Application Of Fractional Calculus Analytical And Numerical Technique Philosophy Essay
This preliminary chapter contains a short exposition of the properties and principles that form the substratum of the topic if fractional calculus. It begins with a brief discussion of these topic so that these may be later applied in the application of Analytical and Numerical analysis.
Fractional Calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders (including complex orders), and their applications in science, engineering, mathematics, economics, and other fields. It is also known by several other names such as Generalized Integral and Differential Calculus and Calculus of Arbitrary Order.
The name "Fractional Calculus" is holdover from the period when it meant calculus of ration order. The seeds of fractional derivatives were planted over 300 years ago. Since then many great mathematicians (pure and applied) of their times, such as N. H. Abel, M. Caputo, L. Euler, J. Fourier, A. K. Grunwald, J. Hadamard, G. H. Hardy, O. Heaviside, H. J. Holmgren, P. S. Laplace, G. W. Leibniz, A. V. Letnikov, J. Liouville, B. Riemann M. Riesz, and H. Weyl, have contributed to this field.
During the last decade Fractional Calculus has been applied to almost every field of science, engineering, and mathematics. Some of the areas where it has made a profound impact include viscoelasticity and rheology, electrical engineering, electrochemistry, biology, biophysics and bioengineering, signal and image processing, mechanics, mechatronics, physics, and control theory. One of the major advantages of fractional calculus is that it can be considered as a super set of integer-order calculus. Thus, fractional calculus has the potential to accomplish what integer-order calculus cannot.
Fractional Calculus is the branch of calculus that generalizes the derivative of a function to non-integer order, allowing calculations such as deriving a function to 1/2 order. Despite "generalized" would be a better option, the name "fractional" is used for denoting this kind of derivative.
The generalized theorems and properties of fractional calculus, which shall be put to use in the application of numerical and analytical techniques are summarized below:
The derivative of a function f is defined as
Iterating this operation yields an expression for the n-st derivative of a function. As can be easily seen -and proved by induction- for any natural number n,
The case of n = 0 can be included as well. Such an expression could be valuable for instance in a simple program for plotting the n-st derivative of a function.
There are some desirable properties that could be required to the fractional derivative:-
1. Existence and continuity for m times derivable functions, for any n
which modulus is equal or less than m.
2. For n = 0 the result should be the function itself; for n > 0 integer
values it should be equal to the ordinary derivative and for n < 0
integer values it should be equal to ordinary integration -regardless
the integration constant.
3. Iterating should not give problems,
5. Its characteristic property should be preserved for the exponential function,
The case of the exponential function is specially simple and gives some clues about the generalization of derivatives.
The above limit exists for any complex number . However, it should be noted that in the substitution of the binomial formula a natural number has been considered. Applying this to the imaginary unit,
Solving this system, the next definition for the sine and cosine derivatives can be obtained:
These relations for the sine and cosine derivatives can be maintained in the generalization of the derivative too.
Applying the above method, we can also calculate the following:
Indeed, the above result of the exponential can be applied to any function that can be expanded in exponentials
Expanding the function in Fourier series,
This method can be useful for calculating fractional derivatives of trigonometric functions.
The case of powers of x also has some simplicity that allows its generalization. The case of integer order derivatives
can be easily generalized to non-integer order derivatives
which can be applied to any function that can be expanded in powers of x
Expanding the function in Taylor series:-
or expanding it in Laurent series,
This can be an useful tool for calculating fractional derivatives. However, we should compare these results of powers with the previous results of exponentials to see if they agree. With the result of exponentials,
but, with the result of powers,
If we compare both results, we see that they only agree for integer values of . We shall see later where these discrepancies come from, and how they can be avoided.
1.3 BINOMIAL FORMULA
The exponential function allows the substitution of the binomial formula, but this is not possible for any given function. For applying this substitution we require the following displacement operator,
whose iteration yields,
which allows the application of the binomial formula for natural numbers and the generalized binomial formula for complex numbers,
so that for any complex number can be generalized
This sheds more light on the derivatives and its generalization. Using the expression of the generalized binomial formula for non-integer numbers,
In the case of integer values the summatory only extends terms and it is equal to the ordinary derivative. Finally, it is obvious that as h goes to 0 the last equation is equivalent to the following
1.4 FUNCTIONS OF THE DERIVATIVE
It is worth to mention that being the expression of the -st derivative of a function the derivative itself is
And being the -st derivative or iterating the differentiation times powering it to , applying other functions to the derivative could be also considered. If the function applied to the derivative can be expanded in powers of x,
The result is a weighted sum of different order derivatives. These functions of the derivative are usually known as "formal differential operators". As an example, the exponential of the derivative applied to the exponential would give the following result that could be valuable for calculating functions of the derivative when both f and g can be expanded in exponentials
If both functions f and g can be expanded in positive powers of x,
1.5 GRUNWALD--LETNIKOV DERIVATIVE
Grunwald-Letnikov derivative or also named Grunwald-Letnikov differintegral, is a generalization of the derivative analogous to our generalization by the binomial formula, but it is based on the direct generalization of the equation. The idea behind is that h should approach 0 as n approaches infinity,
being negative infinity. However, the generalization is done so that any wished less than x can be chosen. The above equation can be generalized now to get a formula equivalent to
In fractional derivatives limits must be considered as they must be in integrals. Limits only vanish with integer order derivatives. This also means that fractional derivatives are nonlocal, which may be the reason that makes this kind of derivatives less useful in describing Nature. We shall see later more about the integration -or better say differentiation- limits. In the case of the derivative generalized by the binomial formula, since n goes to infinity regardless of h, x − a must be infinity, so that the derivative defined is equivalent to the Grunwald-Letnikov derivative with a lower limit of negative infinity.
1.6 RIEMANN—LIOUVILLE DERIVATIVE
Riemann-Liouville derivative is the most used generalization of the derivative. It is based on Cauchy’s formula for calculating iterated integrals. If the first integral of a function, which must equal to deriving it to −1, is as follows
the calculation of the second can be simplified by interchanging the integration order
This method can be applied repeatedly , resulting in the following formula for calculating iterated integrals,
Now this can be easily generalized to non-integer values, in what is the Riemann-Liouville derivative,
Note however that in the above formulas the election of 0 as the lower limit of integration has been arbitrary, and any other number could be chosen. Generally, the election of the integration limits in this and other generalizations of the derivative is indicated with subscripts. The Riemann-Liouville derivative with the lower integration limit a would be
Finally, it can be proved that the Grunwald-Letnikov derivative with any given integration limits is equal to the Riemann-Liouville derivative with the same limits for any complex number with a negative real part. This important result means that our derivative generalized by the binomial formula is equivalent to the Riemann-Liouville derivative with a lower limit of negative infinity provided that the real part of is negative,
This kind of Riemann-Liouville derivative with a lower limit of negative infinity is known as Weyl derivative.
1.7 DOMAIN TRANSFORMS
The Laplace and Fourier transforms that serve to transform to the frequency domain can be used to get generalizations of the derivative valid for functions that allow such transformations. The Laplace transform is defined by
while its inverse transform is
where a is chosen so that it is greater than the real part of any of the singularities of f(x). An important property of the Laplace transform is related to the transform of the n-st derivative of a function,
In the case that the terms in the summatory are zero the relation is particularly simple, and for those kind of functions the derivative can be generalized so that this property holds true for non-integer values of
for which the generalized derivative can be defined as
Keeping in mind the result of the generalized derivative of the exponential, the following development provides an easy understanding of the reasons involved in the above generalization,
On the other hand, the Fourier transform is defined by
while its inverse transform is
This transform also has an analogous property related to the transform of the n-st derivative of a function,
and the derivative can be generalized so that this property holds true for non-integer values of
yielding the following definition of the generalized derivative
This provides an easy understanding of the reasons involved in the generalization,
In these two generalizations the implicit limits of differentiation should be determined. In the case of Laplace transform, the generalized derivative is a Riemann-Liouville derivative with a lower limit of 0, whereas in the case of Fourier transform it is a Weyl derivative. Indeed, if we check for instance the derivative generalized by the Fourier transform in the cases of the sine and cosine functions -calculated in with the generalized derivative of the exponential that we have seen in that also is a Weyl derivative- we will find that they match perfectly.
1.8 CAUCHY INTEGRAL FORMULA
Another way of generalizing the derivative to non-integer order is given by the Cauchy integral formula that plays a key role in complex analysis,
Despite its generalization to any complex number seems straightforward, it must be taken into account that while being n integer there is an isolated singularity at t = z, being it non-integer there is a branch point, what means that the integration contour has to be chosen carefully. Otherwise, the generalization only involves changing the factorial to the gamma function, so defining what is known as the Cauchy-type fractional derivative
where supposing that the branch line starts at t = z and passes through z0, the contour C starts at t = z0, encircles t = z once in the positive sense and returns to t = z0 where now the integrand has a different value. It can be proved that this generalization of the derivative is equivalent to the Riemann-Liouville derivative with a lower limit of z0 for the appropriate values of in which both derivatives are defined.
The generalizations of the derivative suggest that they can be formulated in terms of the convolution, which would be important for the convolution is a very simple operation in the frequency spaces achieved by Laplace and Fourier transforms. The following development shows how this is the case, and how after all the derivative of a function is its convolution with certain function,
which Laplace convolution with f(x) yields the Riemann-Liouville derivative of order −
And since the transform of the function expressed gives the following simple result
and the transform of the convolution is the multiplication of the transforms, in the case that the function f fulfills all the requirements given previously for the simplicity of the Laplace transform of the derivative, we again get the equation which was the key for the generalization by Laplace transform
These last results showing that the generalized derivative is a convolution with certain function, open the interesting question of what other kind of operators would be defined if functions other than that had been chosen. The answer is that the new operators defined would be functions of the derivative. To see this, we can exploit the linearity of the convolution, supposing that the function g can be expanded in powers of x
This proves the equivalence between functions of the derivative and convolutions.
For any convolution there is an equivalent function of the derivative if the function implied can be expanded in powers of x and vice versa. Now, the Laplace transform shows what was expectable,
since the linearity of the Laplace transform combined with the earlier result shows clearly that
It also shows for the Fourier transform that
These are useful tools for the calculation of functions of the derivative. As an example, the following case is considered with the help of Fourier transforms
and directly as
yielding both methods the same result. This case also matches when calculated with Fourier transforms.
1.10 LOCAL OPERATORS
Despite of some interesting results, some important questions about the functions of the derivative are left unanswered. For instance, it would be very interesting to know if integer order derivatives are or not the only kind of functions of the derivative that are local, and if are or not other kind of local operators that are not functions of the derivative.
It seems that in fact the only local functions of the derivative are integer order derivatives and their finite sums, being nonlocality the result of adding infinite terms with displacement functions steadily increasing the distance to the point in which the function is evaluated. However, this impediment can be overcome by defining the derivative in the following way
so that the limit assures locality. In order to avoid another feature of fractional derivatives that sometimes is not wanted -the fact that the derivative of a constant is not zero- it is usual to define the local fractional derivative as
or similar forms. So there can be operators being local and taking an infinite number of evaluations of the function, provided that the points in which these evaluations are carried out -or the value of their displacement functionremains being infinitesimal. The question now is if there are other local operators besides the local fractional derivative and functions of it.
It could be argued that if a function admits its expansion in Taylor series, all the information about the function is indeed in its derivatives, so that any operator giving information about the function will be unnecessary and reducible to a sum of integer order derivatives. However, on the one hand there are functions that are not differentiable but are local fractional differentiable, and on the other hand, even if local operators could be reduced to ordinary derivatives for differentiable functions, that does not assure that ordinary derivatives are the most appropriate local operators for all the tasks.
Characterizing all local operators is important, for locality makes of them tools that could be useful in understanding Nature.
Fractional derivatives satisfy quite well all the properties that one could expect from them, despite some of them are only characteristic of integer order differentiation and some other have restrictions. For instance, the property of linearity is fulfilled, while that of the iteration has some restrictions in the cases that positive differentiation orders are present.
Some properties that include summatories can be generalized changing the summatories into integrals. One such property is the expansion in Taylor series
and the other is the Leibniz rule
These and other generalized properties can be applied to the study of special functions, which often can be expressed in terms of simple formulas involving fractional derivatives. For instance, Gauss’s hypergeometric function can be expressed as
Linear Differential Equations Of Fractional Order
In this section, a summary of the known results is given for the following equation.
The behaviour of the solution of above equation for the case ν € (0,1] is qualitatively different to the behaviour of its solution for the case ν € (1,2]. Therefore the two cases are considered separately.
The case ν € (0,1] is dealt with in section 2.1. We first consider the solution in the limiting case of ν=1. Secondly, we solve it for the case ν € (0,1] using the Laplace transform, this solution is of limited usefulness because it requires an initial condition that is not readily available. Thirdly, we solve it in the case where the initial value of the function is specified.
The case ν € (1,2] is dealt with in section 2.2. We first consider the solution in the limiting case of ν=2. Secondly, we solve it for the case ν € (1,2] using the Laplace transform, this solution is of limited usefulness because it requires two initial condition that involve fraction operators. Thirdly, we solve it in the case where the initial value of the function and its derivative is specified.
2.1 LINEAR FRACTIONAL O.D.E’s OF ORDER v € (0,1]
2.1.1 Limiting Case
The limiting case is of order 1 and has the form
Taking the Laplace transform of this equation gives
rearranging this for gives
Therefore the general solution is
2.1.2 Homogeneous Fractional O.D.E
If v € (0,1), p>0 we have the homogeneous fractional linear O.D.E
Using the Riemann-Liouville Definition of the fractional derivative we have:
Taking the Laplace transform of both sides of (2.1.2) yields
Now, taking the Laplace transform of the Generalized Mittag-Leffler function with α=v and β=v we have
Since x(t) is continuous on (0,∞) we can conclude that this is the unique solution to (2.1.3)
2.1.3 Inhomogeneous Linear Fractional O.D.E. of Order v€(0,1]
In this case the equation has the form
which, by the convolution property for Laplace transforms, introduces the additional term
into the solution so that it becomes
which is uniquely determined upto
2.1.4 Incorporated Initial Conditions
The above solution has the disadvantage of requiring the value of the derivatives of fractional integrals as t →0+ as initial conditions. To avoid having to specify the value of a fractional derivative, as t →0+, as an initial condition we can incorporate x0 in to the original equation. This gives an equation of the form
where t≥0, p>0, v € (0,1], x(0)=x0.
The solution to (2.1.11) is
Taking the Laplace transform of the second representation gives us
Rearranging (2.1.13) gives
Obtaining the inverse Laplace transform of (2.1.14) we have
2.2 Linear Fractional O.D.E’s of Order v € (1,2]
The limiting case is of order 2 and has the form
Taking the Laplace transform of this equation gives
Therefore the general solution is
2.2.2 Inhomogeneous Linear Fractional O.D.E of Order v € (1,2]
This is equivalent to
Taking the Laplace transform of (2.2.4) yields
Obtaining the inverse Laplace transform of (2.2.5) yields
Equivalently we can write (2.2.6) as
2.2.3 Incorporated Initial Conditions
Incorporating the initial conditions into the second order case we have
where t>0, p>0, v € (0,1], x(0)=x0, x’(0)=x’0
Assuming the solution to (2.2.8) is once differentiable and Laplace transformable then we have
Taking the Laplace of (2.2.9) yields
The inverse Laplace transform of (2.2.11) is