MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 12 doc

OTHER DEFINITIONS OF DIFFERINTEGRALS 393 z-phe t Fig. 14.3 Contour C' = C + Co + LI + Lz in the differintegral formula which goes to zero in the limit 60 + 0. For the CO integral to 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 to evaluate the [ f+L, - f +Lz 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 $, dz integral we get (:!$I (14.82) After taking the limit we substitute this into the definition [Eq. (14.74)] to obtain (14.83) Simplifying this we can write (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) dqf(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. 14.3.2 Riemann Formula We now evaluate the differintegral of f (x) = xp, (14.88) which is very useful for finding differintegrals of functions the Taylor series of which can be given. Using formula (14.84) we write SP d6 (14.89) dxq ?I- s 0 (6-x)4+1 dqxp r(q + l) sin(?I-q)(-1)9 and We define - -s 6 - X 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 dqxp r(p + 1)xP-q dxq p > -1 and q < 0. - r(p + 1 - q) ' (14.90) (14.91) (14.92) (14.93) (14.94) (14.95) (14.96) 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 (14.98) Combining this with the results in Equations (14.96) and (14.98) we obtain a formula valid for all q as dqxP r(p + 1)xP-q - p>-1. dxq r(p - q + 1) ' (14.99) This formula is also known as the Riemann formula. It is a generalization of the formula m! - xrn-", 6"X" dxn (m-n)! (14.100) 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 dqf - -E-'[sq&] , q < 0, dxq (14.101) where F(s) is the Laplace transform of f(x). To see that this agrees with the Riemann-Liouville definition we make use of the convolution theorem In this equation we take g(x) as (14.103) OTHER DEFINITIONS OF DIFFERINTEGRALS 397 where its Laplace transform is (14.104) = r( q)s*, (14.105) and also write the Laplace transform of f(x) as For q < 0 we obtain ["'I = -c-"sq&)] , q<o. dxq (14.106) (14.107) (14.108) The subscripts L and R-L denote the method used in evaluating the differin- tegral. Thus the two methods agree for y < 0. For q > 0, the differintegral definition by the Laplace transforms is given as (Section 14.6.1) or dq-lf dxg- 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) = - q-n<O, r(n - q) Jx ,, (z - f(r)dr T)Q-"+~ ' (14.114) (14.115) (14.116) As in the Griinwald and Riemann-Liouville definitions we assume that the [ ]L definition also satisfies the relation [Eq. (14.40)] ( 14.117) where n a is positive integer and q takes all values. We can now write which gives us Similarly we find the other terms in Ekpation (14.110) to write (14.118) (14.119) (14.120) 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 (14.123) we can write a differintegral definition valid for all q values via the Laplace transform as -= dqf -E-l[sQ~(s)]. dxq (14.124) However, because the boundary conditions (14.123) involve fractional deriva- tives this will create problems in interpretation and application. (See Problem 14.7 on the Caputo definition of fractional derivatives.) 14.4 PROPERTIES OF DIFFERINTEGRALS In this section we see the basic properties of differintegals. These properties are also useful in generating new differintegrals from the known ones. 14.4.1 Linearity We express the linearity of differintegrals as d9f2 + d9[fl + f2l - d4f1 - [d(x - ~)]q [d(x - ~)]q [d(X - ~)]q* (14.125) 14.4.2 Homogeneity Homogeneity of differintegrals is expressed as Co is any constant. (14.126) dQ(Cof) =Co dqf [d(z - ~)]q [d(x - ~)]q ' Both of these properties could easily be seen from the Griinwald definition [Eq. (14.39)) 400 FRACTIONAL DERIVATIVES AND INTEGRALS. "DIFFERIN TEGRALS" 14.4.3 Scale Transformation We express the scale transformation of a function with respect to the lower limit a as f + f(rz - ya + 4, (14.127) where y is a constant scale factor. If the lower limit is zero, this means that f(4 -+ f(r.)- (14.128) If the lower limit differs from zero, the scale change is given as dqf(yX) x = z + [a - ay]/y (14.129) dQf(yX) - - [d(z - a)]q "[d(yX - a)]Q' This formula is most useful when a is zero: (14.130) 14.4.4 Differintegral of a Series 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 dQ c O0 - a]P+(j/n) = I- (pn +: + [. - a]P-q+(.i/n) 00 (14.132) where q can take any value, but p + (j/n) > -1, a0 # 0, and n is a positive integer. ) [d(z - .)I" j=o zaj I- ( pn - q;+ j + n 14.4.5 Composition of Differintegrals When working with differintegrals one always has to remember that operations like dqd& = dQdQ, dQ& = dq+Q and d9f =g+ f =d-qg (14.133) PROPERTIES OF DIFFERINTEGRALS 401 are valid only under certain conditions. In these operations problems are not just restricted to the noninteger values of q and Q. When n and N are positive integer numbers, from the properties of deriva- tives and integrals we can write d" dN f dn+N f [d(z - a)]" { [d(z - a,].} = [d(z - a)]n+N (14.134) - - and f (14.135) d-n-N - However, if we look at the operation [d(z d*n - a)]*" { [d(~ diNf - .)ITN 1, (14.136) the result is not always d*niN 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- f (N- yu). (n - l)! 402 FRACTIONAL DERIVATIVES AND INTEGRALS- "DIFFERINTEGRALS" Since we write (14.142) Writing Equation (14.142) for N = 0 gives us n- 1 k! k=O [d(z - a)]-" (14.143) We differentiate this to get f("-")(a). (14.144) [x - a]k- n- 1 (k - l)! k= 1 After N-fold differentiation we obtain f('"-")(u). (14.145) I. - n- 1 (k - N)! k=N For N 2 n, remembering that differentiation does not depend on the lower limit and also observing that in this case the summation in Equation (14.145) is empty, we write dN-nf = f(N-")(z). (14.146) On the other hand for N < n, we use Equation (14.143) to write dN-n n-N-1 (a). (14.147) k! [d(z - f3)lN-n k=O This equation also contains Equation (14.146). In Equation (14.145) we now make the transformation k+k+N (14.148) to write n-N-1 (a)- k! (14.149) [...]... 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... essentially carries information about the delays and the traps present in the system Thus, in a way, memory effects are introduced to the random walk process These theories are called continuous time random walk (CTRW) theories In CTRW, if the integral r =Stq(t)dt, (14.300) 426 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS" r~ 3 '2 Fig 14.6 Random walk and CTRW that gives the average waiting time of... 14.130)] and the semiderivative given in Section 14.5.7 Substituting the above equation into Equation (14.261) gives C + Cexp(a2z) erf(&) - I 2a&C1 exp(a2x) (14.273) thus we obtain a relation between C and C'l as r( L, - = 2aC1 (14.274) J?-; Now the final solution is obtained as (14.275) 14.6.5 Evaluating Definite Integrals by Differintegrals We have seen how analytic continuation and complex integral... solution involves two integration constants and a divergent integral However, this integral can be defined by using the incomplete gamma function y*(c,x), which is defined as 1‘; -y*(c,x)= c-x W ) 0 x’”-l exp(-x’)dx’ (14.265) (14.266) where y*(c,x) is a singlevalued and analytic function of c and x Using the relations y * ( c - 1,x) = zy*(c,z) +eXP(-X) r(c) (14.267) and (14.268) we can determine the... (14.299) In Figure 14.6, the first figure shows the distance covered by a Brown particle In Brownian motion or Einstein random walk, even though the particles are hit by the fluid particles symmetrically, they slowly drift away from the origin with the relation (14.299) In Einstein's theory of random walk steps are taken with equal intervals Recently theories in which steps are taken according to a waiting... behaves like the second curve in Figure 14.7, which has a cusp compared t o a Gaussian APPLICATIONS OF DIFFERINTEGRALS IN SCIENCE AND ENGINEERING 427 Fig 14.7 Probability distribution in random walk and CTRW An important area of application for fractional derivatives is that the extraordinary diffusion phenomenon, studied in CTRW, can also be studied by as the differintegral form of Equation (14.294)... obtain the formula -+ p ( p is positive but this is very useful in the evaluation of some definite integrals As a special case we may choose x = 1 t o write (14.283) Example 14.6 Evaluation of some definite integrals by diflerintegrals: Using differintegrals we evaluate l ' e x p ( 2 - 2t2I3)dt (14.284) Using formula (14.281) with x = 2 and p = 2/3 along with Equation (14.215) we find n (14.285) In. .. 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 + r(j- q)/I'(-q)I'(j I) = Using the properties of gamma functions; Cy=il r ( N - q)/r(l- q ) r ( N ) ,and limN,oo[N*r(N - q... 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 t o 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 k-l . Writing the remaining part of the $, dz integral we get (:!$I (14.82) After taking the limit we substitute this into the definition [Eq. (14.74)] to obtain (14.83) Simplifying this. possible to define a left-handed Riemann-Liouville differintegral as 408 FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS" where k is again an integer satisfying Equation. 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