Volume 7. November 2007. Article ID 1664

Geometric Interpretation of Fractional Integration

Tomas Skovranek

In 2001/2002, Podlubny suggested a solution to the more than 300-years old problem of geometric interpretation of fractional integration (i.e., integration of an arbitrary real order). His geometric interpretation for left-sided and right-sided Riemann-Liouville fractional integrals, and for Riesz potential is given in terms of changing time scale with constant order of integration, and also in a case of varying order of integration with constant time parameter. In this article we present animations of such interpretation.

- Fractional integral
- Fractional calculus
- Geometric interpretation

- Introduction
- Left-sided Riemann-Liouville fractional integral
- Right-sided Riemann-Liouville fractional integral
- Riesz potential
- References

- Left-sided Riemann-Liouville fractional integral
- Right-sided Riemann-Liouville fractional integral
- Riesz potential

*Note:* sample animations use the same function, namely `f` (`t`) = `t` + 0.5 sin (`t`), as the static pictures in the text.

It is generally known that integer-order derivatives and integrals have clear physical and geometric interpretations, which signicantly simplify their use for solving applied problems in various fields of science. However, in case of fractional-order integration and differentiation, which represent a rapidly growing field both in theory and in applications to realworld problems, it was not so until recent time.

Fractional integration and fractional differentiation are generalizations of notions of integer-order integration and differentiation, and include `n`-th derivatives and `n`-fold integrals (`n` denotes an integer number) as particular cases. Because of this, it would be ideal to have such physical and geometric interpretations of fractional-order operators, which will provide also a link to known classical interpretations of integer-order differentiation and integration. For background information, see the following articles in in Wolfram MathWorld

The brief history of the problem of geometric interpretation of fractional order integration is the following [Podlubny 2002].

Since the appearance of the idea of differentiation and integration of arbitrary (not necessary integer) order there was not any acceptable geometric and physical interpretation of these operations for more than 300 years. The lack of these interpretations has been acknowledged at the First International Conference on the Fractional Calculus in New Haven (USA) in 1974 by including it in the list of open problems [Ross, 1975]. The question was unanswered, and therefore repeated at the subsequent conferences at the University of Strathclyde (UK) in 1984 [McBride, 1985] and at the Nihon University (Tokyo, Japan) in 1989 [Nishimoto, 1990]. The round-table discussion ([Kiryakova, 1998], [Gorenflo, 1998], [Mainardi, 1998]) at the conference on Transform Methods and Special Functions in Varna (1996) showed that the problem was still unsolved.

Since the need for the geometric and physical interpretations was generally recognised at the aforementioned meetings, several authors attempted to provide them. Probably due mostly to linguistical reasons, much effort have been devoted to trying to relate fractional integrals and derivatives, on one side, and fractal geometry, on the other ([Nigmatullin, 1992], [Zu-Guo Yu, Fu-Yao Ren, Ji Zhou, 1997], [Fu-Yao Ren, Zu-Guo Yu, Feng Su, 1996], [Monsrefi-Torbati, Hammond, 1998]). This approach was criticized by R. Rutman ([Rutman 1994], [Rutman 1995]). In principle, fractals themselves are not directly related to fractional integrals or fractional defivatives; only description of dynamical processes in fractal structures can lead to models involving fractional order operators.

Besides those "fractal-oriented" attempts, some considerations regarding interpretation of fractional integration and fractional differentiation were presented in [Monsrefi-Torbati, Hammond, 1998]. However, those considerations are, in fact, only a small collection of selected examples of applications of fractional calculus, in which hereditary effects and self-similarity are typical for the objects modelled with the help of fractional calculus. Although each particular problem, to which fractional derivatives or/and fractional integrals have been applied, can be considered as a certain illustration of their meaning, the paper [Monsrefi-Torbati, Hammond, 1998] could not be considered as an explicit answer to the problem of interpretation of fractional integration. A different approach to geometric interpretation of fractional integration and fractional differentiation, based on the idea of the contact of α-th order, has been suggested by F. Ben Adda ([Adda 1997], [Adda 1998]. However, it is difficult to speak about an acceptable geometric interpretation if one cannot see any picture there.

Obviously, there was still a lack of geometric and physical interpretation of fractional integration and differentiation, which is comparable with the simple interpretations of their integer-order counterparts.

A new approach to the solution of this challenging old problem was proposed by Podlubny [Podlubny 2002], who introduced a simple and really geometric interpretation of several types of fractional-order integration: the left-sided and the right-sided Riemann-Liouville fractional integration and the Riesz potential. A specific feature of these interpretations is that they are *dynamic*, that is, they require some kind of animation for a proper illustration.

In this article we recall Podlubny's [Podlubny 2002] explanations and provide animations for geometric interpretation of left-sided and right-sided Riemann-Liouville fractional integrals, and for the Riesz potential.

There is also a very recent and closely related article by Heymans and Podlubny [2006], where a physical interpretation for intial conditions expressed in terms of fractional integrals is suggested.

Let us consider the left-sided Riemann-Liouville fractional integral ([Podlubny, 1999]; [Samko, Kilbas, Marichev, 1987]) of order `α`,

(1) |

and write it in the form

(2) |

(3) |

The function `g`_{t}(`τ`) has an interesting scaling property. Indeed, if we take `t`_{1} = `k` `t` and `τ`_{1} = `k` `τ` (`k` > 0), then

(4) |

Now let us consider the integral (2) for a fixed `t`. Then it becomes simply a Stieltjes integral, and we can utilize G. L. Bullock's idea [Bullock 1988].

Let us take the axes `τ`, `g`, and `f`. In the plane (`τ`; `g`) we plot the function `g`_{t}(`τ`) for 0 ≤ `τ` ≤ `t`. Along the obtained curve we "build a fence" of the varying height `f`(`τ`), so the top edge of the "fence" is a three-dimensional line (`τ`; `g`_{t}(`τ`); `f`(`τ`)), 0 ≤ `τ` ≤ `t`. This "fence" can be projected onto two surfaces (see Figure 1 and its animated version):

- the area of the projection of this "fence" onto the plane (
`τ`;`f`) corresponds to the value of the integral(5) - the area of the projection of the same "fence" onto the plane (
`g`;`f`) corresponds to the value of the integral (2), or, what is the same, to the value of the fractional integral (1).

In other words, our "fence" throws two shadows on two walls. The first of them, that on the wall (`τ`; `f`), is the well-known "area under the curve `f`(`τ`)", which is a standard geometric interpretation of the integral (5). The "shadow" on the wall (`g`; `f`) is a geometric interpretation of the fractional integral (1) for a fixed `t`.

Obviously, if `α` = 1, then `g`_{t}(`τ`) = `τ`, and both "shadows" are equal. This shows that classical definite integration is a particular case of the left-sided Riemann-Liouville fractional integration even from the geometric point of view.

What happens when `t` is changing (namely growing)? As `t` changes, the "fence" changes simultaneously. Its length and, in a certain sense, its shape changes. For illustration, see Figure 2. If we follow the change of the "shadow" on the wall (`g`; `f`), which is changing simultaneously with the "fence" (see Figure 3), then we have a dynamical geometric interpretation of the fractional integral (1) as a function of the variable `t`.

Let us consider the right-sided Riemann-Liouville fractional integral ([Podlubny, 1999], [Samko, Kilbas, Marichev, 1987]),

(6) |

and write it in the form

(7) |

(8) |

Then we can provide a geometric interpretation similar to the geometric interpretation of the left-sided Riemann-Liouville fractional integral. However, in this case there is not any fixed point in the "fence" base - the end, corresponding to `τ` = `b`, moves along the line `τ` = `b` in the plane (`τ`; `g`) when the "fence" changes its shape. This movement can be observed in Figure 4 and its animated version. In the case of the left-sided integral, the left end, corresponding to `τ` = 0, is fixed and does not move. This behavior can be observed in the corresponding animation.

All other parts of the geometric interpretation remain the same: the "fence" changes its shape as `t` changes from 0 to `b`, and the changing shadows of this "fence" on the walls (`g`; `f`) and (`τ`; `f`) represent correspondingly the right-sided Riemann-Liouville fractional integral (6) and the classical integral with the moving lower limit:

(9) |

Obviously, if `α` = 1, then `h`_{t}(`τ`) = `τ`, and both "shadows" are equal. Therefore, we see that not only the left-sided, but also the right-sided Riemann-Liouville fractional integration includes the classical definite integration as a particular case even from the geometrical point of view.

The Riesz potential ([Podlubny, 1999], [Samko, Kilbas, Marichev, 1987])

(10) |

is the sum of the left-sided and the right-sided Riemann-Liouville fractional integrals:

(11) |

The Riesz potential (10) can be written in the form

(12) |

(13) |

The shape of the "fence", corresponding to the Riesz potential, is described by the function `r`_{t}(`τ`). In this case the "fence" consists of the two parts: one of them (for 0 < `τ` < `t`) is the same as in the case of the left-sided Riemann-Liouville fractional integral, and the second (for `t` < `τ` < `b`) is the same as for the right-sided Riemann-Liouville integral, as shown in Figure 5. Both parts are joined smothly at the inflection point `τ` = `t`.

The shape of the "fence", corresponding to the Riesz potential, is shown in some of its intermediate position by the bold line in Figure 5 (see also its animated version). Obviously, Figure 5 can be obtained by laying Figure 4 over Figure 2, which is a geometric interpretation of the relationship (11).

The shadow of this "fence" on the wall (`g`; `f`) represents the Riesz potential (10), while the shadow on the wall (`τ`; `f`) corresponds to the classical integral

(14) |

For `α` = 1, `r`_{t}(`τ`) = `τ`, and both "shadows" are equal. This shows that the classical definite integral (14) is a particular case of the Riesz fractional potential (10) even from the geometric point of view. We have already seen this inclusion in the case of the left-sided and the right-sided Riemann-Liouville fractional integration. This demonstrates the strength of the suggested geometric interpretation of these three types of generalization of the notion of integration.

- Ben Adda, F. (1997).
*Geometric interpretation of the fractional derivative*. Journal of Fractional Calculus, vol. 11, 21-52. - Ben Adda, F. (1998).
*Interpretation geometrique de la differentiabilite et du gradient d'ordre reel*. C. R. Acad. Sci. Paris 326, Serie I, 931-934. - Bullock, G. L. (1988). "A geometric interpretation of the Riemann-Stieltjes integral". American Mathematical Monthly, vol. 95, no. 5, 448-455.
- Fu-Yao Ren, Zu-Guo Yu, Feng Su. (1996).
*Fractional integral associated to the self-similar set of the generalised self-similar set and its physical interpretation.*Phys. Lett. A, vol. 219, 59-68. - Gorenflo, R. (1998).
*Afterthoughts on interpretation of fractional derivatives and integrals.*In: P. Rusev, I. Dimovski, V. Kiryakova (Eds.),*Transform Methods and Special Functions, Varna'96 (Proc. 3rd Internat. Workshop)*, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia (ISBN 954-8986-05-1), 589-591. - Heymans, N., and Podlubny, I. (June 2006).
*Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives*. Rheologica Acta, vol. 45, no. 5, 765–772 . - Kiryakova, V. (1998).
*A long standing conjecture failed?*In: P. Rusev, I. Dimovski, V. Kiryakova (Eds.),*Transform Methods and Special Functions, Varna'96 (Proc. 3rd Internat. Workshop)*, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia (ISBN 954-8986-05-1), 579-588. - Mainardi, F. (1998).
*Considerations on fractional calculus: interpretations and applications.*In: P. Rusev, I. Dimovski, V. Kiryakova (Eds.),*Transform Methods and Special Functions, Varna'96 (Proc. 3rd Internat. Workshop)*, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Soa (ISBN 954-8986-05-1), 594-597. - McBride, A., Roach, G. (Eds). (1985).
*Fractional Calculus.*Research Notes in Mathematics, vol. 138, Pitman, Boston-London-Melbourne. - Monsrefi-Torbati, M., Hammond, J. K. (1998).
*Physical and geometrical interpretation of fractional operators.*J. Franklin Inst., vol. 335B, no. 6, 1077-1086. - Nigmatullin, R. R. (1992).
*A fractional integral and its physical interpretation*. Theoret. and Math. Phys.,vol. 90, no. 3, 242-251. - Nishimoto, K. (Ed.). (1990).
*Fractional Calculus and Its Applications*. Nihon University, Koriyama. - Podlubny, I. (1999).
*Fractional Differential Equations*. Academic Press, San Diego. - Podlubny, I. (2002).
*Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation*. Fractional Calculus and Applied Analysis, vol. 5, no. 4, 367–386. Preprint (2001) - Ross, B. (Ed.). (1975).
*Fractional Calculus and Its Applications*. Lecture Notes in Mathematics, vol. 457, Springer-Verlag, New York. - Rutman, R. S. (1994).
*On the paper by R.R. Nigmatullin "A fractional integral and its physical interpretation"*. Theoret. and Math. Phys., vol. 100, no. 3, 1154-1156. - Rutman, R. S. (1995).
*On physical interpretations of fractional integration and dierentiation*. Theoret. and Math. Phys., vol. 105, no. 3, 1509-1519. - Samko, S. G., Kilbas, A. A., Marichev, O. I. (1987).
*Integraly i proizvodnye drobnogo poryadka i nekotorye ich prilozheniya (Integrals and Derivatives of the Fractional Order and Some of Their Applications)*. Nauka i Tekhnika, Minsk (English transl.: S. G. Samko, A. A. Kilbas, and O. I. Marichev,*Fractional Integrals and Derivatives*. Gordon and Breach, Amsterdam (1993)). - Zu-Guo Yu, Fu-Yao Ren, Ji Zhou (1997).
*Fractional integral associated to generalized cookie-cutter set and its physical interpretation*. J. Phys. A: Math. Gen., vol. 30, 5569-5577.