Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and Its Analysis

Affiliation(s)

^{1}
Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, India.

^{2}
Department of Applied Mathematics, University of Calcutta, Kolkata, India.

^{3}
Reactor Control Systems Design Section, E & I Group, BARC, Mumbai, India.

ABSTRACT

The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. In this paper we have defined fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The H?lder exponent and Box dimension of this new function have been evaluated here. It has been established that the values of H?lder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function. This new development in generalizing the classical Weierstrass function by use of fractional trigonometric function analysis and fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, establishes that roughness indices are invariant to this generalization.

The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. In this paper we have defined fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The H?lder exponent and Box dimension of this new function have been evaluated here. It has been established that the values of H?lder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function. This new development in generalizing the classical Weierstrass function by use of fractional trigonometric function analysis and fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, establishes that roughness indices are invariant to this generalization.

KEYWORDS

Hölder Exponent, Fractional Weierstrass Function, Box Dimension, Jumarie Fractional Derivative, Jumarie Fractional Trigonometric Function

Hölder Exponent, Fractional Weierstrass Function, Box Dimension, Jumarie Fractional Derivative, Jumarie Fractional Trigonometric Function

Cite this paper

Ghosh, U. , Sarkar, S. and Das, S. (2015) Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and Its Analysis.*Advances in Pure Mathematics*, **5**, 717-732. doi: 10.4236/apm.2015.512065.

Ghosh, U. , Sarkar, S. and Das, S. (2015) Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and Its Analysis.

References

[1] Mandelbrot, B.B. (1982) The Geometry of Nature. Freeman, San Francisco.

[2] Peitgen, H. and Saupe, D., Eds. (1988) The Science of Fractal Images. Springer-Verlag, New York.

[3] Ghosh, U. and Khan, D.K. (2014) Information, Fractal, Percolation and Geo-Environmental Complexities. LAP LAMBERT Academic Publishing.

[4] Ross, B. (1977) The Development of Fractional Calculus 1695-1900. Historia Mathematica, 4, 75-89.

http://dx.doi.org/10.1016/0315-0860(77)90039-8

[5] Diethelm, K. (2010) The Analysis of Fractional Differential Equations. Springer-Verlag.

http://dx.doi.org/10.1007/978-3-642-14574-2

[6] Kilbas, A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier Science, Amsterdam, 1-523.

[7] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York.

[8] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon.

[9] Das, S. (2011) Functional Fractional Calculus. 2nd Edition, Springer-Verlag.

http://dx.doi.org/10.1007/978-3-642-20545-3

[10] Jumarie, G. (2007) Fractional Partial Differential Equations and Modified Riemann-Liouville Derivatives. Method for Solution. Journal of Applied Mathematics and Computing, 24, 31-48.

[11] Podlubny, I. (1999) Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, San Diego, 198.

[12] Liang, Y.S. and Su, W. (2007) Connection between the Order of Fractional Calculus and Fractional Dimensions of a Type of Fractal Functions. Analysis in Theory and Applications, 23, 354-362.

[13] Falconer, J. (1990) Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York.

[14] Johensen, J. (2010) Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth. Journal of Fourier Analysis and Applications, 16, 17-33.

http://dx.doi.org/10.1007/s00041-009-9072-2

[15] Zhou, S.P., Yao, K. and Su, W.Y. (2004) Fractional Integrals of the Weierstrass Functions: The Exact Box Dimension. Analysis in Theory and Applications, 20, 332-341.

http://dx.doi.org/10.1007/BF02835226

[16] Kolwankar, K.M. and Gangal, A.D. (1997) Holder Exponent of Irregular Signals and Local Fractional Derivatives. Pramana, 48, 49-68.

http://dx.doi.org/10.1007/BF02845622

[17] Jumarie, G. (2006) Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-Differentiable Functions Further Results. Computers and Mathematics with Applications, 51, 1367-1376.

http://dx.doi.org/10.1016/j.camwa.2006.02.001

[18] Jumarie, G. (2008) Fourier’s Transformation of Fractional Order via Mittag-Leffler Function and Modified Riemann-Liouville Derivatives. Journal of Applied Mathematics and Informatics, 26, 1101-1121.

[19] Erdelyi, A. (1954) Asymptotic Expansions. Dover Publications, New York.

[20] Erdelyi, A., Ed. (1954) Tables of Integral Transforms. Volume 1, McGraw-Hill, New York.

[21] Erdelyi, A. (1950) On Some Functional Transformation Univ Potitec Torino 1950.

[22] Mittag-Leffler, G.M. (1903) Sur la nouvelle fonction Eα(x). Comptes Rendus de l’Académie des Sciences, 137, 554-558.

[23] Hunt, B.R. (1998) The Hausdorff Dimension of Graph of Weierstrass Functions. Proceedings of the American Mathematical Society, 126, 791-801.

http://dx.doi.org/10.1090/S0002-9939-98-04387-1

[24] Wen, Y.Z. (2000) Mathematical Foundations of Fractal Geometry. Shanghai Science and Technology Educational Publishing House, Shanghai.

[25] Zahle, M. and Ziezold, H. (1996) Fractional Derivatives of Weierstrass-Type Functions. Journal of Computational and Applied Mathematics, 76, 265-275.

http://dx.doi.org/10.1016/S0377-0427(96)00110-0

[1] Mandelbrot, B.B. (1982) The Geometry of Nature. Freeman, San Francisco.

[2] Peitgen, H. and Saupe, D., Eds. (1988) The Science of Fractal Images. Springer-Verlag, New York.

[3] Ghosh, U. and Khan, D.K. (2014) Information, Fractal, Percolation and Geo-Environmental Complexities. LAP LAMBERT Academic Publishing.

[4] Ross, B. (1977) The Development of Fractional Calculus 1695-1900. Historia Mathematica, 4, 75-89.

http://dx.doi.org/10.1016/0315-0860(77)90039-8

[5] Diethelm, K. (2010) The Analysis of Fractional Differential Equations. Springer-Verlag.

http://dx.doi.org/10.1007/978-3-642-14574-2

[6] Kilbas, A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier Science, Amsterdam, 1-523.

[7] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York.

[8] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon.

[9] Das, S. (2011) Functional Fractional Calculus. 2nd Edition, Springer-Verlag.

http://dx.doi.org/10.1007/978-3-642-20545-3

[10] Jumarie, G. (2007) Fractional Partial Differential Equations and Modified Riemann-Liouville Derivatives. Method for Solution. Journal of Applied Mathematics and Computing, 24, 31-48.

[11] Podlubny, I. (1999) Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, San Diego, 198.

[12] Liang, Y.S. and Su, W. (2007) Connection between the Order of Fractional Calculus and Fractional Dimensions of a Type of Fractal Functions. Analysis in Theory and Applications, 23, 354-362.

[13] Falconer, J. (1990) Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York.

[14] Johensen, J. (2010) Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth. Journal of Fourier Analysis and Applications, 16, 17-33.

http://dx.doi.org/10.1007/s00041-009-9072-2

[15] Zhou, S.P., Yao, K. and Su, W.Y. (2004) Fractional Integrals of the Weierstrass Functions: The Exact Box Dimension. Analysis in Theory and Applications, 20, 332-341.

http://dx.doi.org/10.1007/BF02835226

[16] Kolwankar, K.M. and Gangal, A.D. (1997) Holder Exponent of Irregular Signals and Local Fractional Derivatives. Pramana, 48, 49-68.

http://dx.doi.org/10.1007/BF02845622

[17] Jumarie, G. (2006) Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-Differentiable Functions Further Results. Computers and Mathematics with Applications, 51, 1367-1376.

http://dx.doi.org/10.1016/j.camwa.2006.02.001

[18] Jumarie, G. (2008) Fourier’s Transformation of Fractional Order via Mittag-Leffler Function and Modified Riemann-Liouville Derivatives. Journal of Applied Mathematics and Informatics, 26, 1101-1121.

[19] Erdelyi, A. (1954) Asymptotic Expansions. Dover Publications, New York.

[20] Erdelyi, A., Ed. (1954) Tables of Integral Transforms. Volume 1, McGraw-Hill, New York.

[21] Erdelyi, A. (1950) On Some Functional Transformation Univ Potitec Torino 1950.

[22] Mittag-Leffler, G.M. (1903) Sur la nouvelle fonction Eα(x). Comptes Rendus de l’Académie des Sciences, 137, 554-558.

[23] Hunt, B.R. (1998) The Hausdorff Dimension of Graph of Weierstrass Functions. Proceedings of the American Mathematical Society, 126, 791-801.

http://dx.doi.org/10.1090/S0002-9939-98-04387-1

[24] Wen, Y.Z. (2000) Mathematical Foundations of Fractal Geometry. Shanghai Science and Technology Educational Publishing House, Shanghai.

[25] Zahle, M. and Ziezold, H. (1996) Fractional Derivatives of Weierstrass-Type Functions. Journal of Computational and Applied Mathematics, 76, 265-275.

http://dx.doi.org/10.1016/S0377-0427(96)00110-0