In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.
A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. T...A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. The variance of scattering intensity is presented to study the fractal characteristics through theoretical analysis and numerical calculations. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated as a slope of linear equation. This conclusion will be applicable for solving the inverse problem of reconstructing rough surface and remote sensing.展开更多
The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The exact box dimension for such functions many important cases is established. We need to point out that, althoug...The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The exact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.展开更多
The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicat...The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.展开更多
This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given bywhere 1 < s < 2, λk> tends to infinity as k→∞ and λk satisfies λk+1/λk≥λ...This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given bywhere 1 < s < 2, λk> tends to infinity as k→∞ and λk satisfies λk+1/λk≥λ>1. The results show thatis a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions. For the fractional Riemann-Liouvtlle differential operator Du and the fractional integral operator D-v, the results show that if A is sufficiently large, then a necessary and sufficient condition for box dimensionof Graph(D-v(B)), 0 < v < s - 1, to be s - v and box dimension of Graph(Du(B)), 0 < u < 2 - s, to bes + u is also lim.展开更多
基金Project supported by National Natural Science Foundation of China.
文摘In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.
基金supported by the National Natural Science Foundation of China (Grant No 60571058)Specialized Research Fund for the Doctoral Program of Higher Education, China (Grant No 20070701010)
文摘A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. The variance of scattering intensity is presented to study the fractal characteristics through theoretical analysis and numerical calculations. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated as a slope of linear equation. This conclusion will be applicable for solving the inverse problem of reconstructing rough surface and remote sensing.
文摘The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The exact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.
文摘The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.
基金Research supported by national Natural Science Foundation of China (10141001)Zhejiang Provincial Natural Science Foundation 9100042 and 1010009.
文摘This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given bywhere 1 < s < 2, λk> tends to infinity as k→∞ and λk satisfies λk+1/λk≥λ>1. The results show thatis a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions. For the fractional Riemann-Liouvtlle differential operator Du and the fractional integral operator D-v, the results show that if A is sufficiently large, then a necessary and sufficient condition for box dimensionof Graph(D-v(B)), 0 < v < s - 1, to be s - v and box dimension of Graph(Du(B)), 0 < u < 2 - s, to bes + u is also lim.
