Introduction of Derivatives and Integrals of Fractional Order and Its Applications

Mehdi Delkhosh

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Introduction of Derivatives and Integrals of Fractional Order and Its Applications

Mehdi Delkhosh

Islamic Azad University, Bardaskan Branch, Department of Mathematics, Bardaskan, Iran

Abstract

Fractional calculus is a branch of classical mathematics, which deals with the generalization of operations of differentiation and integration to fractional order. Such a generalization is not merely a mathematical curiosity but has found applications in various fields of physical sciences. In this paper, we review the definitions and properties of fractional derivatives and integrals, and we express the prove some of them.

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Cite this article:

  • Delkhosh, Mehdi. "Introduction of Derivatives and Integrals of Fractional Order and Its Applications." Applied Mathematics and Physics 1.4 (2013): 103-119.
  • Delkhosh, M. (2013). Introduction of Derivatives and Integrals of Fractional Order and Its Applications. Applied Mathematics and Physics, 1(4), 103-119.
  • Delkhosh, Mehdi. "Introduction of Derivatives and Integrals of Fractional Order and Its Applications." Applied Mathematics and Physics 1, no. 4 (2013): 103-119.

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1. Introduction

1.1. History of Creation

In 1695, Hopital wrote a letter to Leibniz, to this content which, how can you justify when? Leibniz wrote in answer, and suggested a close relationship between derivatives and infinite series (divergent), and he continues: “ is equal to .”, And stated: "This is an obvious paradox that someday something good will result." [21].

In 1819, Lacroix devoted two pages to the discussion of the derivative of arbitrary order, in his book with seven hundred pages. He showed that if then . In particular, this result obtained that [18].

In 1823, Abel provided the first application of the fractional calculus in physical problems, and of course he did not solve this problem. (Tautochrone: It is to determine a curve so that an object under the effect of gravity on it tumbles without friction, the time motion is independent of the starting point) [2].

In 1884, Laurent presented a theory of generalized operators with where is real, and he was confronted with the differentiation and integration of the arbitrary order [19].

In 1892, Heaviside applies derivatives of fractional order in the theory of transmission lines [12].

In 1936, Gemant continued theory of transmission lines of Heaviside and he used the fractional derivatives in elasticity [11].

In 1974, Ross held the “Proceeding of the International Conference on Fractional Calculus and Its Applications ", and report it published in a book [31].

In 1993, Kenneth Miller and Ross published the book: "an Introduction to the Fractional Calculus and Fractional Differential Equations". This book provides good methods [25].

In 1997, Kolwankar, in his doctoral thesis, paid to studies of fractal structures and processes using methods of fractional calculus [15].

In 2000, Hilfer published a book in 9 chapters: "Applications of Fractional calculus in Physics " [13].

In 2003, Falconer published a book in 18 chapters: "Fractal Geometry - Mathematical Foundations and Applications " [10].

Given clarify the importance and numerous applications of fractional derivatives and integrals, in recent years many articles and books on this subject have been published [3, 4].

1.2. Gamma Function

Let be factorial function. In this case, for any positive integer , we have:

(1-1)

Gamma function for any positive is defined as follows:

For the Gamma function can be demonstrated that:

(1-2)

Now, according to equations (1-1) and (1-2) we have:

For example,

With the change of variable :

So we have:

Also, for negative values of x, we use the following definition:

For example,

It can be proved that (which is proved by Weierstrass) [1]

Euler for the Gamma function provided another defined as follows:

Chart for Gamma function is plotted in Figure 1-1. So, we can accept that for :

1.3. Incomplete Gamma Function

Incomplete Gamma function,, is defined as follows [1]:

1.4. Beta Function

Beta function is defined as follows [1]:

And it has the following properties:

1-

2-

3-

1.5. Incomplete Beta Function

Incomplete beta function, , is defined as follows [1]:

2. Integral of Fractional Order (Fractional Calculus)

The derivative operator for all those who have studied the normal calculations is introduced. And we know the derivation of -order of function , i.e. where is a positive integer. (If is a positive integer, then is integral of -order of the function ).

Now the question that arises is: If we have , where is an arbitrary number (rational, irrational or complex), What is the value of ?

Fractional calculus was used initially for solving the Abel integral equations. But later found many applications in different sciences. Such as: Tautochrone, calculation of Heaviside’s operator in physics, Fluid flows and the design of Weir Notch in Civil and Mechanics, Analysis diagrams earthquake in geology and et al.

There are two main methods for defining fractional calculus: One using of integrals and other using of the series. Any method has the features, applications and disadvantages of owning.

2.1. Definitions by Integrals for Integral of Fractional Order

Definition 1. Definition of Riemann [25]

Where indices in indicate, : order of fractional integrals, : the lower limit of integration, and: the upper limit of the integral.

Definition 2. Definition of Liouville [25]

Definition 3. Definition of Weyl [25]

where is very suitable for periodic functions.

Definition 4. Definition of Riemann - Liouville [25]

Let and is a point wise continuous function on and is integral able on any finite interval of . Then for all , we have:

The set of all functions that satisfy in this definition, we show with , and we define, .

2.2. Definitions by Series for Integral of Fractional Order

One of the important definitions for fractional calculus by series, as follows:

Definition 5. Definition of Grunwald [25]

Grunwald defined fractional integral of the function at the point as follows:

We can write it, for each , as follows [15]:

where is Lower limit of integration and .

Now, we show that the above definitions can be obtained with different methods, which shows the importance of these definitions and ideas for a variety of definitions.

2.3. Verification Methods Definitions of Integral of Fractional Order
2.3.1. Method of Iterative Integrals

We consider the ordinary integrals of n-tuple as follows [25]:

(2-1)

Where and is continues function on .

If , we have

By repeating this procedure -times, (2-1) becomes:

(2-2)

It is clear that the right side (2-2) for each where is significant. Thus

That is the definition of Riemann.


2.3.2. Method of Differential Equations

Let is a linear derivative operator, where the coefficients are continuous functions on an interval . Also let is continues on and .

We consider the differential equation:

(2-3)

Unique solution of the equation (2-3) for every is as follows:

(2-4)

where is the Green's function corresponding to [24].

If is any basic set of solution of homogeneous equation , then Green’s function can be written as:

Where is Wronskian:

Now, we assume that , then (2-3) can be written as follows:

(2-5)

And is any basic set of solution of homogeneous equation , thus Green’s function is:

And Wronskian is:

Where is independent of .

We see that can be transformed into a polynomial of degree with a leading coefficient:

We know according to properties of the Green function that:

So, is a zero of order for . Thus

Thus, from the equations (2-4) and (2-5) we have:

Where for each number, where the real part is positive, it is meaningful.

Thus, we obtain the definition of Riemann.

It should be noted that the same result can be achieved by using the Laplace’s transform, i.e. if equation (2-5) becomes

Where and are Laplace’s transforms of and . Thus

And by Convolution Theorem, we have [8]:

Thus, we obtain the definition of Riemann - Liouville.


2.3.3. Definition of Weyl

Let .

The adjoint equation for is as follows:

Then, the solution of non-homogeneous equation

is

(2-6)

Where is the Green's function corresponding to adjoint operator [24].

However, if is the Green’s function of , then [24]:

Thus, equation (2-6) can be written as:

Now, we assume that , then , thus:

Where for each number, where the real part is positive, it is meaningful.

Thus, let is and , then

Thus, we obtain the definition of Weyl.

2.4. Some Examples of Integral of Fractional Order

Example 1:

1-

Where . In particular, if , integral of fractional order for constant is:

2-

3-

Where is the incomplete Gamma function. Mainly, fractional integral of an exponential function to represent the following: [25]

And the same method can obtain: [25]

4-

5-

To calculate other trigonometric functions, we can be used of relations, for example,

6-

7-

8- Let where is a positive number and . Then

Where is incomplete Beta function. In the special case, if then

9- Let is a non-negative integer number

If we consider:

And we put in the above equation, we have:

Where it’s a special case of Leibniz’s formula, the general form is shown later.

In all the examples above, we hypothesized that the lower limit of the integral is equal to zero, the definition of Riemann - Liouville, if that were necessary case where the lower limit of the integral is nonzero, we can use the following technique:

By changing the variable , we have:

Where

10-

11-

12-

13-

In all cases, if , the previous equations are obtained.

3. Derivative of Fractional Order (Fractional Calculus)

The derivative operator for all those who have studied the normal calculations is introduced. And we know the derivation of -order of function , i.e. where is a positive integer.

Now, we describe the case where is an arbitrary number in .

There are two main methods for defining fractional calculus.

3.1. Definitions by Integrals for Derivative of Fractional Order
3.1.1. Using of the Integral of Fractional Order

Definition 6. Let , and , then and derivative of fractional of order , for is

Where is integral of fractional order.

Example 2: if and then , and we have

(3-1)

If and then

Since , thus

(3-2)

By comparing equations (3-1) and (3-2) to conclude that

(3-3)

For and .

To equation (3-3) we are saying Differintegral, Namely, if is a positive number, operation is derivative (differential) and if is a negative number, operation is integration.

Remark 1: if

(3-4)

is definition of Riemann and and is continues function, then derivative of fractional order is exists. However, this does not guarantee the existence of fractional derivatives. For example, let is continues function but it’s not differentiable (Such as the Weierstrass function), and , then

Now, if then (as ), by equation (3-1) we have

But by assumption, is not differentiable.

On the other hand, if is -time differentiable, then the equation (3-3) has existed for all , To prove this, By changing the variable in equation (3-4), where , we have

And

Or

Where it exists for each , Since is a continues by assumption.

Therefore, this method is useful in many cases, but in the general case, it’s not efficient for functions that are not differentiable of ordinary order.


3.1.2. Definitions of Direct

Definition 7. Definition of Riemann [25]

Definition 8. Definition of Riemann - Liouville [25]

Definition 9. Definition of Marchaud [25]

Example 3: Let then

By changing the variable , we have

So after simplification, we have:

3.2. Definitions by Series for Derivative of Fractional Order

Definition 10. Definition of Grunwald [25]

Grunwald defined fractional derivative of the function at the point as follows:

(3-5)

We can write it, for each , as follows [6, 7, 15, 20]:

(3-6)

Where is Lower limit of derivative.

In the definition of Grunwald, We can show that if is a positive number such as , then the equation (3-5) by using

Convert to

Now, we assume , thus

To approximate the derivative of fractional order, on the large , we have:

4. Properties Fractional Calculus

Now, we express some properties of the operators of differentiation and integration of arbitrary order. [15, 25, 27, 32].

4.1. Linear and Homogeneous

From the definitions introduced in the previous, it is clear that the differintegral is held on the following property:

4.2. Change of Arguments

When the argument is changed by a factor, Differintegral is held on the following property: [27]

4.3. The Chain Rule

If is an analytic function and is a sufficiently differentiable function, then the chain rule for fractional derivatives is as follows [27]:

Where can be extended on all nonnegative integer combinations of , such that and .

This is a generalization of the chain rule for ordinary derivatives, but its nature is complex.

4.4. Differintegral by Parts

First consider the case . Let is uniformly convergent on the , where . Then [27]

Also, right hand side is uniformly convergent on .

For , if and are uniformly convergent on the , then

On the .

4.5. Taylor Series

Osler [28] generalized Taylor series of fractional derivatives. He has proven a very general result. We discuss a special case. This case called the – Riemann series.

Where and is an analytic function.

4.6. Composition’s Rule

Now, we are studying the relationship between these two operators:

and

When the , we have:

And Composition’s Rule is clear.

When the , Composition’s Rule is true if

And if it's not true, we have

We can be shown that when , Composition’s Rule is true for any differintegral on function . In fact, if is finite in , then the Composition’s Rule is valid for .

4.7. Dirichlet’s Formula and Extension’s Rule

We know that if is a continuous function on , we have:

To show the Dirichlet formula, we assume that is a continuous function and are positive numbers, then

Example 4: if and , then we have

In addition, if then we have

Where is Beta function.

As a useful application of Dirichlet's formula, we express the extension’s rule for fractional integrals:

Theorem 1: Let is a continues function on and . Then for any [25]

Proof: By definitions, we have

Theorem 1 is established for (or ) If is an operator identity.

Example 5: If , Since is a continues function, we have

And we have and . So,

We obtained the following useful recursive relation:

(Assuming strong )

And, in the same method, we can obtain:

4.8. Derivatives of fractional Integrals and Integrals of Fractional Derivatives

Theorem 2: Let is a continues function on and . Then [25]

a) If is in class , then

b) If is a continues function on . Then for any

Proof: a) let and . Then the derivatives of and are continuous functions on . Therefore, the integration by parts shows that:

We take the limit when tended to zero, and we divided into , then part (a) is proofed.

b) By changing the variable , where , we have

Then for any

Changing the variable will be returned, and then part (b) is proofed.

Example 6: If , According to (a) in theorem 2, we have

By using Example 1 part 3, we have

i.e.

The result is a recursive relation for . Similarly, we can obtain:

Theorem 3: Let is a positive number and is a continues function on and . Then [25]

a) If is in class , then

b) If is a continues function on . Then for any

Where

Theorem 4: Let is a positive number and and has a continuous derivative on . Then for any [25]

Theorem 5: Let are positive integer number and are positive number such that and has -time continuous derivatives on where . Then for any

(4-1)

Where and is sign function. And for any

(4-2)

Proof: if it’s clear. Let , and therefore . By theorem 3, we have:

Since and then the equation (4-1) is true.

For proof equation (4-2), by theorem 4, we have

With-time derivative, we have:

Since and then equation (4-2) is true.□

4.9. Laplace Transforms of the Integral of Fractional Order

We say the function is of exponential order, If there are positive constants such as such that for any

If is in class and it’s of exponential order, then

is exist, for any such that . we call the Laplace transform of , and write:

And also write:

i.e., is the inverse Laplace transform (unique) of .

Functions and are in class , and they are of exponential order. So

One of the very useful properties of the transform is in the Convolution theorem [8]. The theorem shows that the Laplace transform of the convolution of two functions is multiplication the transforms of them. i.e.

Now, if is in class , then fractional integral of of order :

is a Convolution. Therefore, if is of exponential order then

(4-3)
(4-4)

Equation (4-4) is true for , but the equation (4-3) is not true, although

(4-5)

Example 7:

Now, we try to calculate the Laplace transform of fractional integral of the derivative and the Laplace transform of derivatives of fractional integral.

Let is continues function on and is in class and it’s of exponential order, then by equation (4-4) we have

(4-6)

Since, is continues function on then is exist. Therefore, we obtained the Laplace transform of fractional integral of the derivative. This formula for is obvious.

By theorem 2 part (b), we have

(4-7)

Therefore, we obtained the Laplace transform of derivatives of fractional integral.

By equation (4-6), let , we have , but by the equation (4-7), we have . This "unconformity" occurs from this fact that:

(4-8)

By comparing equations (4-5) and (4-8), we conclude that the "" and "" are not commutative.

Thus, we can show that:

4.10. Leibniz’s Formula

Leibniz classical formula (i.e. ) is as follows:

where and are -time differentiable functions.

Now, we want to express the Leibniz formula for fractional operators.

Theorem 6: Let is a continues function on and is an analytic function for any . Then for any and , we have [25]

Osler, a general result, proved to differintegral, where its expressed as follows: [28]

where is arbitrary constant.

5. Fractional Differentiable

Now, we want to express some necessary conditions for that the fractional derivative of a function is exist [15, 30].

Definition 11: is said to have an th derivative, where for integer , if (in the Weyl sense), with , has Peano derivatives at , That is, there exists a polynomial of degree s.t.

Further if

is said to have an th derivative in the sense.

It is clear that this definition of fractional differentiability is not local. Particularly the behavior of the function at also plays a crucial role. The main results can be stated using this notion of differentiability and involve the classes and which are given by the following definitions.

Definition 12: If there exists a polynomial of degree s. t. as then is said to satisfy the condition , Or in simpler terms, is the set of all functions that satisfy in the Holder exponents of an order exponential.

Definition 13: If and Further

is said to satisfy the condition .

Definition 14: is said to satisfy the condition if for some

Now, we express the main results in four theorems (without proof) [15, 30].

Theorem 7: Suppose that satisfies the condition at every point of a set of positive measure. Then exists almost everywhere in if and only if satisfied condition almost everywhere in .□

Theorem 8: The necessary and sufficient condition that satisfies the condition almost everywhere in a set is that satisfies the condition and exists in the sense, almost everywhere in this set.□

Theorem 9: Let , and suppose that . Then if . □

Theorem 10: Let Then if .□

Despite their merits, these results are not really suitable and adequate to obtain information regarding irregular behavior of functions and Holder exponents. We observe that the Weyl definition involves highly nonlocal information and hence is somewhat unsuitable for the treatment of local scaling behavior.

6. Local Fractional Derivatives (LFD)

The definition of the fractional derivative was discussed in the last chapter. These derivatives differ in some aspects from integer order derivatives. In order to see this, one may note, from definition 6, that except when is a positive integer, the th derivative is nonlocal as it depends on the lower limit ‘’. The same feature is also shown by other definitions. However, we wish to study local scaling properties and hence we need to modify this definition accordingly. Secondly from example 1 part 2, it is clear that the fractional derivative of a constant function is not zero. Therefore adding a constant to a function alters the value of the fractional derivative. This is an undesirable property of the fractional derivatives to study fractional differentiability. While constructing local fractional derivative operator, we have to correct for these two features. This forces one to choose the lower limit as well as the additive constant before hand.

The most natural choices are as follows.

1) We subtract, from the function, the value of the function at the point where we want to study the local scaling property. This makes the value of the function zero at that point, canceling the effect of any constant term.

2) The natural choice of a lower limit will again be that point itself, where we intend to examine the local scaling.

Definition 15: If, for a function , the limit

is exists and finite, then we say that the local fractional derivative () of the order (denoted by ), at , exists. [15]

This defines the for . We generalized to include all positive values of as follows. [10]

Definition 16: If, for a function , the limit

exists and is finite, where is the largest integer for which the derivative of at exists and is finite, then we say that the local fractional derivative () of the order (), at , exists.

Definition 17: The critical order , at , of a function is

Sometimes it is essential to distinguish between limits, and hence the critical order, taken from above and below. In that case we define

We will assume unless mentioned otherwise.

The local fractional derivative that we have defined above reduces to the usual derivatives of integer order when is a positive integer, where this study is simple.

It is clear that in our construction local fractional derivatives generalize the usual derivatives in fractional order keeping the local nature of the derivative operator intact, in contrast to other definitions of fractional derivatives. The local nature of the operation of derivation is crucial at many places, for instance, in studying the differentiable structure of complicated manifolds, studying evolution of physical systems locally, etc. The virtue of such a local quantity will be evident in the following section where we show that the local fractional derivative appears naturally in the fractional Taylor expansion. This will imply that the LFDs are not introduced in an ad hoc manner merely to satisfy the two conditions mentioned in the beginning, but they have their own importance.

6.1. Local Fractional Taylor Expansion

In order to derive local fractional Taylor expansion, let [9, 15]

It is clear that

Now, for

provided the last term exists. Thus

(6-1)

i.e.

(6-2)

where is a remainder given by

Equation (6-2) is a fractional expansion of involving only the lowest and the second leading terms. Using the general definition of and following similar steps one arrives at the fractional expansion for (provided exists), given by,

(6-3)

Where

We note that the local fractional derivative (not just fractional derivative) as defined above provides the coefficient in the approximation of by the function , for , in the vicinity of . We further note that the terms on the RHS of equation (6-1) are nontrivial and finite only in the case .

Let us consider the function , where , Then and using equation (6-3) at we get since the remainder term turns out to be zero.

6.2. Geometrical Interpretation of LFD

Clearly, integrals and derivatives of the integer order having simple geometrical and physical interpretation, which can be used to solve application problems in various sciences. But, in integrals and derivatives of the fractional order, it is not easy [15, 25]. The idea of integrals and derivatives of arbitrary order (not necessarily an integer), specific geometric and physical interpretation for these operators, to more than 300 years was not expressed. The lack of this interpretation, in the First International Conference for Fractional Calculus in the New Haven U.S. in 1974 led to that this question be considered as an open problem [32], and the future conferences, such as the University of British Astrakly in 1984 [23] and Nihon University in Japan in 1990 [26], this question is not answered. Although, there were various discussions on this issue, the problem was not solved until 1996. However, later that year, fairly satisfactory answers were given to this question [14, 22, 29].

Whereas, the local fractional Taylor expansion of section 6.1 suggests a possibility of such an interpretation for s. In order to see this note that when is set equal to unity in the equation (6-2) one gets the equation of the tangent. It may be recalled that all the curves passing through a point and having the same tangent form an equivalence class (which is modeled by a linear behavior). Analogously all the functions (curves) with the same critical order and the same will form an equivalence class modeled by (If f differs from by a logarithmic correction then terms on the RHS of equation (6-1) do not make sense like in the ordinary calculus). This is how one may generalize the geometric interpretation of derivatives in terms of ‘tangents’. This observation is useful when one wants to approximate an irregular function by a piecewise smooth (scaling) function.

6.3. Generalization to Higher Dimensional Functions

The definition of the Local fractional derivative can be generalized for higher dimensional functions in the following manner [15, 16, 17].

Definition 18: Let , We define

Then the directional- of at of order , , in the direction is given (provided it exists) by

where the RHS involves the usual fractional derivative of equation (3-6). The directional s along the unit vector will be called th partial-.

7. Applications of Fractional Calculus

7.1. Abel's Integral Equation and the Tautochrone Problem

Abel was the first to solve an integral equation by means of the fractional calculus. Perhaps even more important, our derivation below will furnish an example of how the Riemann-Liouville fractional integral arises in the formulation of physical [25].

Suppose, then, that a thin wire is placed in the first quadrant of a vertical plane and that a frictionless bead slides along the wire under the action of gravity (see Figure 7-1). Let the initial velocity of the bead be zero. Abel set himself the problem of finding the shape of the curve for which the time of descent from to the origin is independent of the starting point. Such a curve is called a tautochrone.

Abel's tautochrone problem should not be confused with the brachistochrone problem in the calculus of variations. That problem was to find the shape of the curve C such that the time of descent of the bead from to would be a minimum. This question was discussed as early as 1630 by Galileo; but it was not until 1696 that Johann Bernoulli formulated and solved the problem of finding "the curve of quickest descent." In this case the brachistochrone is a cycloid.

We now proceed to formulate Abel's problem. Let be the arc length measured along from to an arbitrary point on , and let a be the angle of inclination (see Figure 7-1). Then is the acceleration of the bead, where is the gravitational constant, and

Hence we have the differential equation

With the aid of the integrating factor , we see immediately that

(7-1)

where is a constant of integration. Since the bead started from rest, is zero when , and thus . We therefore may write (7-1) as

The negative square root is chosen since as increases, decreases. Thus the time of descent from to is

Now the arc length is a function of , say , where depends on the shape of the curve . Therefore,

Or

(7-2)

where

(7-3)

If we let

(7-4)

then the integral equation of (7-2) may be written in the notation of the fractional calculus as

(7-5)

But the right-hand side of (7-5) is the Riemann-Liouville fractional integral of of order . This is our desired formulation. It remains then to solve (7-5) and then find the equation of .

Abel attacked the first problem by applying the fractional operator to both sides of (7-5) and writing

(7-6)

Now we know from Theorem 1, that this is legitimate if and are of class . But a constant is certainly of class , and since , we see that also is of class . Thus (7-6) becomes

(7-7)

which is the solution of (7-5) [or (7-2)].

Now, to solve the second part of the problem, that is, to find the equation of , we begin by using (7-3) and (7-4) to write

Thus

Or

(7-8)

But since at the origin .

If we let , then the change of variable of integration reduces (7-8) to

where

These last two equations then imply that

and if we make the trivial change of variable , the parametric equations of become

(7-9)

The solution of our problem is now complete. We see from (7-9) that is a cycloid.

7.2. Heaviside Operational Calculus and the Fractional Calculus

G. W. Hill had the daring to publish in 1877 a paper on the problem of the moon's perigee in which he used determinants of infinite order. Hill's novel method was open to serious questions from the standpoint of rigorous analysis until H. Poincare in 1886 proved the convergence of infinite determinants [25].

A somewhat similar history followed Oliver Heaviside's publication in 1893 of certain methods for solving linear differential equations (known today as the Heaviside operational calculus) except that in this case, a much longer period elapsed before his procedures were put on a firm foundation by T. J. Bromwich in 1919 and J. R. Carson in 1922.

We illustrate Heaviside's methods by applying them to a particular partial differential equation. The partial differential equation we consider is

(7-10)

If is interpreted as temperature, then (7-10) is the heat equation in one dimension. If is interpreted as voltage or current, (7-10) is called the submarine cable equation (Depending on intended equation, constant will change).

Let us assume that (7-10) is the heat equation. Let

(7-11a)

be the initial condition and

(7-11b)

(where is a given constant) be the boundary condition.

We shall solve (7-10) together with (7-11) using Heaviside's arguments. He introduced the letter to represent ,i.e. .

In this notation we may write (7-10) as

(7-12)

Now he assumed that was a constant and treated (7-12) as an ordinary differential equation in . The solution is therefore

(7-13)

where and are independent of . On physical grounds he was led to choose as zero. If we do so, the boundary condition of (7-11b) implies that . Thus

Expanding the exponential in a power series, we obtain

Now Heaviside ignored positive integral powers of and wrote as

(7-14)

Since

(7-15)

Although formula (7-15) is certainly the correct expression for the fractional derivative of a constant of order , Heaviside did not record how he arrived at (7-15). One may speculate on how he deduced this result; however, we choose not to second-guess a genius.

Substituting (7-15) into (7-14) immediately yields

Or

(7-16)

If we note that

then (7-16) reduces to

(7-17)

which is the solution to our problem. ( is Error function)

7.3. Fluid Flow and the Design of a Weir Notch

A weir notch is an opening in a dam (weir) that allows water to spill over the dam, see Figure 7-2, where we have indicated a cross section of the dam and a partial front view. (The sketch is not to scale.) [5, 25, 33].

Our problem is to design the shape of the opening such that the rate of flow of water through the notch (say, in cubic feet per second) is a specified function of the height of the opening. Starting from physical principles we derive the equation for determining the shape of the notch. It turns out to be an integral equation of the Riemann-Liouville type. After formulating the problem, we shall, of course, solve it.

Let the -axis denote the direction of flow, the -axis the vertical direction, and the -axis the transverse direction along the face of the dam. See Figure 7-3, where we have drawn an enlarged view of a portion of Figure 7-2. The axes are oriented as indicated, and is the height of the notch.

The solid square at point I and the solid square at point II are supposed to indicate the same element of fluid as it moves from point I [with coordinates ] to point II [with coordinates ] along the same "tube of flow." Then by Bernoulli's theorem from hydrodynamics

(7-18)

where is the density of water, the acceleration of gravity, and and are the pressure and velocity at point I while and are the corresponding quantities at point II.

If we assume that point I is far enough upstream, is negligible and we may write (7-18) as

(7-19)

Now

and since point II is in the plane of the notch (the shaded area of Figure 7-3b)

Thus is a constant (namely,) times and (7-19) implies that

(7-20)

Referring to Figure 7-4, we see that the element of area (the shaded region in Figure 7-4) is

(7-21)

where we have assumed that the shape of the notch is symmetrical about the -axis. Now is some function of , say

(7-22)

and we may write (7-21) as . Thus the incremental rate of flow of water through the area is , where is the velocity of flow at height , and from (7-20)

The total flow of water through the notch is thus

(7-23)

Equation (7-23) is the desired integral equation for the determination of when is given. In the notation of the fractional calculus we may write it as

(7-24)

To solve (7-24) we first observe that if , then certainly also is of class . Hence

and by Theorem 1

(7-25)

which is the desired solution.

For example, suppose that , (where ; is a constant). Then certainly if and

Thus, by equation (7-25), we have

Where and or .

In particular, if , that is, then

and the notch is V-shaped (Figure 7-5).

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