MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 12 doc

so that Equation 14.90 becomes Remembering the definition of the beta function: we can write Equation 14.92 as Also using the relation 14.86 and between the beta and the gamma functio

OTHER DEFINITIONS OF DIFFERINTEGRALS 393

z-phe

t

Fig 14.3 Contour C' = C + C o + LI + Lz in the differintegral formula

which goes to zero in the limit 60 + 0 For the CO integral t o be zero in the limit 6, 4 0, we have taken q as negative Using this result we can write Equation (14.74) as

Now we have t o evaluate the [ f+L, - f + L z 3 integral we first evaluate the parts of the integral for [-m, 01, which gives zero as

= 0

394 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFfRlNTEGRALS"

Fig 14.4 Contours for the $ + L , , $+,,, , and jC integrals

Writing the remaining part of the $, dz integral we get

(14.84) (14.85)

To see that this agrees with the Riemarin-Liouville definition we use the fol- lowing relation of the gamma function:

and write

(14.86)

d q f ( x ) '(' Ids q < 0 and noninteger (14.87) dxq .I r(-q) (x-6)4+"

OTHER DEF/N/T/ONS OF DIFFERINTEGRALS 395

This is nothing but the Riemann-Liouville definition Using Equation (14.71)

we can extend this definition to positive values of q

so that Equation (14.90) becomes

Remembering the definition of the beta function:

we can write Equation (14.92) as

Also using the relation (14.86) and

between the beta and the gamma functions, we obtain the result as

396 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

Limits on the parameters p and q follow from the conditions of convergence

for the beta integral

For q 2 0, as in the Riemann-Liouville definition, we write

(14.97)

and choose the integer n as q - n < 0 We now evaluate the differintegral inside the square brackets using formula (14.71) as

Combining this with the results in Equations (14.96) and (14.98) we obtain a

formula valid for all q as

for p > -1, where m and n are positive integers For p 5 -1 the beta function

is divergent Thus a generalization valid for all p values is yet to be found

14.3.3 Differintegrals via Laplace Transforms

For the negative values of q we can define differintegrals by using Laplace transforms as

OTHER DEFINITIONS OF DIFFERINTEGRALS 397

where its Laplace transform is

sqF(~) - -(o) - - sn-'-(O)] dxq-n (14.110)

In this definition q > 0 and the integer n must be chosen such that the inequality n - 1 < q 5 n is satisfied The differintegrals on the right-hand side are all evaluated via the L method To show that the methods agree we write

and use the convolution theorem to find its Laplace transform as

(14.112)

= ~ q - ~ F ( s ) , q - n < 0 (14.113)

398 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

-

This gives us the sqf(s) = snx(s) relation

definition [Eqs (14.7O-71)] we can write

Using the Riemann-Liouville

Since q - n < 0 and because of Equation (14 108)' we can write

From the definition of A ( z ) we can also write

A ( z ) = - r(n - q ) Jx ,, (z - f ( r ) d r T ) Q - " + ~ ' q - n < O ,

(14.114)

(14.115)

(14.116)

As in the Griinwald and Riemann-Liouville definitions we assume that the

( 14.117) where n a is positive integer and q takes all values We can now write

PROPERTIES OF DIFFERINTEGRALS 399

Using Equation (14.111) we can now write

(14.122)

which shows that for q > 0, too, both definitions agree

In formula (14.1 lo), if the function f(x) satisfies the boundary conditions

400 FRACTIONAL DERIVATIVES AND INTEGRALS "DIFFERIN TEGRALS"

Using the linearity of the differintegral operator we can find the differintegral

of a uniformly convergent series for all q values as

(14.131)

Differintegrated series are also uniformly convergent in the same interval For

functions with power series expansions, using the Riemann formula we can write

(14.133)

However, if we look at the operation

[ d ( z d*n - a)]*" { [ d ( ~ diNf - .)ITN 1, (14.136)

the result is not always

d * n i N f

[ d ( z - u)]*"?" ( 14.137)

Assume that the function f (z) has continuous Nth-order derivative in the interval [a, b] and let us take the integral of this Nth-order derivative as

We integrate this once more:

and repeat the process n times to get

(14.140)

( - a)"-'

1 - (n - l)! f ( N - y u )

402 FRACTIONAL DERIVATIVES AND INTEGRALS- "DIFFERINTEGRALS"

(14.149)

PROPERTIES OF DIFFERINTEGRALS 403

Because the right-hand sides of Equations (14.149) and (14.147) are identical,

we obtain the composition rule for n successive integrations followed by N differentiations as

(14.150)

To find the composition rule for the cases where the differentiations are performed before the integrations, we turn to Equation (14.142) and write

the sum in two pieces as

Comparing this with Equation (14.147), we now obtain the composition rule

for the cases where N-fold differentiation is performed before n successive integrations as

On the other hand, for

we have to use formula (14.152) Since N = 1 and n = 3, k takes only

the value two, thus giving

(14.154)

404 FRACTIONAL DERIVATIVES AND INTEGRALS: “DIFFERINTEGRALS”

it can be shown that the composition rule [Eq

functions satisfying the condition

(14.155)] is valid only for

In practice it is difficult to apply the composition rule as given in Equation

(14.158) Because the violation of Equation (14.157) is equivalent to the

is satisfied For the remaining cases condition (14.157) is violat,& We now

PROPERTIES OF DtFFERlNTEGRALS 405

check the equivalent condition = 0 to identify the terms responsible for the violation of condition (14.157) For the derivative to vanish, from Equation (14.159) it is seen that the gamma function in the denominator

must diverge for all uj # 0, that is,

[dxl

p + j - Q + 1 = 0, -1, -2, For a given p (> -1) and positive Q, j will eventually make ( p - Q + j + 1)

positive; therefore we can write

p + j = Q - l , Q - 2, , Q - m (14.160)

where m is an integer satisfying

O < Q < m < Q + 1 (14.161) For the j values that make ( p - Q + j + 1) positive, the gamma function

in the denominator is finite, and the corresponding terms in the series satisfy condition (14.157) Thus the problem is located to the terms with the j values satisfying Equation (14.160) Now, in general for an arbitrary diffferintegrable function we can write the expression

d-Q

f (z) - - [dzl-Q [ 5 1= coxQ-' + clzQ-2 + + cmzQ-,, (14.162)

[&]Q where c1, c2, ) c, are arbitrary constants Note that the right-hand side of

Equation (14.162) is exactly composed of the terms that vanish when #

0, that is, when Equation (14.157) is satisfied This formula, which IS very useful in finding solutions of extraordinary differential equations can now be used in Equation (14.158) to compose differintegrals

Another useful formula is obtained when Q takes integer values N in Equa- tion (14.158) We apply the composition rule [Eq (14.158)] with Equation

(14.142) written for n = N , and use the generalization of the Riemann for- mula:

1"11

(14.163)

to obtain

- dq+N f

-

[d(z - U)]"+"

(14.164)

406 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

Example 14.2 Composition of diffeerintegmls: As another exampie we

consider the function

for the values a = 0, Q = 1/2, and q = -1/2 Since condition (14.157)

is not satisfied, that is,

Contrary to what we expect a d-' is not the inverse of 3 d z for x-lI2

Example 14.3 Inverse of differintegmk: We now consider the function

14.4.6 Leibniz's Rule

The differintegral of the qth order of the multiplication of two functions f and

g is given by the formula

where the binomial coefficients are to be calculated by replacing the factorials with the corresponding gamma functions

14.4.7 Right- and Left-Handed Differintegrals

The Riemann-Liouville definition of differintegral was given as

where k is an integer satisfying

k = O for q < 0

k - l < q < k for 4 2 0 ( 14.179) This is also called the right-handed Riemann-Liouville definition If f ( t ) is a

function representing a dynamic process, in general t is a timelike variable The principle of causality justifies the usage of the right-handed derivative because the present value of a differintegral is determined from the past values

o f f ( t ) starting from an initial time t = a Similar to the advanced potentials,

it is also possible to define a left-handed Riemann-Liouville differintegral as

408 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

where k is again an integer satisfying Equation (14.179) Even though for dynamic processes it is difficult t o interpret the left-handed definition] in general the boundary or the initial conditions determine which definition is

t o be used It is also possible to give a left-handed version of the Griinwald definition In this chapter we confine ourselves t o the right-handed definition

14.4.8

We now discuss the dependence of

Dependence on the Lower Limit

DIFFERINTEGRALS OF SOME FUNCTIONS 409

Even though we have obtained this expression for q < 0, it is also valid for all

q (Oldham and Spanier, Section 3.2) For q = 0,1,2, , that is, for ordinary derivatives, we have

which is given as an integral, and the Griinwald definition [Eq (14.39)] which

is given as an infinite series

14.5 DIFFERINTEGRALS OF SOME FUNCTIONS

In this section we discuss differintegrals of some selected functions For an extensive list and discussion of the differintegrals of functions of mathematical physics we refer the reader to Oldham and Spanier

14.5.1 Differintegral of a Constant

First we take the number one and find its differintegral using the Griinwald definition [Eq (14.39)] as

Using the properties of gamma functions; Cy=il r(j - q)/I'(-q)I'(j + I) =

r ( N - q)/r(l- q ) r ( N ) , and limN,oo[N*r(N - q ) / r ( N ) ] = 1, we find

(14.189)

When q takes integer values, this reduces to the expected result

arbitrary constant C, including zero, the differintegral is (see Problem For an 14.7)

(14.190)

410 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

We now use the Riemann-Liouville formula to find the same differintegral

(14.200)

DIFFERINTEGRALS OF SOME FUNCTIONS 411

For the other values of q we use formula (14.40) t o write

This is now valid for all q

14.5.3

Here there is no restriction on p other than p > -1 We start with the

Riemann-Liouville definition and write

Differintegral of [Z - u]” ( p > -1)

(14.206)

When we use the transformation z’ - a = v, this becomes

Now we make the transformation v = ( x - a)u t o write

Using the definition of the beta function [Eq (13.151)] and its relation with

the gamma functions, we finally obtain

(14.209) (14.210)

where q < 0 and p > -1 Actually, we could remove the restriction on q and

use Equation (14.210) for all q (see the derivation of the Riemann formula with the substitution x -+ x - a )

412 FRACTIONAL DERIVATIVES AND INTEGRAlSr"DIFFERINTEGRA1S"

We now use Equation (14.132) and the Riemann formula (14.99), along with

the properties of gamma and the beta functions t o find

where B, is the incomplete beta function

14.5.5 Differintegral of exp(fz)

We first write the Taylor series of the exponential function as

(14.213)

(14.214)

and use the Riemann formula (14.99) t o obtain

where y* is the incomplete gamma function

MATHEMATICAL TECHNIQUES WITH DIFFERINTEGRALS 413

14.5.7 Some Semiderivatives and Semi-integrals

We conclude this section with a table of the frequently used semiderivatives and semi-integrals of some functions:

414 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

In this equation we can replace the upper limit in the sum by any number greater than n-1 We are now going to show that this expression is generalized for all q values as

(14.222)

where n is an integer satisfying the inequality n - 1 < q 5 n

as

We first consider the q < 0 case We write the Riemann-Liouville definition

and use the convolution theorem

MATHEMATICAL TECHNlQUES WITH DIFFERINTEGRALS 415

Since q - n < 0, from Equations (14.223-225) the first term on the right-hand side becomes s q E { f } When n - 1 - k takes integer values, the term,

dn- 1- k

- &n-I-k [-I dxq-n (O),

under the summation sign, with the q - n < 0 condition and the composition formula [Eq (14.226)], can be written as

dq-1-k dxq- 1-k ' (O), which leads us to

n- 1 dq-1-k k=O

'(O), 0 < q # 1,2,3 (14.229) dxq-1-k

We could satisfy this equation for the integer values of q by taking the condi- tion n - 1 < q 5 n instead of Equation (14.227)

Example 14.4 Heat transfer equation: We consider the heat transfer equation for a semi-infinite slab:

get

a q x , s) sF(x, s) - T ( x , 0) = K

ax2 '

(14.233)

(14.234)

416 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

We now use Equation (14.229) and choose n = 1:

Using the other boundary condition [Eq (14.231)] the second term on the right-hand side is zero; thus we write

MATHEMATICAL TECHNIQUES WlTH DIFFERINTEGRALS 417

Using this in the surface heat flux expression we finally obtain

(14.246)

The importance of this result is that the surface heat flux is given in terms of the surface temperature distribution, that is T(0, t ) , which is experimentally easier to measure

14.6.2 Extraordinary Differential Equations

An equation composed of the differintegrals of an unknown function is called

an extraordinary differential equation Naturally, solutions of such equations involve some constants and integrals A simple example of such an equation can be given as

(14.247)

Here Q is any number, F ( x ) is a given function, and f(x) is the unknown function For simplicity we have taken the lower limit a as zero We would like to write the solution of this equation simply as

(14.248)

dQ dx-Q and - dxQ are not the inverses However, we have seen that the operators -

of each other, unless condition

d-Q

(14.249)

is satisfied It is for this reason that extraordinary differential equations are

in general much more difficult to solve

A commonly encountered equation in science is

418 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"

Solutions of

dnz

dt"

- = (Fa)%, n = 1,2' (14.253)

are the Mittag-Lef€ler functions

which correspond to extrapolations between exponential and power depen- dence A fractional generalization of Equation (14.253) as

is frequently encountered in kinetic theory, and its solutions are given in terms

of the Mittag-Leffler functions as

Ed(%) = -[COS( $6) +cash( s)]

A frequently encountered Mittag-Leffler function is given for the 4 = 1/2

value and can be written in terms of the error function as

E1p(.t&) = exp(z) [I +erf(&&] , z > 0 (14.257)