and consider the new difference equation

92 Chapter 2 Obviously, for E = 1 this new differencc equation is the same as (2.15.2).

W e look for the solution of (2.15.3), (1.3.1) having the form

(2.15.4)

u ( k )

=

For this, it is necessary to have

l

m =O

L

i=o

and

m =O

Thus, 011 equating the c:ocfficicnts of e r n , ' r n =

0, 1,

. . .

we find

t,he infinite system of initial value problems

n-l

(2.15.5) ~ r v i ( k ) ? P ( k + i )

+o.,(k)v.O(k+n)

= 0, U 0 ( u + i - 1) = ?/,i, i=O

l < i < n

n,- 1 n,- 1

(2.15.6).

c c ~ i ( k ) ~ " ( k + ⁱ⁾ + u , ( k ) ~ " ( k +

^71.)

⁼ c c i(k )u"- '( k + ^i),

i=O i=O

w . m ( a . + i - l ) =

0,

l l i l n ,

7 1 1 = 1 , 2 , . . .

. This infinite system can be solved recursively. Indeed, from our initial assumption the solution

~ , ' ( k )

of (2.15.5) can be obtained explicitly, and t h u s the term ci(k)uo(k+i) in

(2.15.6)l

is known; consequently the solution

ul(k)

of the nonhomogeneous initial value problem (2.15.6)1 can be obtained by the method of variation of parameters. Continuing in this way the functions u2

( k ) ,

u 3

( k ) ,

. . . can similarly be obtained. Finally, the solution of the original problem is obtained by summing the series (2.15.4) for E

= 1.

The above formal perturbativc procedure is not only applicable for the initial value problem (1.2.4), (1.3.1) but also can be employed to a variety of linear as well as nonlinear problems. The implementation of this powerful tcchniquc consists in the following three basic steps:

(i) Conversion of the given problem into a perturbation problem by introducing the small parameter E.

(ii) Assumption of the solution in the form of a perturbation series and the computation of the coefficients of that series.

(iii) Finally, obtaining the solution of the original problem by summing the perturbation series for the appropriate value of c.

It is clear that the parameter E i n the original problem can be in- troduced in an infinite number of ways, however the perturbed problem is meaningful only if the zeroth order solution, i.e.

u o ( k )

is obtainable explicitly. Further, in a large number of applied problems this parameter occurs naturally.

The pertltrhation method naturally leads to the question: Under what conditions docs the perturbation series converge and actually represent a solution of the original prol)lcm? Unfortunately, often perturbation series arc divergent, however this is not necessarily

bad

tm:ausc a good approxi- mation to the solut,iorl when c is very small can be obtained by surrlrning only first few tcrms of the series.

Example 2.15.1. Consider the initial value problem

(2.15.7)

~ ( k +

²⁾ - 2 ~ ( k

+

¹⁾

+ - ~ ( k )

3 ₄ =

0,

?L(()) = 1, ~ , ( l ) =

-

₂ ¹

for which u ( k )

=

1/2' is the unique solution. W e convert (2.15.7) into a perturbation problem

(2.15.8) A % ( k ) = E

(+) ^{, u ( 0 )}

^{= 1, u(1) =}

^-

₂ ¹

and assume that its solution can be written as perturbation series (2.15.4).

This leads to an infinite system of initial value problems 1

L P V , O ( k )

=

0, 7 r 0 ( 0 )

=

1, UO(1) = - 2 1

A2um(k) = -um-'(k), 4

~ " ( 0 )

= um,(l) = 0,

m

= 1,2, which can be solved recursively, to obtain

94 Chapter 2 Thus, the solution w.(k, F) of the perturbation problem (2.15.8) appears

as

Example 2.15.2. Consider Airy's differential cqlmtiorl y"-ty = 0, t

2 0

together with the initial conditions

y(0)

= 1,

y'(0)

=

0.

The simplest difference equation approxinlatiorl to this initial v a l w problcrn is

(2.15.9)

4k- +

^{2) -}

^{h ( k} +

¹⁾

+

7 1 ( , q -

( k +

1 ) ~ 4 k )

= 0,

k- E

IN

= ~ ( 1 ) = 1

where h

>

0 is an arbitrary constant step size, and

u(k)

approximates the solution y ( t ) at t k = kh.

W e convert (2.15.9) into a perturbation problem

~ ~ ~ ( k ) = € ( ( I C + 1)h,3u(k)),

v . ( ~ ) = v . ( ~ ) = I

and assun~e that its solution can be written as perturbation scrics (2.15.4).

This leads to an infinite system of initial value problems A2uO(k) = 0,

uO(0)

= UO(1) = 1

~ 2 ~ 7 y r c ) =

(IC +

1 ) ~ ~ 7 " ( k ) ,

w,yo)

= T / ~ m y l ) =

0,

= 1,2,. . . which can be solved recursively, to obtain

w,O(k)

=

1

?l2@)

= - ( k

^z

+

^1)(5)(2k

+

^{l ) h 6}

6!

2

u 3 ( k )

= - ( k

9!

+

1)(7)(14k2

+

^{7k - 6)h9}

...

Thus, a uniform approximation to the solution

u,(k)

of (2.15.9) can be taken as

u.(k)

²¹ 1

+ ^-p

_3! ¹

+

^{p h 3}

+ ^-(k

_6! ²

+

^{p ( 2 k}

+

^{1 ) ~}

+

^{? ( k} 2

+

1)(7)(14k2

+ ^{7 k}

^-

^6)hZ9.

This approximation is exact for k E W(0,7).

In many practical problems one often meets cases where the parameter

E is involved in the diffcrcnce equation in such a way that the method of regular perturbation cannot be applied. In the literature such problems are known as singular pcrturbation problems, and to understand these we consider the following:

Example 2.15.3. For the initial value problem

explicit solution can be writtcrl as

(2.15.11) 1

u,(k)

=

-

1°C [ ( C Y 1 - ^COO)

+

^{((Yo - Nl)Ek]}

,

for which it follows that

Suppressing the small parameter E in (2.15.10), the resulting dcgencrate first order equation is

(2.15.12) V O ( k

+

^{2) -}

^7P(k +

¹⁾

^{= 0.}

Obviously, for (2.15.12) the initial conditions

~ ~ ( 0 )

= ^{N O ,} t ~ ~ ( 1 )

=

cul are inconsistent unless a0

=

( ~ 1 . Thus, (2.15.10) is said to be in the singularly perturbed form, and a

boun,dary

^{layer. occurs at} k

=

0.

If WC seek the solution of (2.15.10) in the regular perturbation series form (2.15.4), then it leads to the system of first order difference equations

u.O(k +

^{2) -}

^uyk +

^{1) =}

^0,

I P ( 0 ) = (Yo, VP(1) = Nl

u,”(k +

^{2) -}

^u”(k +

¹⁾

⁼

^7r”-l(k

+

^{1) - u.”-l(k),}

^{u”(0) = u,”(l)}

⁼

^0,

m .

=

1,2,. . .

96 Chapter 2 which can be solved only if the initial conditions are consistent, i.e. ^(YO

=

01. Further, in such a case it is easy to obtain

u o ( k )

= (YO,

u " ( k ) =

0,

'm, =

1,2,. . .

,

and hence (2.15.4) reduces to just

u ( k ) =

^(YO which is indeed a solution of (2.15.10).

Now ignoring the terms with coefficients of t and higher powers of c in (2.15.11), the zeroth order approximate solution appears as

(2.15.13)

u ( k ) =

^a1

+ ^{t y o o}

^{- 01).}

The first part of this solution, i.e. cy1 is called the 0 u . k solution,, as it is valid outside the boundary layer. This satisfies only one of the initial conditions

~ ( 1 )

= cy1. The second part of this solution, i.e. ((YO - 0 1 ) is called the in,n.er S O h t i O T L which recovers the lost initial condition

u(0)

=

cy0.

The preserlce of 6' in (2.15.13) sllggests that the inner solution has thc transformation , u J ( ~ ) = u ( k ) / c ' . Using this transformation in (2.15.10) anti dividing throughout with '+'F lcads to thc difference equation (2.15.14) fW(k

+

^{2) -}

⁽¹ + ^E)W(k + ¹⁾ +

^{Wl(k) =}

^0.

Putting f = 0 in the above equation gives the degenerate equation

-zO(k+1) + z O ( k )

=

0.

This equation is solved with the initial condition

.yo) = u ( 0 )

- CY1 = (Yo - CY1

to obtain

zo(k)

= (YO - al, which is the same as the inner solution Thus the total zeroth order solution of (2.15.10) is composed of the outer and inner solutions and given by

u ( k ) =

7 / 0 ( k )

+ ^€%O(k).

Utilizing the above ideas we write the solution w,(k) of (2.15.10) as the sum of two solutions

(2.15.15)

u ( k ) =

v ( k )

+

^E%(k),

where v ( k ) and

z ( k )

are the outer and inner solutions. Substituting (2.15.15) in (2.15.10) and separating the terms, we obtain two equations (2.15.16)

w(k +

^{2) -}

⁽¹ +

^E)ll(k

+ ¹⁾ + ^{W ( k ) = 0}

and

(2.15.17) EZ(k

+

^{2) - (1}

+

E ) Z ( k

+

¹⁾

+

^{Z ( k ) =}

^0.

For solving these equations, we assunle that

W W

(2.15.18) v ( k ) =

c trn.7lrn,(k),

Z ( k ) =

c

ErnZ"(k).

m.=O m,=O

Snbstitnting the above series solutions

i n

(2.15.16) and (2.15.17) respec- tively leads to the systems

Q ( k +

^{2) - vO(k}

+ ¹⁾

⁼

⁰

(2.15.19)

P ( k +

²⁾ -?,"(/c

+ ¹⁾

⁼ ?,""(k

+

^{1) -}

V'"(k),

m . = 1,2,. . .

Finally, the series solution of (2.15.10) is written as

m ^W

(2.15.22)

u ( k )

=

c

Ern.Vrn,(k)

+

fk

c

E ? P ( k ) .

m,=O m.=O

The above systems (2.15.19) ^~ (2.15.21) can easily be solved to obtain approximations of

u,(k)

up to any order. For example, tllc zeroth order approximation is the same as (2.15.13), as it should be, and the first order approximation appears as

u.(k) [cy1 - ((Yo - cy1)€]

+

E k [ ( Q O - Q l )

+

^{f(O0 - ..l)].}

2.16. Problems

2.16.1. Show that the functions u l ( k )

=

c ( #

0)

and

uz(k) =

l/(k+l)(2)

98

Chapter

2

satisfy the nonlinear difference equation

A 2 u ( k ) + ^{3(k + 1 ) k}

( k +

³⁾ u(k)Au,(k)

= 0,

but

ul(k) + ^u,z(k)

^does not satisfy the given difference equation. (This shows that the principle of superposition holds good only for the linear equations.)

2.16.2. Let XI,. . .

,

^{X ,} be the (not necessarily distinct) eigenvalues of an

n

x

n,

matrix

A.

Show that

(i) the eigenvalues of

AT

are

X1,

. . .

,

^X,

(ii) for any c:onstant CY the eigenvalues of

(YA

are 0x1, . . .

,

^cwXn

(iii)

C:=l

X i

= T r A = cy=,

^aii

(iv) X i

=

det

A

(v) if

A-'

exists then the eigcrlvalucs of

A-'

are l/Xl, . . .

,

l / X , (vi) for any polynomial

Pk

(t) the eigcnvallm of

Pk ( A )

are

Pn:

(XI), . . .

,

P k ( X , , )

(vii) if

A

is upper (lower) triangular,

i.e.

a..;l =

0,

i

>

j (i

<

j ) , then the eigcrlvalues of

A

are the diagonal elements of

A

(viii) if

A

is real and X1 is complex with thc c:orrespondirlg eigenvector v l , then thcre exists at least one i, 2

5

i

5

^R such that X i =

XI

and for such an i , V' is the corresponding eigenvector.

2.16.3. (i) Let the

n,

x

n

matrix

A(t)

=

( a , ; j ( t ) )

be such that

(ii) An

(n

-

1)

^X

(n

- 1) determinant obtained by deleting ith row and j t h column of a given

n

x

n

matrix

A

is called the m,inor Z,ij of the element ai,. W e define the cofactor of a;j as ~ ~

=

i(-l)i+jEaj. Show j

that (2.16.2)

n

(2.16.3)

c

^aijcukj

^{= 0}

^{if i}

^{# k .}

j=1

2.16.4.

Let the functiorls

ui(k.),

1 5

i

5

n,

be defined on IN and l i r r i ~ ~ ~ ( ~ / , ~ ( k ) / ~ / , ~ + ~ ( k ) )

=

0,

1

5

i

5 n.

-

1. Show that these functions arc linearly independent.

2.16.5.

Let u(k) be

a

complex solution of the homogeneous system (1.2.12) on H. Show that both the rcal and imaginary parts of u(k) are solutions of (1.2.12).

2.16.6.

Let

U ( k , k.0) and

V ( k ,

ko)

be the principal fundamental matrix solutions of (1.2.12) and (2.7.1) respectively. Show that V T ( k , k o ) U ( k ,

ko)

=

1.

2.16.8.

Let the notations and hypotheses of Lernrrla 2.8.2 be satisficd.

Dans le document BOARD AND (Page 104-112)