Boundary problems of ordinary differential equations | 4

ABSTRACT

Boundary problems of ordinary dierential

equations

Linear dierential equations

Let us rst consider equations of second order According to chapter equa

tions of motion and other models combine the acceleration xt of a phe

nomenon with some external inuence like forces The most general linear

dierential equation of second order apparently has the form

xy fx

Here p

and f are in most cases given notvanishing continuous func

tions If fx is zero the equation is called homogeneous We now rst

solve the homogeneous equation

Let

yx C

x C

be the general solution where y

and y

are fundamental solutions and C

and C

are constants of integration In the next section we will discuss meth

ods how one may nd y

and y

Let y

x be a particular solution of the

inhomogeneous equation then its general solution has the form

yx C

x C

x y

The next step is to nd a particular solution y

x of We do this by

replacing the constants C

and C

by nonconstant functions C

x and C

This method is called the method of variation of parameters of constants

because the constants in are now allowed to vary Instead of the

unknown function yx we now have two new functions additionally The

setup

x C

delivers y

x and y

x Since the new functions C

x and C

x are quite

arbitrary we may require two new conditions

From and we then obtain

and from and one gets

fxp

Since is assumed to be a solution of insertion of

and into demonstrates that is actually a solution of

Now we determine the two stillunknown functions C

x and C

from and We obtain

xfx

x y

xfx

x y

Integrations yield

d C

The denominator appearing in is called the Wronskian determinant

W x y

x y

If this determinant vanishes then the two solutions y

x and y

x are lin

early dependent

for C

If one knows two independent solutions y

x and y

then also the particular solution y

is known

x y

If one solution y

x of is known then the second solution of

may be found

x y

exp

xdx

yx zx exp

may transform away the y

x term in Inserting into

one obtains

z fx exp

and neither y

nor z

appears The homogeneous equation now takes

the form

x Ixzx

The invariant

is a means to classify dierential equations of second order Thus the general

solutions of two dierential equations having the same invariant dier only by

a factor

The method just described can be applied on two examples We consider

the inhomogeneous equation

y expx

Its solution is given by

y C

expx C

expx expx

where C

and C

are constants The same result can be obtained by the

Mathematica command

DSolve[y [x]-3*y[x]+2*y[x]==Exp[5*x],y[x],x] We now consider the boundary problem twopoint problem

y y

of the equation

y expx

The general solution of is given by

y C

expx C

expx expx

Now the integration constants C

and C

can be determined by inserting

into Again the solution can be obtained by the

Mathematica command

Problems

For constant C

insert into and obtain the resulting

dierential equations for y

Does the solution of

satisfy these equations derived by you Answer should be yes Try to

use Mathematica for this calculation

Calculate the invariant I for the two equations and

Answer and Try to reproduce the solutions

and by using and

In Mathematica the Wronskian can be dened by a determi

nant Since a determinant is an operation on a matrix we rst have

to dene a matrix We use the solutions y

x and y

x contained in

M={{Exp[x],Exp[2*x]},{Exp[x],2*Exp[2*x]}} then Det[M] results in ex Calculate the Wronskian for the solution The answer should read e

Calculate the Wronskian for

a sinx cosx Answer cos

x sin

b sinx sinx Answer Why

c x

Answer x

Solve p

x p

y for constant p

In order to

delete previous denitions for y we use

Clear[y];DSolve[p0*y[x]+p1*y[x]+p2*y[x]==0,y[x],x] The result looks complicated Simplify[%] does not help very much But is the result correct Can we verify the output of the calculation by

inserting it into the dierential equation To do so we bring the result

into the input form by using a new function ux The new function

must have x as an independent variable guaranteeing that ux is a

global function giving values for any x

The following example will clear the situation

u[x]=4*xˆ2 gives u[2]=u[2] but v[x_]:=4*xˆ2 gives v[2]=16 To verify the solution of the dierential equation we use again

solution

y xexp

p p x

exp

p p x

Solving linear dierential equations

As a rst example of the solution of a boundary value problem we consider

the linear dierential equation

which has the general solution

yx A sinxB cosx

A sinx or cosx would be particular solutions The general solution admits

both initial or boundary value problems If we choose the initial conditions

onepoint conditions

then the integration constants A and B can be obtained from

A sin

B cos

A cos

B sin

On the other hand if we choose the boundary conditions two point conditions

we obtain from

cosx

sinx

cosx

sinx

cosx

Thus the general solution of the boundary value problem and

is given by the sum of two particular solutions

cosx

sinx

cosx

sinx

cosx

sinx

cosx

sinx

cosx

A warning is now necessary not all arbitrary boundary problems can be

solved If we assume that

then the general solution is not able to satisfy these equations From

one obtains the contradiction

A and A

If we replace by

y y

we get an innity of solutions since B but A remains undetermined

The nonvanishing boundary conditions and are

called inhomogeneous Adversely the vanishing conditions

are called homogeneous

We now have the same situation that we discussed in section for partial

dierential equations A boundary problem is called homogeneous if the dif

ferential equation and the boundary conditions are both homogeneous If the

dierential equation or the boundary condition or both are inhomogeneous

then the boundary problem is said to be inhomogeneous

Having solved the homogeneous equation we now consider the in

homogeneous equation of oscillations

y fx

A particular solution of the homogeneous equation is given by yx A cosx

In order to solve the inhomogeneous equation we use the method of variation

of constants In analogy to we write

geneous equation for Ax

cosx A

sinx fx

Application of the Mathematica command

A x C

C Sec K

Cos K f K dK

Sec K

This apparently means that Mathematica cant solve To solve the

equation step by step we consider the corresponding homogeneous equation

tanx

The substitution ux A

x u

x A

x gives the separable equation

tanxdx

Integration yields

x ux C exp

tanxdx

C cos

It seems that Mathematica is not able to integrate equation Since

we do not need Ax itself we can now solve the inhomogeneous equation

directly by inserting A

x Cxcos

x into it The result after

some short algebra is

x fx cosx

and

fx cosxdx

This result may also be derived with the help of Mathematica

f x Sec x C

Finally we obtain

uxdx

cos

fx cosxdx

and

fx cosxdx

cos

xdx cosxA cosxB sinx

Here the rst term is the particular solution of the inhomogeneous equation

and the other two terms represent the general solution of the homo

geneous equation Solution is a consequence of the theorem that the

solution of an inhomogeneous linear equation consists of the superposition of

a particular solution of the inhomogeneous equation and the general solution

of the homogeneous equation

For the special function fx A sinx D where A D and are given

constants we now solve the boundary problem

Here y

and y

are given constant values With the function fx given the

solution takes two forms For resonance between the eigenfrequency

and the exterior excitation frequency ie for the solution is

x sinxB sinx

where CB and are constant Solutions of this type are not able to satisfy

They are called secular and play a role in approximation theory

The second form of the solution is valid for and reads

sinx

B sinx

If one combines this solution with the boundary conditions one gets

sinx

B sinx

sinx

B sinx

These equations determine the integration constants B and

Since we now know that inhomogeneous problems of linear equations can be

reduced to a homogeneous problem we restrict ourselves to discuss methods

Here the functions p

and p

are the functions p

and p

from

renamed The dierential equation is called to be of the Fuchsian

type if the functions are regular rational with exception of poles local regular

singular points To make this clear we consider the Euler equation

ax x

yx bx x

x cyx

which is a special case of and where a b and c are constants A

point x

is called an ordinary point if ax x

and a singular point if

axx

Near an ordinary point solutions of can be found using

the method of power series

Near a singular point the Frobenius

method will be used

Instead of we can consider Now bx x

ax x

will

be replaced by p

x Thus if ax x

then p

x singular point

pole A function regular everywhere but with one pole at x

can no longer

be expanded into a power series but it can be represented by a Laurent

series

x x

All these considerations could better be done in

the complex plane z x iy

If a

for n m a

one says that the point x

is a pole

a regular singular point of order m Singular points that are not poles are

called irregular singular or essential singular Equation is thus called

a Fuchs equation if xp

x and x

x are regular for x that means

that p

x has a pole of rst and p

x of second order respectively These

regular rational functions can be expanded

x a

For L we may write

According to Frobenius the singularity can be split o and the solution of

can be rewritten as a socalled Frobenius series

yx x

is called the index of the series The series is convergent Since the method

of power series is just the special case of the Frobenius method we

will discuss only the latter

We will now solve using the Frobenius method yields

x x

nn an a

since a convergent power series may be dierentiated

Inserting and into and using we obtain

nn a

A power series vanishes only if all coecients vanish For n

reads

since a

cancels The case n will be treated later Equation

is the socalled indicial equation We now apply the method on a special form

of the Euler equation We use x

a b c so that

ba ca

x xy

x yx

Then the indicial equation reads

Its solutions are

so that the solution of is

given by the superposition of two particular solutions

yx AxBx

The command

DSolve[xˆ2*y[x]+3*x*y[x]-3*y[x]==0, y[x],x] delivers the same result It would be easy to show that this solution satises

for instance the initial conditions Also the boundary conditions

and can be satised by

In the case that the indicial equation has real repeated roots

n n the Frobenius method delivers only the rst

solution y

x The second solution will then contain an essential singularity

like a logarithmic term This solution may be derived from For

this equation reads

x y

exp

xdx

exp

xdx

exp

lnx

where P

is a regular power series that does not vanish for x Assume

that the rst solution y

x has the form

x x

where P

x is a regular power series that does not vanish for x Then

the integrand in may be written in the form

exp

xdx

since

n Expanding the regular power series P

the integral becomes

xdx

m n

lnx m n

For x

a b

the Euler equation has the

solution y C

lnx For

the command

DSolve[xˆ2*y[x]+2*x*y[x]+2*y[x]==0, y[x],x] yields an expression containing power of x with complex exponents which is

equivalent to

yx x

cos

lnx

sin

lnx

If the roots of the indicial equation are complex they must be conjugate Then

the solution of may be expressed in terms of trigonometric functions

Up to now we have investigated only the case n We use the Bessel

equation

to demonstrate the procedure for n Inserting into we

receive for n replaced by

We can expect the existence of a logarithmic solution This solution and

the determination of c

will be discussed later on Making the replacement

we can join the two sums into one to receive

For we thus obtain the twotermed recurrence relation

is still unknown If we choose then yields

Furthermore we nd c

so that only

appears and the series representing the Bessel functions contains only the

power

The command

produces the solution

where the space replaces the representing multiplication As an exercise

the reader is invited to solve the following equations using the Frobenius

method and Mathematica

Equation Solution Indicial Equ

y y

sinx

y y J

ix I

y y C

essential singularity x a

y y C

y c

ny c

Due to the recurrence formulae one has c

for and therefore

convergence

Apparently the singularities appearing in dierential equations help to clas

z a

z A

z a

then the equation is called a B

ocher equation Nearly all bound

ary value problems one may come across in physics engineering and applied

mathematics are of this type n l m

We now discuss some special cases Four singularities are to be found in

z a

z A

z a

Heine equation one pole of rst order two poles of second order and one

pole at innity

z a

q p p z

z z a

z a

Lam

e wave equation or Lam

e equation for for and

z a

z A

z a

Wangerin equation Three singularities are contained in to

z a

z A

z a

two poles of rst order in the nite domain and one pole of fourth order in

innity and in

z a

z A

equation y q cos xy has three singularities two are essential

The hypergeometric equation is the grandmother of many equations used in

physics and engineering It reads

c a b z

z z

This important equation has poles at and The values of the index

are

c at the location x

c a b at x

and

b at innity The Legendre wave equation

and the Legendre equation are children of exhibiting three

singularities

Two singularities are found in

z a

and

z a

but even the simple equation

has two poles at and Furthermore some wellknown and important equa

tions have two singularities the con uent hypergeometric equation Kummer

equation

c z

which is a daughter of It has one pole at z and an essential

singularity at ! c is its indicial equation The Bessel

equation and

the Bessel wave equation

as well as the generalized Bessel equation

y z

is a cylinder function like J

have also two singularities Other children

of are the Whittaker equation

solved by Whittaker functions or the Gegenbauer equation

One singularity will be found in

z a

which comprises the Euler equation The most simple linear dierential

equation of second order is given by

The equation y

z a has a pole at z a and the solutions y

z a Also the Weber equation

which is a grandchild of has one pole but y

ky has an

essential singularity at z Also and have one singularity

This is a consequence of the Liouville theorem which expresses the fact that

all functions yz of a complex variable z x iy must either have one or

more singularities or be a constant

Problems

Now solve the initial value problems using

y,{x,0,Pi}] InterpolatingFunction ff gg

In order to plot the result we use now

Plot[Evaluate[y[x]/.%],{x,0,Pi}] In order to plot the values of yx must be known Evaluate replaces the denition of a new function as vx in problem of section

The phrase y[x]/.% has the meaning replace yx by the result of the last calculation ie the solution of the initial value problem

Now use Mathematica to solve the inhomogeneous boundary value prob

lem numerically for x

bsol=NDSolve[{y[x]+y[x]==0,y[0]==1.,y[2.]==2.}, y[x],{x,0,Pi}] Here we have given a name to the calculation Plotting is now possible

Now solve the homogeneous boundary problem

Clear[y]; ts=NDSolve[{y[x]+y[x]==0,y[0]==0,y[Pi]==0}, y[x],{x,0,Pi}] and plot the result If this does not work look at the values yx by

Find the indicial equation or for the following equations

solve them according to the Frobenius method and verify the result

with Mathematica Take some of the equations on page

DSolve[y[x]+mˆ2*y[x]==0,y[x],x] ffy x C Cos mx C Sin mxgg

DSolve[y[x]+y[x]/x-(1+nˆ2/xˆ2)*y[x]==0,y[x],x] ffy x BesselJ nix C BesselY nix C gg

modied Bessel function

y x x

y x

BesselI

BesselK

This does not work Why For numerical solution a range must be

given see problem

Solve the initial value problem of an equation of third order

Solve and giving

y x C HypergeometricF a b c x

C HypergeometricF a c b c c xgg

y x

exp a x C

y x C HermiteH

C HypergeometricF

y x C Cos x C Sin x

x Cos x

x Cos x CosIntegral x

Cos x CosIntegral x x CosIntegral x Sin x

x Sin x

x Cos x SinIntegral x

x Sin x SinIntegral x x

Sin x SinIntegral x

compare to

y[x],x] y x exp q

HermiteH

p q

q x

exp q

HypergeometricF

p q

Dierential equations of physics and engineering

Prior to the discussion of boundary problems it seems to be useful to inves

tigate some of the dierential equations of physics and engineering in more

detail A large class of partial dierential equations allows separation into

ordinary dierential equations Many of these ordinary dierential equations

are children or grandchildren of the hypergeometric dierential equation

In spherical problems the separation of the pertinent partial dierential

equation like eg Helmholtz equation leads to the Legendre

equation

cot

ll y

sin

where is the polar angle in spherical coordinates The solutions of

are usually called spherical functions The substitution cos x gives rise to

the equation

This is the special case of the Gegenbauer equation It is

easy to see that this equation has poles at the location x For m it

has the recurrence relation

is dened by The case m will be treated later With the

substitution x x one obtains from the new

equation

ll y

This equation has a pole at and is the special case a l b l c

transcendental spherical functions P

In order to write down the general

solution of we need a second solution But due to the relations

c and

cab which are valid for the hypergeometric

equation one obtains

for both poles This means that the

second solution is identical with P

Due to the mother the hypergoemetric

dierential equation its solutions are closely related So the function cos is the

elliptic and cosh the hyperbolic child Whereas the function of the rst

kind P

corresponds to cos the hyperbolic part is given by the Legendre

function Q

of the second kind

cosh t

dt x

This expression is a consequence of the possibility to represent the members

of the hypergeometric family by integrals see later The general solution of

the Legendre equation is now given by

yx C

x C

If the parameter l is a natural number a positive integer then the P

generate into polynomials and the Q

go over into elementary transcendental

functions For n l the recurrence relation breaks down and the

solutions are given by the Legendre polynomials

x P

x x cos

cos

cos cos

These polynomials as well as other polynomials are important for physical

and engineering problems and may be represented by

a a Rodriguez formula

b using a generating function

c or by an integral representation

These possibilities oer many practical applications The Rodriguez for

mula for the Legendre polynomial is given by

and their generating function f is

fx u

ux u

Cauchy integral

t z

For gt t

one obtains

t z

Using z x cos t cossin expi tx

i sin expi dt sin expi etc one obtains the Laplace integral

representation

cos

cos i sin cos

Using the integral representation of the solution of the hypergeometric equa

tion the transcendental functions may be represented by

cos t

dt x

compare

Up to now we have considered only the special case m For m

we consider The Frobenius method creates the solutions that are

called associated Legendre polynomials P

x and associated Legendre

functions respectively Mathematica denes all these various functions

x LegendreP[l,x] P

x LegendreP[l,m,x] P

z LegendreP[n,z] P

z LegendreP[n,m,z] Q

z LegendreQ[n,z] Q

but has not been able to produce these functions by solving equations

We give some formulae for the associated polynomials

sin

cos

sin sin

There exists an important attribute of these polynomials This feature is

very important for applications The attribute is called orthogonality and is

described by

xdx

xdx for l k

xdx

l m"

Other children of the hypergeometric equation are the Chebyshev polyno

mials and Chebyshev functions if x is replaced by complex z

ChebyshevT[n,x] T n

ChebyshevU[n,x] U n

which satisfy the equation

x xT

x n

and are given by T

which xes the c

in their

recurrence relation and explicitely by

x T

x x T

x x

x T

x x

Chebyshev polynomials of rst kind are often used in making numerical

approximation to functions They satisfy the orthogonality relations

for m n

The polynomials U

x of the second kind and the Chebyshev functions are

by Laguerre orHermite In contrast to the orthogonality relations

of the Legendre polynomials the orthogonality relations contain a

weighting function x

The same is true for the associatedLaguerre

polynomials They satisfy the Laguerre dierential equation

x k xL

x nL

and the Rodriguez formula

x! L

x! k

For k and the polynomials L

x are designated by L

x For

they are given by

x L

x x L

x x k

x L

k x

k k

n k"

nm"k m"m"

k n

The orthogonality relations again contain a weighting function They read

xdx

n k"

This has the important practical consequence that all functions fx that are

quadratic integrable can be expanded into a Laguerre series

expxx

with

n"n k"

fx expxx

xdx

Application of DSolve on yields the result in the form of a con u ent hypergeometric Kummer function For k one obtains an

analogous result but for n the results are retrieved

Additionally a transcendental function appears that seems to have no prac

tical applications

Hermite polynomials H

x have similar important properties They sat

isfy the Hermite equation

hypergeometric function Solution by the Frobenius methods leads to the

recurrence relation

and the Rodriguez formula reads