Mathematics
Modelling with First-order Differential Equations
Modelling with first-order differential equations involves using mathematical equations to represent and analyze real-world phenomena that change continuously over time. These equations capture the rate of change of a quantity with respect to another, allowing for predictions and understanding of various natural and artificial processes. The solutions to these equations provide valuable insights into the behavior of dynamic systems.
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10 Key excerpts on "Modelling with First-order Differential Equations"
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Each of these statements involves a rate of 40 CHAPTER 2 First-Order Differential Equations change (derivative) and consequently, when expressed mathematically, leads to a differential equation. The differential equation is a mathematical model of the process. It is important to realize that the mathematical equations are almost always only an approximate description of the actual process. For example, bodies moving at speeds comparable to the speed of light are not governed by Newton’s laws, insect populations do not grow indefinitely as stated because of eventual lack of food or space, and heat transfer is affected by factors other than the temperature difference. Thus you should always be aware of the limitations of the model so that you will use it only when it is reasonable to believe that it is accurate. Alternatively, you can adopt the point of view that the mathematical equations exactly describe the operation of a simplified physical model, which has been constructed (or conceived of) so as to embody the most important features of the actual process. Sometimes, the process of mathematical modeling involves the conceptual replacement of a discrete process by a continuous one. For instance, the number of members in an insect population changes by discrete amounts; however, if the population is large, it seems reasonable to consider it as a continuous variable and even to speak of its derivative. Step 2: Analysis of the Model. Once the problem has been formulated mathematically, you are often faced with the problem of solving one or more differential equations or, failing that, of finding out as much as possible about the properties of the solution. It may happen that this mathematical problem is quite difficult, and if so, further approximations may be indicated at this stage to make the problem mathematically tractable. For example, a nonlinear equation may be approximated by a linear one, or a slowly varying coefficient may be replaced by a constant.
eBook - PDF- Dennis Zill(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
84 3 Modeling with First-Order Differential Equations 3.1 Linear Models 3.2 Nonlinear Models 3.3 Modeling with Systems of First-Order DEs C H A P T E R 3 I N R E V I E W I n Section 1.3 we saw how a first-order differential equation could be used as a mathematical model in the study of population growth, radioactive decay, cooling of bodies, mixtures, chemical reactions, fluid draining from a tank, velocity of a falling body, and current in a series circuit. Using the methods discussed in Chapter 2, we are now able to solve some of the linear DEs in Section 3.1 and nonlinear DEs in Section 3.2 that frequently appear in applications. Fotos593/Shutterstock.com Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 3.1 LINEAR MODELS 85 INTRODUCTION In this section we solve some of the linear first-order models that were introduced in Section 1.3. GROWTH AND DECAY The initial-value problem dx dt 5 kx , x ( t 0 ) 5 x 0 , (1) where k is a constant of proportionality, serves as a model for diverse phenomena involving either growth or decay. We saw in Section 1.3 that in biological applica-tions the rate of growth of certain populations (bacteria, small animals) over short periods of time is proportional to the population present at time t. Knowing the popu-lation at some arbitrary time t 0 , we can then use the solution of (1) to predict the population in the future—that is, at times t . t 0 . The constant of proportionality k in (1) can be determined from the solution of the initial-value problem, using a subse-quent measurement of x at a time t 1 . t 0 . In physics and chemistry (1) is seen in the form of a first-order reaction —that is, a reaction whose rate, or velocity, dx y dt is directly proportional to the amount x of a substance that is unconverted or remain-ing at time t. The decomposition, or decay, of U-238 (uranium) by radioactivity into Th-234 (thorium) is a first-order reaction.
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Each of these statements involves a rate of change (derivative) and consequently, when expressed mathematically, leads to a differential equation. The differential equation is a mathematical model of the process. 42 CHAPTER 2 First-Order Differential Equations It is important to realize that the mathematical equations are almost always only an approximate description of the actual process. For example, bodies moving at speeds compa- rable to the speed of light are not governed by Newton’s laws, insect populations do not grow indefinitely as stated because of eventual lack of food or space, and heat transfer is affected by factors other than the temperature difference. Thus you should always be aware of the limita- tions of the model so that you will use it only when it is reasonable to believe that it is accurate. Alternatively, you can adopt the point of view that the mathematical equations exactly describe the operation of a simplified physical model, which has been constructed (or conceived of) so as to embody the most important features of the actual process. Sometimes, the process of mathematical modeling involves the conceptual replacement of a discrete process by a contin- uous one. For instance, the number of members in an insect population changes by discrete amounts; however, if the population is large, it seems reasonable to consider it as a continuous variable and even to speak of its derivative. Step 2: Analysis of the Model. Once the problem has been formulated mathematically, you are often faced with the problem of solving one or more differential equations or, failing that, of finding out as much as possible about the properties of the solution. It may happen that this mathematical problem is quite difficult, and if so, further approximations may be indi- cated at this stage to make the problem mathematically tractable.
eBook - PDFDifferential Equations
An Introduction to Modern Methods and Applications
- James R. Brannan, William E. Boyce(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
These features can be interpreted in terms of the physical behavior of the systems that the differential equations model. Furthermore, it is often easy to vary parameters in the mathe- matical model over wide ranges, whereas this may be very time-consuming or expensive, if 56 Chapter 2 First Order Differential Equations not impossible, in an experimental setting. Nevertheless mathematical modeling and exper- iment or observation are both critically important and have somewhat complementary roles in scientific investigations. Mathematical models are validated by comparison of their pre- dictions with experimental results. On the other hand, mathematical analyses may suggest the most promising directions for experimental exploration, and they may indicate fairly precisely what experimental data will be most helpful. In Section 1.1 we formulated and investigated a few simple mathematical models. We begin by recapitulating and expanding on some of the conclusions reached in that section. Regardless of the specific field of application, there are three identifiable stages that are always present in the process of mathematical modeling. ▶ Construction of the Model. In this stage, you translate the physical situation into mathematical terms, often using the steps listed at the end of Section 1.1. Perhaps most critical at this stage is to state clearly the physical principle(s) that are believed to govern the process. For ex- ample, it has been observed that in some circumstances heat passes from a warmer to a cooler body at a rate proportional to the temperature difference, that objects move about in accordance with Newton’s laws of motion, and that isolated insect populations grow at a rate proportional to the current population. Each of these statements involves a rate of change (derivative) and consequently, when expressed mathematically, leads to a differen- tial equation. The differential equation is a mathematical model of the process.
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Each of these statements involves a rate of 40 CHAPTER 2 First-Order Differential Equations change (derivative) and consequently, when expressed mathematically, leads to a differential equation. The differential equation is a mathematical model of the process. It is important to realize that the mathematical equations are almost always only an approximate description of the actual process. For example, bodies moving at speeds comparable to the speed of light are not governed by Newton’s laws, insect populations do not grow indefinitely as stated because of eventual lack of food or space, and heat transfer is affected by factors other than the temperature difference. Thus you should always be aware of the limitations of the model so that you will use it only when it is reasonable to believe that it is accurate. Alternatively, you can adopt the point of view that the mathematical equations exactly describe the operation of a simplified physical model, which has been constructed (or conceived of) so as to embody the most important features of the actual process. Sometimes, the process of mathematical modeling involves the conceptual replacement of a discrete process by a continuous one. For instance, the number of members in an insect population changes by discrete amounts; however, if the population is large, it seems reasonable to consider it as a continuous variable and even to speak of its derivative. Step 2: Analysis of the Model. Once the problem has been formulated mathematically, you are often faced with the problem of solving one or more differential equations or, failing that, of finding out as much as possible about the properties of the solution. It may happen that this mathematical problem is quite difficult, and if so, further approximations may be indicated at this stage to make the problem mathematically tractable. For example, a nonlinear equation may be approximated by a linear one, or a slowly varying coefficient may be replaced by a constant.
eBook - ePubAdvanced Problem Solving with Maple
A First Course
- William P. Fox, William C. Bauldry(Authors)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
kv. We then would have the modelF =F p+F d+F bF =m a = my ″=− m g + k vwith the same initial conditions y(0) = h, y′(0) = 0.In this chapter, we will consider a variety of models, concepts, and techniques that give the reader some of the basic tools needed in solving and analyzing first-order differential equations. Since differential equations are used often in solving problems related to continuous change, we devote a more thorough approach in this chapter.Section 2.2 provides some models that we will solve and analyze. These models come from a variety of disciplines in science and engineering, including chemistry, physics, biology, fluids mechanics, Newtonian Mechanics, environmental engineering, and financial mathematics. Modeling techniques are discussed that help determine the necessary coefficients in the models.Section 2.3 introduces slope fields. Slope fields provided a qualitative view of solutions of first-order differential equations. Recall that we said this book discusses the modeling of change. Slope fields provide information about rates of change since they are derived from information given by the derivative.Section 2.4 covers the analytical method of solution with Maple. Existence and uniqueness are addressed from conceptual and theoretical bases. The focus is on the initial-value problem and the question as to whether or not the initial value problem has a unique solution, and whether new functions can be defined by the initial-value problem.Section 2.4 investigates the special class of ODEs called linear differential equations.Section 2.5
eBook - ePub- Wim van Drongelen(Author)
- 2018(Publication Date)
- Academic Press(Publisher)
Chapter 9 Differential Equations Introduction Abstract In this chapter we review ordinary differential equations (ODEs) as a tool to model dynamics. We present examples of how to formulate them based on the dynamical system that needs to be modeled, and demonstrate the mathematical techniques one can employ to solve the equation analytically. We show how to solve linear differential equations with and without a forcing term, the so-called inhomogeneous and homogeneous ODEs, respectively. To illustrate the analysis of these equations, ODEs with first-order derivatives (e.g., d c / d t) and second-order derivatives (e.g., d 2 c / d t 2) are used in the examples. Next, we show how higher-order ODEs can be represented as a set of first-order ones, and how this leads to a formalism in matrix/vector notation that can be efficiently analyzed using techniques from linear algebra. To complete the overview of the available tools for solving ODEs, the final part of this chapter briefly refers to application of Laplace and Fourier transforms (see also Chapter 12) to solve them. Keywords Characteristic equation; Dynamics; Eigenvalue; Forcing term; Linear homogeneous equation; Linear inhomogeneous equation; Ordinary differential equation (ODE) 9.1. Modeling Dynamics When modeling some aspect of a neural system, we can represent static variables with algebraic expressions, e.g., the concentration c of some chemical in the brain is 10 units, c = 10. Of course, we can make these expressions a bit more complicated; for instance, the concentration could also depend on some combination of other substances a 1 and a 2 : e.g., c = 5 a 1 + a 2 + 10. It is important to realize that the variables do not change with time or space. For that reason the value of this type of equation is limited to situations where we study properties that remain constant over the range and duration of our interest
eBook - PDFModeling and Data Analysis
An Introduction with Environmental Applications
- John B. Little(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
As we have seen, there are many interesting 8 Technical Note: The guiding principle here is that we take A · n k · c n , where k is the smallest non-negative power such that this trial solution is not a solution of the associated homogeneous equation. 150 7. Discrete Time Dynamic Modeling and Difference Equations situations that can be modeled this way. But if you pursue the study of applied mathematics, you should know that there is a somewhat parallel theory of differ-ential equations that uses calculus to start from information about rates of change of functions of a continuous time variable to derive information about the functions themselves. This is also an extremely important modeling tool, and a core area of more advanced applied mathematics. An introduction to differential equations is included in most Calculus 2 courses and almost all college and university math-ematics major programs offer more advanced courses in differential equations as well. In the following, we will assume that you are familiar with some of the basic ideas about differential calculus (in particular, the interpretation of the derivative as an instantaneous rate of change), and the notation Q ( t ) = dQ dt for derivatives. In case you are wondering why the equations we have studied are called dif-ference equations , there is an explanation. Historically, difference equations were first studied as approximations to differential equations, in which derivatives are replaced by “finite differences” such as Q ( x ) . = Q ( x + h ) − Q ( x ) h or Q ( x ) . = Q ( x + h ) − 2 Q ( x ) + Q ( x − h ) h 2 (no limits as h → 0). Once this is done, algebraic equations are solved to derive approximate values of Q at a discrete set of input x -values. This circumstance accounts for the way the perhaps perplexing name, “difference equation,” arose historically. Example 7.5 discusses another way to understand this, at least for first order difference equations.
eBook - ePub- William P. Fox, Robert E. Burks(Authors)
- 2021(Publication Date)
- Chapman and Hall/CRC(Publisher)
A fraction k would now become infected. There are many refinements to the above model. For example, we can consider that a segment of the populations is not susceptible to the disease, that the infection period is limited, that infected students are removed from the dorm to prevent interaction with uninfected students, and so forth. d i d t = k i (400 − i), k > 0 3.1.2 Some Classical First-Order Models in Differential Equations Decay: d Q d t = − k Q, k > 0 Growth: d Q d t = k Q, k > 0 Newton’s Law of Cooling: d T d t = k (T − T 0), k > 0 Chemical Mixtures, Drug Dissemination,. Dialysis: d A d t = R i n − R o u t L-R Series Circuit: L d I d t + R i = E (t) Falling Bodies, Newton’s Law of Σ Forces = 0 m d v d t = F w e i g h t + F r e s i s t a n c e = m g − k v, k > 0 Chemical. Reactions: d X d t = k (α − X) (β − X)... Spread of a Disease or Rumor: d N d t = k N (L − N), k > 0, L > 0 Filling a Tank: Tank with cross-sectional area A, filled from the top at rate K, while draining from the bottom through an orifice of area α, with height of the tank h : d h d t = (k − α a 2 g h) A 3.2 Slope Fields and Qualitative Assessments of Autonomous First-Order ODE We have used the modeling process to create a number of models that are important to differential equations. However,. sometimes we can predict outcomes of models graphically without actually solving them. This is true when we are looking for behavior rather than specific values. For example, will a specific population survive is a question that can be answered graphically. In calculus, we used derivative information to analyze a function or just to gain more information. Knowing where a function is increasing or decreasing, locations of extrema, concavity, and so forth can be important information. Interpreting the derivative as the slope of the line tangent to the graph of the function is useful in gaining information about the solution to the differential equation
- Norman W. Loney(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
11 2 First-Order Ordinary Differential Equations 2.1 LINEAR EQUATIONS The examples of linear first-order differential equations occur frequently in chemical engineering practice through unsteady-state mass balances or first-order chemical reaction problems� Here, we review a few methods for solving first-order ordinary differential equations� Following each method are examples demonstrating the application of that method� Also, the notion of translating prose into mathematical symbolism is introduced in Section 2�4� A brief recap of the definition of linear equations in the context of differential equations is presented here� Following the recap are the examples of unsteady mass balances, which lead to linear first-order problems� Also presented are examples involving chemical reactions that can be treated as linear first-order problems� In this chapter, attention is focused on differential equations of the form ′ ρ = ρ ( , ) f t (2�1) where f is a given function of t and ρ � By linear equations , we mean any equation that can be expressed in the polynomial form ρ + ρ + + ′ ρ + ρ + = --a t a t a t a t a t g t n n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( 1) 2 1 (0) 0 (2�2) where ρ (�) symbolizes the derivative of ρ with respect to t � Consequently, the equation ρ + ρ + = a t a t a g t ( ) ( ) ( ) 2 (1) 1 (0) 0 (2�3) is a linear first-order differential equation and is more familiar in the following form: ′ ρ + ρ + = a t a t a t g t ( ) ( ) ( ) ( ) 2 1 0 (2�4)
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