Initial Value Problem Differential Equations

12 Key excerpts on "Initial Value Problem Differential Equations"

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  Introduction to Computation and Modeling for Differential Equations

  • Lennart Edsberg(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  3 NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS As we have seen in Chapters 1 and 2, ordinary differential equation(ODE) prob- lems occur in numerous applications and therefore important for modeling processes whose evolution depends on one variable, usually time t or one spatial variable x. In this chapter, we treat the initial value problem (IVP). The general formulation of an IVP is du dt = f (t, u), u(t 0 ) = u 0 , t 0 < t ≤ t end (3.1) Often the solution depends on parameters, denoted by the vector p, that occur in the right hand side function: du dt = f (t, u, p), u(t 0 ) = u 0 , t 0 < t ≤ t end (3.2) The initial values u 0 can also be regarded as parameters in case the dependence of the initial values is studied: u = u(t, p, u 0 ) Example 3.1. (Newton’s law for a particle) In Exercise 2.1.4, the following IVP was introduced m  y = −mg − c  y|  y|, y(0) = y 0 ,  y(0) = v 0 (3.3) This ODE models a particle thrown vertically from the position y 0 and with initial velocity v 0 . The particle is influenced by gravity and an air resistance force being Introduction to Computation and Modeling for Differential Equations, Second Edition. Lennart Edsberg. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc. 38 NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS proportional to the square of the velocity. When the solution is studied as a function of, e.g., c, we make a parameter study of the problem. The solution of this problem depends on t and the parameters m, c, g, y 0 , and v 0 , i.e., y = y(t, m, c, g, y 0 , v 0 ).

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  Numerical Methods

  Fundamentals and Applications

  • Rajesh Kumar Gupta(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  14.1.7 Initial Value Problem (IVP) and Boundary Value Problem (BVP) To describe any physical phenomenon, we must define the initial or boundary conditions or both. For example, we take a very simple model of population growth. The rate of change of population is directly proportional to population itself. If X(t) is population of any town at any time t, then the rate of change of population dX dt       at any time t is given by the following first order ordinary differential equation dX dt kX = where constant k depends on nature of the population. First Order Ordinary Differential Equations: Initial Value Problems 581 Say, we want to compute the population of the town after five years i.e. X(5), then we must have the present population. i.e., X(0). Let X(0) = 10000. Now the model is complete for the computation of the population of the town at any time t. dX dt kX X = = ; ( ) 0 10000 The condition X ( ) 0 10000 = is an initial condition. The conditions are known as initial conditions if these are defined at a single point. If the conditions are at more than one point then these conditions are known as boundary conditions. Differential equation with initial conditions is known as initial value problem (IVP), and with boundary conditions is known as boundary value problem (BVP). For examples, the differential equations (i) – (iii) are IVPs, and equations (iv)–(v) are BVPs i) dy dx x y y + = = 2 3 1 2 ; ( ) ii) d y dx x dy dx y y y 2 2 3 0 1 0 2 + + = = ′ = ; ( ) , ( ) iii) d y dx x d y dx xy x y y y 3 3 2 2 0 1 0 2 0 2 + + = = ′ = ′′ = sin cos ; ( ) , ( ) , ( ) iv) x d y dx x dy dx y y y 2 2 2 1 3 0 1 1 3 + − + = = = ( ) ; ( ) , ( ) v) d y dx x d y dx xy x y y y 3 3 2 2 0 1 1 2 3 4 + + = = = = − sin cos ; ( ) , ( ) , ( ) 14.1.8 Existence and Uniqueness of Solutions It is not necessary that each differential equation has a solution, and also if a solution exists it may not be unique.

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  Numerical Solution Of Ordinary And Partial Differential Equations, The (3rd Edition)

  • Granville Sewell(Author)
  • 2014(Publication Date)
  • World Scientific

    (Publisher)

  1 Initial Value Ordinary Differential Equations 1.0 Introduction Differential equations are often divided into two classes, ordinary and par-tial, according to the number of independent variables, and studied sepa-rately. A more meaningful division, however, is between initial value prob-lems, which usually model time-dependent phenomena, and boundary value problems, which generally model steady-state systems. The differences be-tween initial value and boundary value problems, and the similarities within each of these classes, are even more striking when numerical methods for these problems are considered. The identification of initial value problems with time dependency and of boundary value problems with a steady-state condition is helpful in under-standing some of the differences in the properties of these two types of prob-lems. For example, the solution at any time of a time-dependent problem logically depends only on what has gone on before and not on future events. For a steady-state problem, on the other hand, the solution values at dif-ferent spatial points may be interdependent. Thus it is not surprising that a time-dependent (initial value) problem can be solved numerically by marching forward in time from the given initial values, while a system of simultaneous algebraic equations must generally be solved to find the solution to a steady-state (boundary value) problem. It is also clear why initial value problems almost always have unique solu-tions, while boundary value problems sometimes have many or no solutions. Consider, for example, the general second-order equation m u tt = f ( t, u , u t ), which may be thought of as modeling Newton’s second law, applied to an object whose coordinates are given by the vector u ( t ). With initial conditions u (0) = u 0 , u t (0) = u 1 , under very reasonable smoothness assumptions on the 27 28 1. INITIAL VALUE ORDINARY DIFFERENTIAL EQUATIONS force field f , this problem will always have a unique solution.

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  Ordinary Differential Equations

  Principles and Applications

  • A. K. Nandakumaran, P. S. Datti, Raju K. George(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 General Theory of Initial Value Problems 4.1 Introduction 4.1.1 Well-posed problems In this chapter, we address the problem of the existence and uniqueness of solutions of initial value problems (IVP). For this purpose, our first task is to ensure that the given differential equation has a solution. A mathematical model originating from a real life system may exhibit more than one solution starting from the same initial condition, though a unique solution is expected. This may be due to rough approximations and assumptions made while making a mathematical model of the physical system. On the other hand, a mathematical model may not have a solution at all. Similarly, it is also important to study the behaviour of the solution with respect to the initial data as the initial data is usually measured by using some devices and is bound to have some small errors. Continuous dependence of solutions on initial data guarantees that a small error in the initial data does not cause a drastic change in the solution of the system. According to the French mathematician Jacques Hadamard, if an initial value problem arising from a physical phenomenon qualifies the above mentioned tests, namely, a solution exists (existence problem), the solution is unique (uniqueness problem) and the solution depends continuously on the initial conditions (stability problem) in appropriate norms, then the IVP is said to be well-posed. Otherwise, the problem is ill-posed. In this chapter, we will address these issues and prove results which ensure the well-posedness of an IVP, under suitable assumptions. We consider the following IVP 100 Ordinary Differential Equations: Principles and Applications ˙ y = f (t , y), y(t 0 ) = y 0 , (4.1.1) where, f : D ⊂ R 2 → R is a function not necessarily linear and is assumed to be continuous in an open connected set D of R 2 containing the initial point (t 0 , y 0 ).

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  Classical and Modern Numerical Analysis

  Theory, Methods and Practice

  • Azmy S. Ackleh, Edward James Allen, R. Baker Kearfott, Padmanabhan Seshaiyer(Authors)
  • 2009(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 7 Initial Value Problems for Ordinary Differential Equations 7.1 Introduction In this chapter, we are interested in approximating the solution of the fol-lowing initial-value problem (IVP) for a first-order system of differential equa-tions. We seek to approximate y : [ a, b ] → R n that satisfies y ( t ) = f ( t, y ( t )) , a ≤ t ≤ b, y ( a ) = y 0 , (7.1) where f is a given R n -valued function of n + 1 real variables and y 0 is a given vector in R n . That is, we seek n functions y i ( t ), 1 ≤ i ≤ n , defined for a ≤ t ≤ b such that y i ( t ) = f i ( t, y 1 ( t ) , y 2 ( t ) , . . . , y n ( t )) , 1 ≤ i ≤ n, a ≤ t ≤ b, y i ( a ) = y 0 ,i . For example, for n = 2, ⎧ ⎨ ⎩ y 1 ( t ) = cos t + y 1 ( t ) y 2 ( t ) = f 1 ( t, y 1 , y 2 ) y 2 ( t ) = y 2 ( t ) − y 1 ( t ) = f 2 ( t, y 1 , y 2 ) y 1 (1) = 2 , y 2 (1) = 0 . However, for a general function f , problem (7.1) need not have a solution on the interval [ a, b ]. For example, in R 1 , consider y ( t ) = | y ( t ) | 3 / 2 , 0 ≤ t ≤ 3 , y (0) = 1 . The function y ( t ) = (1 − t/ 2) − 2 is a solution to this IVP on the interval [0 , b ], 0 < b < 2; this solution blows up as t → 2. In fact, a solution to this IVP does not exist over the entire interval [0 , 3]. Additionally, even if a solution to the IVP exists over all of [ a, b ], the solution may not be unique. As an example of this phenomenon, consider the IVP y ( t ) = | y ( t ) | 1 / 2 , 0 ≤ t ≤ 1 , y (0) = 0 . 381 382 Classical and Modern Numerical Analysis Clearly, y ( t ) = 0, 0 ≤ t ≤ 1 is a solution. Also, it is easily shown that y ( t ) = 0 , 0 ≤ t ≤ 1 / 2 , ( t − 1 / 2) 2 / 4 , 1 / 2 ≤ t ≤ 1 is also a solution. Hence, solutions to this IVP are not unique on [0 , 1]. Problem (7.1) has a unique solution in some interval 1 about t = a if f 1 , f 2 , · · · , f n are continuous and possess continuous first partial derivatives as stated in the following theorem. Details and a proof can be found in many books on differential equations, such as [94].

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  Numerical Methods with Chemical Engineering Applications

  • Kevin D. Dorfman, Prodromos Daoutidis(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  This is different from the term “initial guesses” that we used before. Initial conditions are not approximations; they reflect true values of y at a specific value of x . Often, you will see these conditions given at x = 0, which is a rather literal interpretation of an “initial” value. However, we can put the conditions at any single value x = x 0 . If you like to have the data at the origin, you can always make a change of variable ˜ x = x − x 0 such that the initial conditions are at the value ˜ x = 0. • Boundary value problems (BVP) : Here, the auxiliary conditions are given at differ-ent values of the independent variable. They are called boundary conditions. Similar to the initial value problems, where you probably associate the initial conditions with the specification of a physical system at the start of a process, you most likely think of boundary value problems as having conditions specified at the boundaries of some physical system, such as fixing the velocity to be zero on the walls of a tube. Although this will indeed be the case for the vast majority of problems that we will encounter here, a boundary value problem has no more need to specify conditions on a physical boundary than an initial value problem needs to specify conditions at x = 0. The concept of a boundary value problem is more general and can be thought of as “not an initial value problem.” We will use this classification system of IVPs and BVPs to guide our exploration of the numerical solution of ODEs. In the present chapter, we will focus on IVPs. We begin with single equations, which will allow us to introduce the concepts in a fairly straightforward way. This is not to say that single equations are unimportant, and we will demonstrate the utility of numerically solving a single equation in the context of a continuous stirred tank reactor. We will then move on to solving systems of first-order ODEs.

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  Numerical Analysis for Engineers and Scientists

  • G. Miller(Author)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  10 Ordinary differential equations Ordinary differential equations (ODEs) have the form dy dx = f (x , y ) (10.1) where y and f may be vectors or scalars. They differ from simple integrals by the dependence of f on y . Since f does depend on y , we cannot solve a problem like (10.1) without knowing something about y . This gives rise to two classes of problem. Initial value problems (IVPs) are ODEs where the value of y is given somewhere, e.g., dy dx = f (x , y ) y (x 0 ) = y 0 . Boundary value problems (BVPs) are ODEs where y is not completely specified any- where, but instead some constraint is provided, e.g., dy dx = f (x , y ) r ( y (a), y (b)) = 0. The plan of this chapter is to discuss first IVPs in various forms, and then to show how the solution to BVPs can be derived from IVP methods. 10.1 Initial value problems I: one-step methods To begin the discussion, here is an IVP algorithm similar to the left Riemann sum for integration. It is called the forward Euler method: y i +1 = y i + h f (x i , y i ) x i +1 = x i + h . (10.2) This has the form of a simple one-step method: y i +1 = y i + h (x i , y i ) x i +1 = x i + h . 276 Ordinary differential equations To analyze these methods we need some new concepts. The first is the local dis- cretization error, τ , defined as τ(x , y ; h ) = (x , y ; h ) − (x , y ; h ), (10.3) where (x , y ; h ) is the one-step method, and where (x , y ; h ) =  y(x +h)−y(x ) h if h  = 0 f (x , y ) otherwise. (10.4) By its construction, an algorithm like y i +1 = y i + h (x i , y ; h ) would give the exact solution, ignoring numerical errors. τ is measure of error such that h τ is the error of a single integration step assuming the initial value is exact. The way one applies (10.3) is to use Taylor series to express all y relative to some reference point.

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  An Introduction to Differential Equations

  Deterministic Modeling, Methods and Analysis(Volume 1)

  • Anil G Ladde, G S Ladde;;;(Authors)
  • 2012(Publication Date)
  • WSPC

    (Publisher)

  In short, x 0 takes values in R , and it is the value of the solution function at t = t 0 (the initial/given time). Moreover, x 0 is an initial (given) state of a dynamic process. The problem of finding a solution of (2.18) is referred to as the initial value problem (IVP). Its solution is represented by x ( t ) = x ( t, t 0 , x 0 ) for t ≥ t 0 and t, t 0 ∈ J . Definition 2.3.2. A solution to the IVP (2.18) is a real-valued deterministic process or function x defined on J into R such that x ( t ) and its differential dx ( t ) satisfy the scalar differential equation in (2.18), and x ( t ) satisfies the given initial condition ( t 0 , x 0 ). In short: (i) x ( t ) is a solution to (2.15) (Definition 2.3.1) and (ii) x ( t 0 ) = x 0 . Example 2.3.4. Verify that x ( t ) = t + c is a solution to the following differential equation: dx = dt . Then determine a value of the constant c so that x ( t ) is the solution to the IVP: dx = dt , x (0) = 5. Solution procedure . By imitating the procedure described in Example 2.3.1, we conclude that x ( t ) = t + c is the general solution to the given differential equation. We need to find a constant c so that x ( t ) = t + c is the solution to the given IVP: dx = dt , x (0) = 5. For this purpose, we substitute t = 0 in x ( t ) = t + c , and obtain x (0) = 0 + c . We know by Definition 2.3.2 that x (0) = 5. Hence, 5 = 0 + c . Now, we solve for c , and obtain c = 5. Finally, we substitute for c = 5 into the expression x ( t ) = t + c , and we have x ( t ) = t + 5. This is the desired solution to the given IVP. Example 2.3.5. Determine the value of a constant c so that x ( t ) = sin t + c is the solution to the IVP: dx = cos t dt , x ( π ) = 0. Solution procedure . By imitating the procedure described in Example 2.3.2, we conclude that x ( t ) = sin t + c is a solution to dx = cos t dt. We follow the procedure outlined in Example 2.3.4 to determine the value of the constant c .

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  Elements of Matrix Modeling and Computing with MATLAB

  • Robert E. White(Author)
  • 2006(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 6 Linear Initial Value Problems First and second order linear initial value problems are studied. Initial value problems require finding a function of time that satisfies an equation with deriv-atives and an initial condition. Examples include the motion of projectiles in Subsection 2.3.3, mixing tanks in Subsection 3.7.4 and population models in Subsection 4.4.3. Additional applications include time dependent models of tuned circuits and resonant mass-spring systems. Particular solutions are found by both undetermined coe cients and variation of parameters. The system for-mulations are developed for the second order problems, and numerical solutions are given by the M ATLAB di erential equation solvers. Chapters six and seven are an introduction to di erential equations, and this topic is covered in more detail in [1] and [7] 6.1 First Order Linear A first order linear initial value problem for an unknown function = ( ) with given ( ) is = 0 = + ( ) and (0) = 0 This can be written in di erential operator form where ( ) 0 and ( ) = ( ) The operator is called linear because ( + ) = ( + ) 0 ( + ) = 0 + 0 = ( ) + ( ) 243 244 CHAPTER 6. LINEAR INITIAL VALUE PROBLEMS and for all real numbers ( ) = ( ) 0 ( ) = ( 0 ) = ( ) The objective is to find ( ) given , an initial condition (0) and the function ( ) . When ( ) is a constant, the solution is easy to find by defining + and noting 0 = 0 + 0 and so that 0 = + becomes ( ) 0 = In this case ( ) = (0) and, hence, ( ) + = ( (0) + ) ( ) = ( (0) + ) If ( ) is not a constant, then the solution process is more complicated. 6.1.1 Motivating Applications A simple model for the vertical velocity of a projectile is derived from Newton’s law of motion 0 = where is the mass, is the gravitational constant and is the air resistance. In this case = , = , ( ) = and (0) = (0) is the initial velocity. The velocity as a function of these parameters is ( ) = ( (0) + ) ( ) Another application is to circuits .

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  Numerical Methods for Chemical Engineering

  Applications in MATLAB

  • Kenneth J. Beers(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  Finally, we present a robust method, based upon IVP solvers, to study how the solution to a set of nonlinear algebraic equations depends upon its parameters, parametric continuation. Initial value problems of ordinary differential equations (ODE-IVPs) IVPs arise when we study the dynamics of a system governed by a set of first-order ODEs, such as the batch reactor kinetics for the network of two elementary reactions A + B → C r R1 = k 1 c A c B C + B → D r R2 = k 2 c C c B (4.6) At t 0 = 0, we start with the initial concentrations c A (t 0 ) = c A0 c B (t 0 ) = c B0 c C (t 0 ) = c D (t 0 ) = 0 (4.7) The time evolution of the system follows the set of first-order ODEs dc A dt = −r R1 dc B dt = −r R1 − r R2 (4.8) dc C dt = r R1 − r R2 dc D dt = r R2 We wish to use a general notation system for IVPs, and so define a state vector, x, that completely describes the state of the system at any time sufficiently well to predict its future behavior; here, x =  c A c B c C c D  T (4.9) We then write the ODE system, substituting for the reaction rates, as ˙ x 1 = −k 1 x 1 x 2 = f 1 (x; k 1 , k 2 ) ˙ x 2 = −k 1 x 1 x 2 − k 2 x 3 x 2 = f 2 (x; k 1 , k 2 ) (4.10) ˙ x 3 = k 1 x 1 x 2 − k 2 x 3 x 2 = f 3 (x; k 1 , k 2 ) ˙ x 4 = k 2 x 3 x 2 = f 4 (x; k 1 , k 2 ) We collect the parameters of the system into a parameter vector Θ = [k 1 k 2 ] T (4.11) and write (4.10) in the standard ODE-IVP form ˙ x = f (x; Θ) x(t 0 ) = x [0] (4.12) We next show that this problem formulation is quite general by considering the following: How do we express the system in the form of (4.12) if the function vector is itself time- dependent? What if we have ODEs of higher order than one? 156 4 Initial value problems To resolve the first question, let us say that we have a problem with time-dependent function values ˙ x = f (t , x; Θ) x(t 0 ) = x [0] (4.13) Expanding the state vector to include time, y =      x 1 .

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  Numerical Methods for Engineers and Scientists

  • Joe D. Hoffman, Joe D. Hoffman, Steven Frankel(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  If the auxiliary conditions are specified at two different values of the independent variable, the end points or boundaries of the domain of interest, the differential equation is a boundary-value ODE. Figure II.I il1ustrates the solution of an initial-value ODE. The initial value of the dependent variable is specified at one value of the independent variable, and the solution domain D(t) is open. Initial-value ODEs are solved by marching numerical methods. 326 Part II y(t) Yo to Figure 11.1 Initial-value ODE. Figure 11.2 Boundary-value ODE. x Figure 11.2 illustrates the solution of a boundary-value ODE. The boundary values of the dependent variable are specified at two values of the independent variable, and the solution domain D(x) is closed. Boundary-value ODEs can be solved by both marching numerical methods and equilibrium numerical methods. 11.4 CLASSIFICATION OF PHYSICAL PROBLEMS Physical problems fall into one of the following three general classifications: 1. Propagation problems 2. Equilibrium problems 3. Eigenproblems Each of these three types of physical problems has its own special features, its own particular type of ordinary differential equation, its own type of auxiliary conditions, and its own numerical solution methods. A clear understanding of these concepts is essential if meaningful numerical solutions are to be obtained. Propagation problems are initial-value problems in open domains in which the known information (initial values) are marched forward in time or space from the initial state. The known information, that is, the initial values, are specified at one value of the independent variable. Propagation problems are governed by initial-value ordinary differential equations. The order of the governing ordinary differential equation may be one or greater. The number of initial values must be equal to the order of the differential equation.

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  Ordinary Differential Equations and Integral Equations

  • C.T.H. Baker, G. Monegato, G. vanden Berghe, J.D. Pryce(Authors)
  • 2001(Publication Date)
  • North Holland

    (Publisher)

  JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 125 (2000) 3 1 ^ 0 = = = = = www.elsevier.nl/locate/cam Initial value problems for ODEs in problem solving environments L.F. Shampine*'*, Robert M. Corless b a Mathematics Department, Southern Methodist University, Dallas, TX 75275, USA b Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada Received 21 June 1999; received in revised form 28 February 2000 Abstract A program is presented for solving initial value problems for ODEs numerically in Maple. We draw upon our experience with a number of closely related solvers to illustrate the differences between solving such problems in general scientific computation and in the problem solving environments Maple and MATLAB. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Initial value problems; ODEs; Problem solving environment; PSE 1. Introduction The problem solving environments (PSEs) Maple [8] and MATLAB [7] are in very wide use. Although they have much in common, they are clearly distinguished by the emphasis in Maple on algebraic computation and in MATLAB on numerical computation. We discuss here a program, IVPsolve, for solving numerically initial value problems (IVPs) for systems of first-order ordinary differential equations (ODEs), y' = f(x,y), in Maple. We draw upon our experience with a number of closely related solvers to illustrate the differences between solving IVPs in general scientific computation (GSC) and in these PSEs. The RKF45 code of Shampine and Watts [10,11] is based on the explicit Runge-Kutta formulas F(4,5) of Fehlberg. It has been widely used in GSC. Translations of this code have been the default solvers in both Maple and MATLAB. Neither takes much advantage of the PSE. In developing the MATLAB ODE Suite of solvers for IVPs, Shampine and Reichelt [9] * Corresponding author. E-mail address: [email protected] (L.F. Shampine).

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