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Problems of Local Fractional Definite Integral of the One-variable Non-differentiable Function
... Proof. see [24] Theorem 6. [28] Suppose that f (x) ∈ C α [a, b], then there is a function ...
... Proof. see [28] ...
... x f (x). According to the (theorem 3.2.9 in [28]), we get ...
The work that we have done in this paper is the coupling method between the local fractional derivative and the Natural transform (we can call it the local fractional Natural transform), where we have provided some essential results and properties. We have applied this method to some linear local fractional differential equations on Cantor sets to get nondifferentiable solutions. The results show this transform’s effectiveness when we combine it with this operator.
... The classical local fractional derivative of the Hölder continuous functions defined on cantor's set were discussed and Hölder exponent is exactly the order of local fractional derivative of the functions. It is important, as a useful tool, to deal with the non-differentiable functions, which are irregular in the real world, defined on fractal set [2][3][4][5][6][7][8][9][10][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. ...
... More recently, local fractional calculus was modified, and a short definition of the local fractional derivative and integral were proposed [22][23][24][25][26]. Local fractional transforms based on the local fractional calculus were discussed [24]. ...
... Property 3.2.1 [23] Suppose that ( ) ( ) [ ] ...
Local fractional calculus (LFC) handles everywhere continuous but nowhere differentiable functions in fractal space. This note investigates the theory of local fractional derivative and integral of function of one variable. We first introduce the theory of local fractional continuity of function and history of local fractional calculus. We then consider the basic theory of local fractional derivative and integral, containing the local fractional Rolle's theorem, L'Hospital's rule, mean value theorem, anti-differentiation and related theorems, integration by parts and Taylor' theorem. Finally, we study the efficient application of local fractional derivative to local fractional extreme value of non-differentiable functions, and give the theory of local fractional extreme value, containing local fractional extreme value theorem, Fermat's theorem, increasing/decreasing test and the derivative test. It is of great significances for us to process optimization problems of the non-differentiable functions on Cantor set.
... Its inverse was given by [9] [7,[9][10][11][12][13][14][15][16][17][18][19] ...
... . For more detail for local fractional calculus, see [7][8][9][10][11]. However, the inversion of local fractional derivative, (1.3) was modified in [12,13], and the transform is called the Yang-Laplace transforms [14]. ...
... (2.13) Theorem 1 [7,10,12,13] Suppose that ...
The Yang-Laplace transforms [W. P. Zhong, F. Gao, In: Proc. of the 2011 3rd International Conference on Computer Technology and Development, 209-213, ASME, 2011] in fractal space is a generalization of Laplace transforms derived from the local fractional calculus. This letter presents a short introduction to Yang-Laplace transforms in fractal space. At first, we present the theory of local fractional derivative and integral of non-differential functions defined on cantor set. Then the properties and theorems for Yang-Laplace transforms are tabled, and both the initial value theorem and the final value theorem are investigated. Finally, some applications to the wave equation and the partial differential equation with local fractional derivative are studied in detail.
... Local fractional calculus [3] [4] [5] [7] [8] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] is a generalization of differentiation and integration of the functions defined on fractal sets. In the last years has found use in studies of viscoelastic materials, as well as in many fields of science and engineering including electerical networks, probability, electromagnetic theory, diffusive transport and fluid flow [1] [2] [6] [9] [20-41].There are many definitions of local fractional derivatives and local fractional integrals (also called fractal calculus) [1- 19]. ...
... Hereby we write down Gao-Yang-Kang definitions as follows. Gao-Yang-Kang local fractional derivative is denoted by [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] ...
... and local fractional integral of ( ) f x denoted by [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] ( ...
This paper deals with the theory and applications of the local fractional Mellin transform of the real order . We define
the local fractional Mellin transform and its inverse transform. This is followed by several examples and the basic operational
properties of local fractional Mellin transform. We discuss applications of local fractional Mellin transforms to local fractional
boundary value problems.
... Local fractional calculus has been revealed a useful tool in areas ranging from fundamental science to engineering in the past ten years [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]. It is important to deal with the continuous functions (fractal functions), which are irregular in the real world. ...
... Theorem 1 [7] [14] ...
... Theorem 3 [7] [14] ...
Local fractional calculus deals with everywhere continuous but nowhere
differentiable functions in fractal space. The Yang-Fourier transform based on
the local fractional calculus is a generalization of Fourier transform in
fractal space. In this paper, local fractional continuous non-differentiable
functions in fractal space are studied, and the generalized model for the
Yang-Fourier transforms derived from the local fractional calculus are
introduced. A generalized model for the Yang-Fourier transforms in fractal
space and some results are proposed in detail.
... There are many definitions of local fractional calculus [2][3][4][5][6][7][8][9][10][11]. Hereby we write down Gao-Yang-Kang's local fractional derivatives [2][3][4][5][6] ...
... There are many definitions of local fractional calculus [2][3][4][5][6][7][8][9][10][11]. Hereby we write down Gao-Yang-Kang's local fractional derivatives [2][3][4][5][6] ...
... and Gao-Yang-Kang's local fractional integrals [2][3][4][5][6] , (1.3) and local fractional Laplace transforms [3] , denoted by (1.4) as new tools to deal with local fractional differential equations and local differential systems, were proposed. More recently, a new imaginary unit proposed in [2,3]. ...
This paper presents the fractional trigonometric functions in complex-valued
space and proposes a short outline of local fractional calculus of complex
function in fractal spaces.
... Kang local fractional derivative is denoted by [4] [5] [6] [7] [8] ...
... Properties of the operator can be found in [6]. We only need here the following results: ...
... f x g x C a b α α = ∈ , then we have [6] ( ) ( ) ( ) ( ) a b I f x g b g a α = − . (2.8) ...
In the present paper, a generalized local Taylor formula with the local
fractional derivatives (LFDs) is proposed based on the local fractional
calculus (LFC). From the fractal geometry point of view, the theory of local
fractional integrals and derivatives has been dealt with fractal and
continuously non-differentiable functions, and has been successfully applied in
engineering problems. It points out the proof of the generalized local
fractional Taylor formula, and is devoted to the applications of the
generalized local fractional Taylor formula to the generalized local fractional
series and the approximation of functions. Finally, it is shown that local
fractional Taylor series of the Mittag-Leffler type function is discussed.
... see [38] ...
... According to the (theorem 3.2.9, [38]), we get ...
The objective of our work is to couple the Elzaki transform method and the local fractional derivative which is called local fractional Elzaki transform, where we have provided important results of this transformation as local fractional Laplace-Elzaki duality, Elzaki transform of the local fractional derivative and the local fractional integral and the local fractional convolution, also we have presented the properties of some special functions with the local fractional derivative sense. The Elzaki transform was applied to solve some linear local fractional differential equations in order to obtain non-differentiable analytical solutions. The results of the solved examples show the effectiveness of the proposed method. Subject Classification (2020): 44A05, 44A20.
... Fractal calculus (also called local fractional calculus) has played an important role in not only mathematics but also in physics and engineers [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Local fractional derivative [6][7][8] were written in the form ...
... Local fractional integral of f (x) [6][7]9] was written in the form ...
Based on the local fractional calculus, we establish some new generalizations
of H\"{o}lder's inequality. By using it, some results on the generalized
integral inequality in fractal space are investigated in detail.
... Local fractional calculus has played an important role in not only mathematics but also in physics and engineers [1][2][3][4][5][6][7][8][9][10][11][12]. There are many definitions of local fractional derivatives and local fractional integrals (also called fractal calculus). ...
... , and local fractional integral of f x , denoted by [5][6]8] More recently, a motivation of local fractional derivative and local fractional integral of complex functions is given [11]. Our attempt, in the present paper, is to continue to study local fractional calculus of complex function. ...
This paper presents a short introduction to local fractional complex
analysis. The generalized local fractional complex integral formulas,
Yang-Taylor series and local fractional Laurent's series of complex functions
in complex fractal space, and generalized residue theorems are investigated.
... In 2009, theory of the local fractional derivative and local fractional integral were developed by Yang and coauthors. [21][22][23] Now, this theory is called Yang's local fractional calculus, which was applied in the different fields, such as fractal Hilbert-type inequalities, 24 generalized (h-m)-convexity on fractal sets, 25 ...
Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of real numbers, some new Gronwall–Bellman inequalities are discussed and established on the fractal space. Our results are a powerful supplement to the existing literature.
... Local fractional calculus [1] [2] [3] [4] [5] [6] [7] [8] played an important role in fractal mathematics and engineering, especially in nonlinear phenomena. Very recently, Yang [9] [10] [11] given ...
–The fractional calculus does with the theory of real (or imaginary) order integral and differential operators
and it stands for a natural instrument to model nonlocal phenomena, either in space or time, involving different scales.
Local fractional calculus (also called fractal calculus) has played a significant part not only in mathematics but also in
physics and engineering. The main purpose of this paper is to establish local fractional improper integrals and an analogue of the
classical Dirichlet-Abel test for local fractional improper integrals in fractal space. In the paper, we study local fractional improper
integrals on fractal space. By some mean value theorems for local fractional integrals, we prove an analogue of the classical
Dirichlet-Abel test for local fractional improper integrals in fractal space.
... Local fractional integral of ( ) f x [6] [7] [9] were written in the form ...
The theory of calculus was extended to local fractional calculus involving fractional order. Local fractional
calculus (also called Fractal calculus) has played a significant part not only in mathematics but also in physics and
engineers. The main purpose of this paper is to further extend some mean value theorems in Fractal space, by Abel's lemma,
definition of Local fractional integrals and using some properties of Local fractional integral . In the paper, we present some
properties of Local fractional integral. By using it, we establish the generalized first mean value theorem and the
generalized second mean value theorem for Local fractional integrals in Fractal space.
... Over the past ten years, local fractional calculus [1][2][3][4][5][6][7][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]. ...
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.
... Recently, the theory of Yang's fractional sets of element sets [6][7][8][9][10][11][12] was introduced as follows: ...
Starting with real line number system based on the theory of the Yang's
fractional set, the generalized Young inequality is established. By using it
some results on the generalized inequality in fractal space are investigated in
detail.
... where local fractional integral of f t is denoted by [3][4][5][6][7][8] L . In the present paper, our arms are to get some assurance that local fractional integral of f can be reasonably 1 1 ...
It is suggest that a new fractal model for the Yang-Fourier transforms of
discrete approximation based on local fractional calculus and the Discrete
Yang-Fourier transforms are investigated in detail.
In this work we focus on presenting a method for solving local fractional differential equations. This method
based on the combination of the Aboodh transform with the local fractional derivative (we can call it local
fractional Aboodh transform), where we have provided some important results and properties. We concluded
this work by providing illustrative examples, through which we focused on solving some linear local fractional
differential equations in order to obtain nondifferential analytical solutions.
Anderson's inequality (Anderson, 1958) as well as its improved version given by Fink (2003) is known to provide interesting examples of integral inequalities. In this paper, we establish local fractional integral analogue of Anderson's inequality on fractal space under some suitable conditions. Moreover, we also show that the local fractional integral inequality on fractal space, which we have proved in this paper, is a new generalization of the classical Anderson's inequality.
In the last year fractional calculus has been used in studies of electromagnetic theory, as well as in many fields of science
and engineering involving diffusive transport, fluid flow, rheology, electrical networks, viscoelastic materials and probability. This
paper deals with the theory of the local fractional Stieltjes transform. We derive the Stieltjes transform. This is followed by several
examples and the basic operational properties of Stieltjes transforms
Based on the local fractional calculus, we establish some new generalizations of Hölder’s inequality. By using it, some related results on the generalized integral inequality in fractal space are investigated in detail.
A new possible modeling for the local fractional iteration process is proposed in this paper. Based on the local fractional
Taylor’s series, the fundamentals of local fractional iteration of the continuously non-differentiable functions are derived
from local fractional calculus in fractional space.
Keywordslocal fractional Taylor’s series–local fractional iteration–continuously non-differentiable function–local fractional calculus–fractional space
In this paper, a new modeling for the local fractional Laplace’s transform based on the local fractional calculus is proposed
in fractional space. The properties of the local fractional Laplace’s transform are obtained and an illustrative example for
the local fractional system is investigated in detail.
Keywordslocal fractional Laplace’s transform–local fractional calculus–fractional space–local fractional system
A new modeling for the local fractional Fourier's transform containing the local fractional calculus is investigated in fractional space. The properties of the local fractional Fourier's transform are obtained and two examples for the local fractional systems are investigated in detail.
This paper presents a better approach to model an engineering problem in
fractal-time space based on local fractional calculus. Some examples are given
to elucidate to establish governing equations with local fractional derivative.
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