Chapter 1 - Introduction

04/18/2023 NGUYỄN THỊ HOÀI 1

Chapter 1

Differential Equations -

Introductions

1.1. Some Basic Mathematical Models

1.2. Numerical Approximations

1.3. Classification of Differential

Equations


§1.1 Some basic mathematical models

Many of the principles, or laws of the natural world are

statements or relations involving rates at which things happen. In

mathematical terms, the relations are equations and the rate are

derivatives.

Equations containing derivatives are differential equations.

A differential equation that describes some physical process is

often called a mathematical model of the process.



Problem 1 (A Falling Object) Suppose that an object is falling in the

atmosphere near sea level. Formulate a differential equation that

describes the motion.

Solution. Let us use t to denote time, v to represent the velocity of

the falling object. The velocity changes with time, so v is a function of t,

here t is the independent variable and v is the dependent variable. Let

us measure time t in second (s) and velocity in meters/second(m/s). We

will assume that v is positive in the downward direction.



The physical law that describes the motion of everyday

objects is Newton’s second law

F = ma = m (dv/dt),

here m is the mass of the object measuring in kilograms (kg),

a is its acceleration which is measured in meters/second2

(m/s2) and F is the net force exerted on the object which is

measured in Newton (N).



Consider the two forces that act on the object as it falls. The

first one is gravity force, that equals to mg, where g is the

acceleration due to gravity and the second one is drag force that

has the value γv, where γ is a constant called the drag coefficient

(kg/s). Thus gravity acts in the downward direction and drag acts

in the upward direction, so the net force F exerted on the falling

object is

F = mg – γv m(dv/dt) = mg – γv .



The last equation is a mathematical model of an falling

object in the atmosphere near sea level. This model contains

the three constants m, g, and γ. The constants m and γ

depend on the particular object, while g is the same for all

objects. γv

mg

m



Problem 2 (Heating and Cooling) Consider a building as a partly

insulated box that is subject to external temperature fluctuations.

Construct a model that describes the temperature fluctuations inside

the building.

Solution Let u(t) and T(t) be the internal and external

temperatures, respectively, at time t. From the Newton’s law of

cooling, which states that the rate of change of u(t) is proportional to

the temperature difference u(t) – T(t), we obtain the differential

equation



du/dt = – k(u(t) – T(t)),

where k is a positive constant, the minus sign in the right-

hand side indicated that du/dt must be negative if u(t) > T(t).

Let us suppose that u and T are measured in degrees

Fahrenheit and that t is measured in days. Then k must have

the units of 1/day.



Constructing Mathematical Models

1. Identify the independent and dependent variables and

assign letters to represent them.

2. Choose the units of measurement for each variable.

3. Relate the basic principle with investigating problems.



4. Express the principle or law in step 3 in terms of the variable

chosen in step 1. It may require the introduction of physical

constants or parameters (such as drag coefficient in P. 1) and the

determination of appropriate values for them.

5. Make sure that each term in the equation has the same physical

units.

6. The result of step 4 is either a single differential equation or a

system of several differential equations.



Definition An equation of the form

F (x, y, y’, y’’…, y(n)) = 0,

(1)

where x as an independent variable; y = y(x) is an unknown

function and y’, y’’, …, y(n) are derivatives of this function, is

called a differential equation.

The order of the highest derivative of y in (1) is called the

order of differential equation.



Example yy’ = exyy2sinx - first order differential equation,

y’’ + 2(y’)3 = 1 - second order differential equation,

(1) - n-th order differential equation.



• The equation (1) is called a linear differential equation if F

is a linear function of the variables y, y’, y’’, …, y(n) . The general

linear differential equation of order n is

a0(x) y(n) + a1(x) y(n-1) +…+ an(x)y = g(x),

where a0(x), a1(x), …an(x), g(x) are given functions.

• A solution of differential equation (1) is a function φ(x) such

that φ’, φ’’…, φ(n) exist and satisfy

F(x, φ, φ’, φ’’…, φ(n)) = 0.



• The geometrical representation of a solution of (1) is called

the integral curve. Each integral curve is defined by the

formula y = φ(x) or Φ(x,y) = 0 or by parameterization

equation x = x(t), y = y(t).



Direction Fields

A slope field (or direction field) is a graphical

representation of the solutions of a first-order differential

equation. It is useful because it can be created without solving

the differential equation analytically. The representation may

be used to qualitatively visualize solutions, or to numerically

approximate them.



How to plot the direction fields

Consider the first order general differential equations of the form

The direction field of this equation is an infinite set of line

segments, passing through a point P(t,y) in ty – plane and whose

slope is the value of f at that point (thus each line segment in the

direction field is tangent to the graph of the solution passing

through that point).

),( ytfdt

dy



From the direction field for a differential equation, we can

Sketch of solutions. Since the arrows in the direction

fields are tangents to the solutions to the DEs we can use

these as guides to sketch the graphs of solutions to the

differential equation.

Long Term Behavior. Direction fields can be used to find

information about this long term behavior of the solution (how

the solutions behave as t increases).



Example 1. Sketch the direction field for the following

differential equation. Sketch the set of integral curves for this

differential equation.

xyy '


§1.2 Numerical Approximations

Many DEs do not have solutions that can be expressed in

term of simple functions. Just as there is no formula in terms

of a and b for the integral

there is generally no way to solve DEs exactly. For this reason,

we must rely on numerical methods that produce

approximations to the desired solutions.

b

a

t dte2



We will describe four families of methods for numerically

solving some DEs:

1) The Taylor series (TS) method,

2) Linear multistep methods (LMMs),

3) The Runge – Kutta (RK) methods and

4) Finite difference methods for boundary value

problems.



We aim to solve all initial value problems (IVPs) of the form

that posses a unique solution on some specified interval.

The first three methods can all be interpreted as generalizations

of the world’s simplest method: Euler’s method. In this chapter

we will start with Euler’s method.

)1(

.)(

),,(

00

yty

ytfdt

dy



Euler’s method

We use h to refer to a “small” positive number called the

“step size”: we will seek approximations to the solution of the

IVP (1) at particular time t1 = t0 + h, t2 = t0 + 2h, …, tn = t0 +

nh, … i.e., approximations to the sequence of numbers ф(t1),

ф(t2), …, ф(tn), …, rather than an approximation to the

solution y = ф(t) of the IVP (1).



Step 1. Start with the point P0(t0, y0), the equation of the

tangent line to the graph of the solution y = ф(t) of the IVP (1)

at the initial point (t0, y0) is

the approximation value of ф(t1) = ф(t0 + h) is

))(,( 0000 ttytfyy

hytfyttytfyy ),())(,( 000010001



Step 2. Continue with the point P1(t1, y1), the equation of

the tangent line to the graph of the solution of the DE in (1)

that passes through (t1, y1) is

the approximation value of ф(t2) = ф(t0 + 2h) is

))(,( 1111 ttytfyy

hytfyttytfyy ),())(,( 111121112



Continue this process we can obtain the approximation value

of the solution ф(t) of the IVP (1) at the moment t = tn+1

according to the formula

hytfyttytfyyt nnnnnnnnnn ),())(,()( 111

))(,()(

))(,(

111 nnnnnnn

nnnn

ttytfyyt

ttytfyy



Example 1. Use a step size h = 0.3 to develop an

approximate solution to the IVP

over the interval 0 ≤ t ≤ 0.9.

Solution The exact solution of this IVP problem is

1)0(

0),()21()('

y

ttytty

2

2

1

4

1exp)( tty



At t = 0, the equation of the tangent line to the solution curve

passing through (0, 1) is y = 1 + t, hence ф(0.3) ≈ 1.3

At t = 0.3, the equation of the tangent line to the solution

curve passing through (0.3, 1.3) is y = 0.52t + 1.144, hence

ф(0.6) ≈ 1.456

At t = 0.6, the equation of the tangent line to the solution

curve passing through (0.6, 1.456) is y = –0.2912t + 1.63072,

hence ф(0.9) ≈ 1.36864



The results of computing the numerical solutions over the

interval [0,3] with h = 0.3, 0.15, and 0.075



h yn Global errors (GEs) GE/h

0.3 y3 = 1.3686 ф(0.9) – y3 = – 0.2745 – 0.91

0.15 y6 = 1.2267 ф(0.9) – y6 = – 0.1325 – 0.89

0.075 y12 = 1.1591 ф(0.9) – y12 = – 0.0649 – 0.86

Exact ф(0.9) = 1.0942

Numerical results at t = 0.9 with h = 0.3, 0.15 and 0.075

Here global error (GE) at t = tn is given by

nnn yte )(



Analysing the Numbers

1. Twice as many steps are needed to find the solution at t = 3.

2. The computed points lie closer to the exact solution curve. This

illustrates the notion that the numerical solution converges to the

exact solution as h → 0.

3. It is seen from the table that halving h results in the error

being approximately halved. This suggests that the GE is

proportional to h.



The final column in the table suggests that the constant of

proportionality in this case is about −0.9 : en ≈ −0.9h, when

nh = 0.9.

If we require that |en| < 0.0005 then h < 0.0005/0.9 ≈

0.00055. Consequently, the integration to t = 0.9 requires

about n = 0.9/h ≈ 1620 steps.


§1.3 Classification of DEs

Classification of Differential Equations

1. From the number of independent variables in unknown

function, we have an ordinary differential equation

(ODE) and a partial differential equation (PDE).

2. From the number of unknown functions, we need a single

equation or a system of equations.


§ 1.3 Classification of DEs

3. The order of a differential equation is the order of the

highest derivative that appears in the equation. We have the

first order differential equation, the second order

differential equation, etc.

4. If F is a linear function of y, y’, y’’…, y(n) , then the ordinary

differential equation (1) is said to be linear, otherwise (1) is a

nonlinear equation.



Some Important Questions

1. The existence of a solution?

2. The uniqueness of solution?

3. How can we determine a solution?



Consider the first order differential equation of the form

there are some important subclasses of this equation that can

be solve analytically

),,( yxfdx

dy

1. Linear equation;

2. Separable equation;

3. Homogeneous

equation;

4. Autonomous equation;

5. Exact equation;


Summary

S1. Differential Equations, Direction Fields, Some

Mathematical Models.

S2. Euler’s Method for Approximating the Values of

Exact Solutions at Some Moment.

S3. Classification of DEs: from the number of

independent variables, from the number of dependent

variables, from the order of highest derivative, from

the linearity of given equations .

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Chapter 1 - Introduction