which is the sarrle as

V(P +

^{1) - U ( k - 1)}

+

^kU(k)

v ( 4

^-

u(k)

^{- -}

^Q,

where

(Y

is an arbitrary constant. Therefore,

U ( k )

and V(!) satisfy the first order ordinary difference equations

(a - k ) U ( k ) = U ( k -

1)

V(P +

^{1) =}

^QV([),

156 Chapter 3 which can

bc

solved to obtain

Summing over cy now gives

where c is an arbitrary function of a.

Laplace's Method. This method is applicable when the sum or thc differencr of the argllnlents of all w.(k

+

^{T, I!}

+

^{7 )} that appear i n a partial difference equation is a constant. For example, in the diffcrcrlce equation (3.13.9) thes~lnloftheargurrlentsofall zr(k,L), u ( k + I , P - l ) , u ( k - l , l + l ) is a constant,

k- +

^{1. If WC set}

^k + ⁼

^{' r n} and dcfirlc 71(

k ) = U( k ,

m -

k ) ,

then equation (3.13.9) 1m:ornes

7 i ( k )

=

p7(k

+ ¹⁾ + ^qv(k

^{- l),}

which is a sccond order ordinary difference cquation whose solution is

(3.13.15)

However, since an arbitrary constant can be considered a function of an- other constant, we recover the solution of (3.13.9) by replacing c1 and c2 in (3.13.15) by

c l ( k + e)

^and

^cz(k + e).

There arc several norllinear partial difference equations which can be reduced to linear equations by means of special transformations. As an cxample, we shall consider Riccati's

ezten.ded form,,

which is a system of nonlinear partial difference equations

(3.13.16)

u,(k + ^{l , e} +

^{l) =}

7I(k

+

^1,e

+

^{1) =}

o u ( k , e) +

B v ( k ,

t ) + ^y

p ( k ,

e) +

^qv(k,

^l) +

^T

p u ( k , P ) +

v v ( k , ! ) + V

pu(k, e) + ^q71(k, e) + r '

where

cy, /?,

y, p, v, 7 , p, q and

r

are constants.

From (3.13.16) it follows that if X,

E

^and

C

are undetermincd multi- pliers, thcn

(3.13.17)

u(k+ l,[+

1) - v(k

+ ^1, e + ¹⁾

cru(k,e) + ^pv(k,e) + ^y

^- p U ( k , e )

+ U v ( k , e ) +

⁷⁷

1

pu(k, e) + ^{p ( k , P)} +

^r

- -

- _- Xu(k

+

^1,

e +

¹⁾

+ ^@(lC +

^1,

e +

¹⁾

+ <

(CYX+~LE+pC)'U.(k,e)+(px+ul+qC)1~(lC,e)+(yX+77€+rC)' Now suppose that X, and

C

are chosen so that

(3.13.18)

where

h,

is an unknown constant. The c:ondition that X,

(

and

C

^are

not to be

zero

demands that (3.13.19)

This is a cubic equation in

h,

and provides three values of

h,

say,

h,l, h.2

and h 3 , and for each of these values WC can find the corresponding values of X,

<

^and

C

from the system (3.13.18), i.e.

h,i

^"+(Xi,

ti, Ci), ⁱ

= 1,2,3.

This allows us to replace (3.13.17) by the new set of equations (3.13.20)

U l ( k + l,[+ 1)

-

U2(k+

l,[+

1)

-

U3(k+ l,[+

1)

h,l

U1

( k e)

^-

h2U2(k,O

^-

hf3U3(k, P) '

where

U i ( k , e )

= Xiw,(k,C)

+ & v ( k , P ) +

^{<i, i = 1,2,3.}

From the equations (3.13.20) it follows that

Ul(k+ l , e +

1) - h1

U~(k,e) U3(k + ^l,! + ¹⁾

^h3

U 3 ( k , [ )

- -~

(3.13.21)

U2(k + ^l,[ + ¹⁾

^h2

^Uz(k,e)

U3(k +

^1,

e +

^{1) h3}

^U3(k, e)

which are of the form U (

k + ^1, + ¹⁾

^{= /?U(}

^{k , l),}

and hence can be solved to obtain

- - ^"

158 Chapter 3

where cl, c 2 and c3 are arbitrary functions of

k

-

L.

Thus, on using the expressions for

Ui(k,e),

i

= 1 , 2 , 3

in (3.13.22), we find

3.14. Wave Equation

From

Example 3.6.1 it is clear that the function ? / , ( I C )

= A

sin wkh,, IC E

IN

is a solution of the second order differerlce cquation

tL(k + ¹⁾

^{- 271.(k)}

+ ^t/,(k

^{- 1)}

₊

²

W ?/,(/c) = 0.

4w-2 sin2 iwh,

Further, as h, -+

0, ~ ( k )

tends to y ( t ) = A s i n w t ant1 the above difference equation converges to t,hc differential equation y”

+

^{w 2 v}

⁼

^0.

The function @ ( x , t )

= A

sir1 w(x+ct) is a solution of t,llo

one

dirrlcnsiorlal wave equation

(3.14.1)

4+t =

c2#)2z.

To obtain the partial difference equation corresponding to this solution, WC

use the discretization x

=

kh,l, t

=

th2,

k , e ^E

^{W where h,l}

^>

^{0, h.2}

^{> 0}

are step-sizes, and represent 4(z,t) = #(kh,l,th,2) =

u,(k,l!).

Thus. it follows that

and

u ( k , e +

^{1) - 2V,(k,l)}

+

^U(k,

^e

^-

¹⁾

~ w - ~ c - ~ sin2

+

w2c2u(k,P) =

0,

whence

As h l ,

h2 ^"+0 this linear second order partial difference equation converges to (3.14.1). The function 4(3:,t) = Asinw(3: - ct) is also a solution of both the equations (3.14.1) and (3.14.2).

The analysis can be extended to a solution consisting of a sum of two terms, e.g.

$(x,

t)

= Al

sinwl(3:

+ ^ct) + ^A2

^{sin w2(z}

+

ct). Indeed from the relation

$ ( X

+

^{h,l,t) - (coswlh,l}

+

cosw2h,l)@(~,t)

+

^{+ ( X - h,l,t)}

= (coswlhl - cosw2hl)[Al sinwl(z

+ ^ct)

^- A:!sinw2(5

+

ct)]

follows the required partial difference equation

~ ( k , / ! +

1) - ( C O S W ~ C ~ , ~

+

coswz(:h,z)u,(k,P)

+

~ ( k , ! - 1)

(COS w l ~ h , 2 - COS W Z C ~ , ~ )

(3.14.3)

-

~ . ( k +

^{l,[) -} (<:oswlh,l

+

~ o s w z h , l ) ~ ( k , P )

+ ^{~ ( k}

^{- l,!)}

- ( c o s u ~ ~ , ~ - ~ 0 ~ w 2 h . l )

Since as h,, "-f0 arid h2 ^-+ 0, cosw1&

+

^{cos w2&2 -+} 2, coswlch,z - c o s ~ ~ h , ~ ^-+ h,:(w:-w;)/2 difference equation (5.143) converges to (3.14.1).

For the general solution

u(x,

t)

= Erl ^Ai

^sinwi(3:

+

^ct) of the Wave eqtlation (3.14.1) it is not possible to find a simple partial difference equa- tion. The appearancc of W in the dcrlonlirlator 4wP2 sin2 i w h loses the advantagc er1,joyed by the usual approximation term h2.

C O S W ~ C ~ , ~ -+ C2h,:(W; - wT)/2, coswlh,l

+

<:oswzh,l -+ 2, coswlh,l -

3.15. FitzHugh-Nagumo's Equation

The partial differential equation

was considered by FitzHugh and Nagurno in modelling the propagation of a nerve pulse [6], has the solitary~-wave solution

(3.15.2) 4 ( x ,

t ) = A

tanh(v3:

+

^wt).

To obtain the partial difference equation corresponding to this solution, once again we use the discretization 3:

=

kh,l,

t =

l?h,2,

k , l

E W where

h l ,

h,2

>

0 are step-sizes, and represent +(x, t)

=

4(kh,l, /!ha) =

u ( k , l?).

From

u.(k,

P) =

A

tanh(vkh1

+

wlhz) it follows that

u(k,l?+

1)

- u ( k , / ! )

=

A t a n h ~ h , z ( A - ~ 7 ~ ( k , / ! ) ' l , , ( k , l ? +

1) -l),

160

Chapter 3

which is the same as (3.15.1). Thus, the partial difference equation (3.15.3) is the best discretization of the partial differential equation (3.15.1).

3.16. Korteweg-de Vries’ Equation

The partial differential equation

(3.16.1)

4x.m

^- 34:

+ 4t = 0,

which, when differentiated with respect to z and the substitution = ^t/j is made, gives the Korteweg-de Vries equation in the usual form

(3.16.2) l j j z X 2 - 6 4 4 ~ ~

+

^li/t

^{= 0.}

W e shall determine a partial difference equation for the single--soliton solution

#(x, t )

= -2w tanh(wz - 4w3t) of thc equation (3.16.1)

by

consid- which, when used with an obvious extension of (3.19.26) gives the required partial difference equation

Thc limiting partial differential equation is (3.16.4)

When

A =

-2w and q = -4w3 so that 4 ( x , t )

=

-2w tanh(wn: - 4w3t), and the equation (3.16.4) is the same as (3.16.1). Thus, in this case the partial difference equation (3.16.3) is the best discretization of the partial differential equation (3.16.1).

3.17. Modified KdV _Equation

Consider the modified Kortewcg de Vries cquation [6,37] i n the form (3.17.1)

4m.Z + _W24X + 4t

=

0

for which the solitary wave solution is

4(n:, t )

= wsech

(wn:

- w 3 t ) . W e shall deterrrlinc differential and difference equations satisfied

by

& ( x , t ) = Ascch (wx

+

^{qt) and u ( k ,}

^{P) =}

Asech (wkhl

+ ^q!h,2).

The limiting partial differential equation is

(3.17.3) ^W"

+

6 ~ ~ A - ~ 4 ' 4 , -

-4t

77

^{= 0.}

When A = ^W and q = -w3 so that

4(x,t)

= wtanh(wz - w3t), and the equation (3.17.3) is the same as (3.17.1). Thus, in this rase the

162 Chaptcr 3 partial difference equation (3.17.2) is the best discretization of the partial differential equation (3.17.1).

3.18. Lagrange’s Equations

Consider a holonornic mechanical system, with q = q(t) =

q l ( t ) ,

. . .

, qn,(t)

the generalized coordinates, with potential energy V ( q ) , and kinetic en- ergy

(3.18.1)

in which aij = a,j.i. Lagrange’s equations of motion give the nonconscrva- tive generalized forces as

(3.18.2)

W e shall discrctizc these equations so that the following two properties arc preserved.

n (3.18.3) - ( T + V ) ^d =

I F , . & . ,

dt ,.=l

which leads to the energy integral T

+

^{V =} constant, when the power of the norlcorlscrvative generalized forces, given by the right side of (3.18.3) is zero.

(Pz)

If the variable q,. is cyclic, i.c. T and V both are independent of q,, then (3.18.2) reduces to

(3.18.4)

or in integrated form (3.18.5)

Thus, if the impulse of the nonconservative force corresponding to the cyclic variable, given by the left side of (3.18.5) is zero, then the generalized mo- mentum corresponding to this variable, given by the right side of (3.18.5), is also zero.

Since the discretization of (3.18.2) will involvc partial differences, first we shall provide a general formula for the partial differences. For this, let

(3.18.6)

If f is independent of q,,., then (3.18.7)

If the time t is discretized at t,irrlc instants

t ( k ) , k

=

0 , 1 , .

^{. .} then the aim is to choose a discrete analog of the partial derivative, which we shall denote by

(3.18.8)

so that the disc:rctc analogs

of

(3.18.6) and (3.18.7) arc satisfied. 111 (3.18.81,

A,.

will b e called the (forward) partial difference operator. The tlisc:ret,e analog

of

(3.18.6) is

164

Chapter

3

Dans le document BOARD AND (Page 168-177)