(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  4

  Modeling with Differential Equations

  Goals and Expectations

  The following chapter is written toward students who have completed an introductory differential equations course. However, examples within the chapter have been written to include guidance for solving differential equations with computer software. Because of the potential use of technology to solve differential equations, the chapter could also be used by students who have not yet completed a differential equations course, but are somewhat familiar with computational software. Note that the Mathematica commands and MATLAB code presented are intended solely as examples, and not as instructional components for the respective software.

  Goals:

  • Section 4.1 (Introduction and Terminology): To introduce important terminology that is used when discussing differential equation models.
  • Section 4.2 (First Order Differential Equations): To construct and solve models using first order differential equations.
  • Section 4.3 (Systems of First Order Differential Equations): To construct and simulate models using a system of first order differential equations.
  • Section 4.4 (Second Order Differential Equations): To construct and solve models utilizing second order differential equations.

  4.1 Introduction and Terminology

  In calculus, we learned that the derivative

  dx

  dt

  is the ratio of the differential dx to the differential dt and that this derivative represents the change in the variable x with respect to the change in the variable t . For this derivative, the variable x is the dependent variable, and the variable t

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  Mathematical Modeling and Numerical Methods in Chemical Physics and Mechanics

  • Ali V. Aliev, Olga V. Mishchenkova, Alexey M. Lipanov(Authors)
  • 2016(Publication Date)
  • Apple Academic Press

    (Publisher)

  Dynamic Systems 287 5.4.1.6 Solution of Stiff Systems of Differential Equations The considered methods of solving ordinary differential equations and equation systems can be ineffective when solving stiff systems of equations. From those discussed in Section 5.1, the equation systems describing chemical kinetics can be referred to such systems (Eqs. (5.11)–(5.19)). The systems of ordinary differential equations, in which the processes with sig-nificantly different scales are available, belong to stiff ones. In this case, the integration step by time should be selected in such a way as to provide the calculation accuracy of all the components forming the problem solu-tion. In a number of cases, this requires the selection of an unacceptably small integration step ∆ t . In [193] there is the following definition of the stiffness of the system of ordinary differential equations: The system of n differential equations d dt A y y = with the constant matrix A with the dimensionality n n × is called stiff in the interval t t t ∈ ( , ) max 0 , if in this interval Re ( ) λ k t < 0 , k=1, 2, …, m, and correlation s t t t k k ( ) max R e ( ) min Re ( ) = λ λ is great. Here λ k t ( ) – own values of the matrix A, Re ( ) λ k t – their real part, s(t) – stiffness number. The more is the stiffness number, the more is the degree in which the properties of the system stiffness of ordinary differential equations are expressed. The simplest algorithms that allow solving the problems described by the stiff systems of differential equations are the computational algorithms of inexplicit type being the modification of Euler method y y f y ( ) ( ) ( , ( ) ) t t t t t t i i i i i i + -≈ + + Δ Δ 1 1 (5.70) The Eq. (5.70) differs from the Eq. (5.51) by the right side which should be calculated at the time moment t i + 1 , for which the function value is not found yet.

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  eBook - PDF

  A Course in Ordinary Differential Equations

  • Stephen A. Wirkus, Randall J. Swift(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 5 Fundamentals of Systems of Differential Equations Earlier we briefly considered systems of equations when converting an n th-order equation to a system of n first-order equations. But systems of differ-ential equations arise in their own right—whenever there is more than one dependent variable for an independent variable. For example, one might con-sider a system with two or more interacting species with the population sizes changing over time. The most common of these are known as Lotka-Volterra models and are discussed in detail in Section 6.5. One of these models is a predator–prey system in which the prey population growth depends upon the number of predators that kill the prey. Similarly, the rate at which the predator population grows depends on the size of their food supply, namely, the prey population. In general, these conditions produce nonlinear equa-tions that are very difficult to solve analytically. This is just one scenario we can consider. In this chapter, we discuss methods for solving these types of systems. The objects of study of the first few sections of this chapter are linear systems of equations, which are differential equations of the form dx 1 dt = a 11 ( t ) x 1 + a 12 ( t ) x 2 + · · · + a 1 n ( t ) x n + f 1 ( t ) dx 2 dt = a 21 ( t ) x 1 + a 22 ( t ) x 2 + · · · + a 2 n ( t ) x n + f 2 ( t ) . . . . . . . . . . . . . . . dx n dt = a n 1 ( t ) x 1 + a n 2 ( t ) x 2 + · · · + a nn ( t ) x n + f n ( t ) , (5.1) where the variables x i , f i and coefficients a ij are all functions of t . The sit-uation when an n th-order system is derived from an n th-order equation is simply a special case. In the event that the coefficients a ij are constant and the f i are zero, we refer to system (5.1) as an autonomous system of n first-order equations and there are special techniques that will apply.

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