Geometrical Interpretation of fractional derivative and integrals
This is a good question with many possible answers. In addition to the incisive answer already given by @Pavel Osinenko, more can be observed.
A thorough study of the interpretation of fractional derivatives ls given in
M.H. Tavassoli A. Tavassoli, M.R.O. Rahimi, The geometric and physical interpretation of fractional order derivatives of polynomial functions, Diff. Geom.-Dynamical Sys. 15, 2013, 93-194:
http://www.mathem.pub.ro/dgds/v15/D15-ta.pdf
See, for example, Fig. 1 (graph of f(x) = x^3 with triangles formed with fractional order derivatives, p. 97. On the same page, see the detailed geometric interpretation of g(x) = x^4 + x^3 with triangles formed with fractional order derivatives.
A very good study of both fractional derivates and fractional integrals is given in
I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calc. & Applied Anal. 5, 2002, no. 5, 367-386:
http://people.tuke.sk/igor.podlubny/pspdf/pifcaa_r.pdf
It is shown that the classical definite integral is a particular case of the left-sided Riemann-Liouville fractional integration from a geometric point of view (See, e.g., Fig. 2 and Fig. 3, pp. 371-372).
vixra.org/pdf/1206.0005v1.pdf has a simple description and comparison with ordinary derivative. A particular example with an illustration can be found at http://mathoverflow.net/questions/153542/geometric-interpretation-of-the-half-derivative.
This is a good question with many possible answers. In addition to the incisive answer already given by @Pavel Osinenko, more can be observed.
A thorough study of the interpretation of fractional derivatives ls given in
M.H. Tavassoli A. Tavassoli, M.R.O. Rahimi, The geometric and physical interpretation of fractional order derivatives of polynomial functions, Diff. Geom.-Dynamical Sys. 15, 2013, 93-194:
http://www.mathem.pub.ro/dgds/v15/D15-ta.pdf
See, for example, Fig. 1 (graph of f(x) = x^3 with triangles formed with fractional order derivatives, p. 97. On the same page, see the detailed geometric interpretation of g(x) = x^4 + x^3 with triangles formed with fractional order derivatives.
A very good study of both fractional derivates and fractional integrals is given in
I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calc. & Applied Anal. 5, 2002, no. 5, 367-386:
http://people.tuke.sk/igor.podlubny/pspdf/pifcaa_r.pdf
It is shown that the classical definite integral is a particular case of the left-sided Riemann-Liouville fractional integration from a geometric point of view (See, e.g., Fig. 2 and Fig. 3, pp. 371-372).
This question has many answers. It is very easy to learn from the following reference
Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calc. Appl. Anal. 5 (2002) 367-386 by I. Podlubny.
One you can understood that, the geometric and physical meaning of fractional derivative or integral is different from their interpretation.
read the book fractional differential equations by I. Podlubny.
and Functional fractional calculus by Shantanu Das
See the books on "Fractional Order Differential Equations" have written by Igor Podulbny, Miller and Ross.
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