Published in Advances in Computer Science and its Applications 1(1)2012 Cite: Yang, X.J., Expression of generalized Newton iteration method via generalized local fractional Taylor series, Advances in Computer Science and its Applications, 1(2) (2012) 89-92.
1
Expression of Generalized Newton Iteration Method via Generalized Local Fractional Taylor Series Yang Xiao-Jun Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu,221008, P. R. China Email:
[email protected] Abstract –Local fractional derivative and integrals are revealed as one of useful tools to deal with everywhere continuous but nowhere differentiable functions in fractal areas ranging from fundamental science to engineering. In this paper, a generalized Newton iteration method derived from the generalized local fractional Taylor series with the local fractional derivatives is reviewed. Operators on real line numbers on a fractal space are induced from Cantor set to fractional set. Existence for a generalized fixed point on generalized metric spaces may take place. Keywords –Newton iteration method, Generalized local fractional Taylor series, Local fractional derivative, Real line number, fractal space, Generalized fixed point
(3) x , z
1. Introduction Over the past ten years, local fractional calculus [1-8] played an important part in fractal mathematics and engineering, especially in nonlinear phenomena. More recently, Newton iteration method based on local fractional calculus was presented in [6]. A procedure for the Newton iteration method is not easy enough for an engineer to master it. In order to express the generalized Newton iteration method, this paper proposes operators on real line numbers on a fractal space. Taking the definitions of local fractional derivatives and local fractional integrals into account and using generalized local fractional Taylor series [11, 15, 16], it is easy to understand the generalized Newton iteration method. This paper is organized as follows. In section 2, operators on real line numbers on a fractal space are investigated. Generalized local fractional Taylor series are derived from local fractional calculus in section 3. Generalized Newton iteration method is discussed in section 4. The conclusions are in section 5.
2. Operators on real line numbers on a fractal space At first, we start with operator on real line numbers in fractal space. Definition 1 Let be real number. There exist responding real line numbers on a fractal set E with fractional dimension , noted by 1 .
x
, y y , z .
The pair ( E , ) is a generalized metric space.
{numbers on a fractal set E with 1 }. Remark. 2 Let E n . Remark.1
1
n 2 2 ,2 x , y xi yi i 1 is a metric on n , where
x x1 , x2 ,..., xn n
and
y y1 , y2 ,..., yn n .
Similarly, if E n n n , we have 1
2 2 n ,2 z1 , z2 z1,i z2,i i 1 is a metric on complex space n , where
z1 z1,1 , z1,2 ,..., z1, n n and
z2 z2,1 , z2,2 ,..., z2,n n .
There is a simple example that for z
x i y
z x12 x2 2 is referred [9]. Remark.3 In special case of
1 it follows that
n
Definition 2 A metric for a fractal set E with fractional dimension , 0 1 , is a map
both n ... and are classical complex
: E E 1 such that for all x , y , z E .
spaces and that both 1 and
x
(1) x , y (2)
, y
0 with equality if x y ,x ;
y ;
1
1
n 1 1 1 1 ... 11 are classical complex spaces. 1 n
Author 1, et al., ACSA, Vol. 1, No. 1, pp. 1-2, March 2012
If
2
a , b , c belong to the set 1 of real line numbers,
j 0,..., N 1 , t0 a, t N b , is a partition of the
then
(1) a
b and a b belong to the set ;
interval
(2) a b b a a b b a ;
(3) a b c
a
(5) a (6) a
and
b c a b c ; b c a b a
For any x a, b , there exists a I x
by f x I x
I x a, b , we
0
The above is operators on real line numbers on a fractional space.
3. Generalized local fractional Taylor series
f x , denoted
a, b . Remark 4. If f x Dx a, b , x have f x C a, b .
c ;
0 0 a a and a 1 1 a a .
(7) a
I f x b Ia f x if a b .
a b
f x 0 if a b
I
a a
(4) a b b a ab ba ;
a, b .
Here, it follows that
b c ;
where t j t j 1 t j , t maxt1, t2 , t j ,... and t j , t j 1 ,
Remark 5. Above definition is deferent from Jumarie [10] and Kolwankar [1,4]. Meanwhile, is fractal dimension and there exist the following relations
a b a b
3.1 Review for local fractional derivatives and local fractional integrals
and
a b a b .
Definition 3 A non-differentiable function
f : , x f x
is called to be local fractional continuous, when
f x f x0
with
(1)
x x0 , for , 0 and , . If
f x is local fractional continuous on the interval a, b , we denote f x C a, b .
In order to take the above into account, we have the equality y y c 1 1 1 dt dt dt , x c x 1 1 1 which provides 1 1 1 y x y c c x . 1 1 1 In this case we have the following relation
y x
Definition 4 The local fractional derivative of
f x of order at x x0 is defined [6-9] f
d f x x0 dx
where
x x0
lim
x x0
,
(2)
0
denoted by
x Dx
f x Dx
y x
y x
f x ,
I
a b
y x ,
(4)
f x of
a, b is defined [6-9, 12-16] ,
x y .
(5)
3.2 Generalized local fractional Taylor series Lemma 1 (Generalized local fractional Taylor theorem) k1 Suppose that f x C a, b , for k 0,1,..., n , 0
f x
jN 1 lim f t j t j 1 t 0 j 0
x y
a, b .
b 1 f t dt 1 a
y x y x y ,
Hence
Definition 5 Local fractional integral of order in the interval
It follows from (4) that
y c c x .
When c 0 , we can find that
0
For any x a, b , there exists
f
y x
f x f x 1 f x f x .
y c c x .
Furthermore,
f x f x0
xx0
1 , then we have[11] n1 n f x0 f k n1 f x xx0 xx0 1 n1 k0 1k k
(3)
with a
f
k 1
x0 x b , x a, b , where
k 1 times x Dx ...Dx f x .
Theorem 2 (Generalized local fractional Taylor series)
(6)
Author 1, et al., ACSA, Vol. 1, No. 1, pp. 1-2, March 2012
Suppose that f
k1
3
Notice: Both x and
x C a, b ,
for k 0,1,..., n , 0
1 , then we have
continuous functions. Hence, from (12) we may obtain a new fixed point.
f x0 k f x x x0 ,0 1 k 0 1 k k
with a
(7)
x0 x b , x a, b ,
k 1 times where f x Dx ...Dx f x . Proof. From(6), taking the limit n , we have the
k 1
result. As a direct result, we have the local fractional MCLaurin’s series, which is read as
f x
f k 0
k 0 1 k
x k , 0 1 .
(8)
5 Remarking conclusions In the present paper, a new iteration method via the local fractional Taylor series with local fractional derivatives, a generalized local fractional iteration method, is studied. The method is to solve the equations of non-differential functions defined on fractal sets. Real line numbers on fractal sets are investigated, and it may be a new subject on fractal geometrics. As classical iteration method, there may be a new fixed point method in a fractal space. It is to say, the existence for a generalized fixed point on generalized metric spaces [15-16] may take place.
6. References
4. Generalized Newton iteration method
[1]
In this section, an iteration method via the local fractional derivative for a non-differential function is called generalized local fractional iteration method. This method is to solve the local fractional equation. Suppose that : , x x , is
[2] [3] [4] [5]
a th continuously non-differentiable function, * for 0 1 . There exists x , xk 1 such that
[6]
x* xk x* xk 1
1
.
*
(9) [7]
x x
k 1
L x* xk 1
[8]
We have the inequality
[9]
x* xk Lx* xk1 Lk x* x0 Lk x* x0 , where
x L 1 for any x a, b . 1
[10]
. For any positive integer p we can find that
Therefore xk x as k
[11]
*
[12]
xk p xk xkp xkp1 xkp1 xkp2 xk1 xk
[13]
(10)
Lp1 xk1 xk Lp2 xk1 xk xk1 xk Lp1 Lp2 1 xk1 xk . It follows from (10) that *
x xk
[14]
[15] [16]
1 xk 1 xk as p , 1 L
Hence, there exists a local fractional iteration process (11) xk+1 xk ,
which converges the root of x 0 .
x are local fractional
K.M. Kolwankar, A.D. Gangal, Local Fractional Fokker–Planck Equation, Phys. Rev. Lett., 80(1998) 214-217. A. Babakhani, V. Daftardar-Gejji, On Calculus of Local Fractional Derivatives, J. Math. Anal. Appl., 270(2002) 66-79. Y. Chen, Y.Yan, K. Zhang, On the local fractional derivative, J. Math. Anal. Appl., 362(2010)17-33. F.B. Adda, J. Cresson, About Non-differentiable Functions, J. Math. Anal. Appl., 263(2001) 721-737. A.Carpinteri, B. Chiaia, P. Cornetti, Static-kinematic Duality and the Principle of Virtual Work in the Mechanics of Fractal Media, Comput. Methods Appl. Mech. Engrg., 191(2001) 3-19. F. Gao, X.J., Yang, Z. X., Kang, Local Fractional Newton’s Method Derived from Modified Local Fractional Calculus, In: The 2th Scientific and Engineering Computing Symposium on Computational Sciences and Optimization, pp.228-232, IEEE Computer Society, 2009. X.J. Yang, F. Gao, The Fundamentals of Local Fractional Derivative of the One-variable Non-differentiable Functions, World SCI-TECH R&D., 31(2009) 920-921. X.J.Yang, L. Li, R.Yang, Problems of Local Fractional Definite Integral of the One-variable Non-differentiable Function, World SCI-TECH R&D., 31(2009) 722-724. X.J.Yang, Z.X. Kang, C.H. Liu, Local Fractional Fourier’s Transform Based on the Local Fractional Calculus, In: The 2010 International Conference on Electrical and Control Engineering, pp.1242-1245. IEEE Computer Society, 2010. G. Jumarie, On the Representation of Fractional Brownian Motion as an Integral with Respect to (dt)a, Appl. Math. Lett., 18 (2005)739-748. X.J. Yang, Generalized local Taylor’s formula with local fractional derivatives. arXiv:1106.2459v2. X.J. Yang, Local fractional partial differential equations with fractal boundary problems, Advances in Computational Mathematics and its Applications, 1(1) (2012) 60-63. G.S. Chen, Mean value theorems for Local fractional integrals on fractal Space, Advances in Mechanical Engineering and its Applications, 1(1) (2012) 5-8. G.S. Chen, Local fractional improper integral on fractal space, Advances in Information Technology and Management, 1 (1) (2012) 4-8. X.J. Yang, Local Fractional Integral Transforms, Progress in Nonlinear Science, 4(2011) 1-225. X.J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher Limited. 2011.
