scholarly journals Non-local Integrals and Derivatives on Fractal Sets with Applications

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 542-548 ◽  
Author(s):  
Alireza K. Golmankhaneh ◽  
D. Baleanu
Keyword(s):  
Differential Equations ◽  
Cantor Sets ◽  
Physical Models ◽  
Scaling Properties ◽  
Fractal Sets ◽  
Fractal Differential Equations ◽  
Non Local

AbstractIn this paper, we discuss non-local derivatives on fractal Cantor sets. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared. Related physical models are also suggested.

scholarly journals Fractal Logistic Equation

2019 ◽  
Vol 3 (3) ◽  
pp. 41 ◽  
Author(s):  
Alireza Khalili Golmankhaneh ◽  
Carlo Cattani
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Euler Method ◽  
Logistic Equation ◽  
Fractal Sets ◽  
Logistic Equations ◽  
Fractal Calculus ◽  
Approximate Analytical Solutions ◽  
Fractal Differential Equations ◽  
Modeling Growth

In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics.

scholarly journals Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

2019 ◽  
Vol 3 (2) ◽  
pp. 25 ◽  
Author(s):  
Alireza Khalili Golmankhaneh ◽  
Cemil Tunç
Keyword(s):  
Differential Equations ◽  
Coordinate Systems ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Fractal Sets ◽  
Fractal Calculus ◽  
Symmetry Properties ◽  
Fractal Differential Equations ◽  
Canonical Coordinate

In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results.

scholarly journals Random Variables and Stable Distributions on Fractal Cantor Sets

2019 ◽  
Vol 3 (2) ◽  
pp. 31 ◽  
Author(s):  
Alireza Khalili Golmankhaneh ◽  
Arran Fernandez
Keyword(s):  
Probability Theory ◽  
Shannon Entropy ◽  
Random Variables ◽  
Stable Distributions ◽  
Distribution Functions ◽  
Cantor Sets ◽  
Physical Models ◽  
Fractal Sets ◽  
Fractal Calculus ◽  
Related Distribution

In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results.

scholarly journals Non-local Solvable Birth–Death Processes

2021 ◽  
Author(s):  
Giacomo Ascione ◽  
Nikolai Leonenko ◽  
Enrica Pirozzi
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Limit Distribution ◽  
Spectral Decomposition ◽  
Strong Solutions ◽  
Stochastic Representation ◽  
Discrete Approximations ◽  
Local Difference ◽  
Non Local ◽  
Birth Death

AbstractIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

An implicit multistep numerical method for real-time simulation of stiff systems

SIMULATION ◽  
2021 ◽  
pp. 003754972110216
Author(s):  
Zhang Lei ◽  
Li Jie ◽  
Wang Menglu ◽  
Liu Mengya
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Real Time ◽  
Physical System ◽  
Physical Models ◽  
Stiff Systems ◽  
Real Time Simulation ◽  
Stiff Ordinary Differential Equations ◽  
Root Finding ◽  
Time Simulation

Simulating a physical system in real-time is widely used in equipment design, test, and validation. Though an implicit multistep numerical method excels at solving physical models that are usually composed of stiff ordinary differential equations, it is not suitable for real-time simulation because of state discontinuity and massive iterations for root finding. Thus, a method based on the backward differential formula is presented. It divides the main fixed step of real-time simulation into limited minor steps according to computing cost and accuracy demand. By analyzing and testing its capability, this method shows advantage and efficiency in real-time simulation, especially when the system contains stiff equations. A simulation application will have more flexibility while using this method.

scholarly journals On local fractional operators View of computational complexity: Diffusion and relaxation defined on cantor sets

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 755-767 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Zhi-Zhen Zhang ◽  
Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Computational Complexity ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Systems ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Fractional Partial Differential Equations ◽  
Comparative Results

This paper treats the description of non-differentiable dynamics occurring in complex systems governed by local fractional partial differential equations. The exact solutions of diffusion and relaxation equations with Mittag-Leffler and exponential decay defined on Cantor sets are calculated. Comparative results with other versions of the local fractional derivatives are discussed.

scholarly journals Uniqueness and Stability Results on Non-local Stochastic Random Impulsive Integro-Differential Equations

2021 ◽  
Vol 2 (3) ◽  
pp. 9-20
Author(s):  
VARSHINI S ◽  
BANUPRIYA K ◽  
RAMKUMAR K ◽  
RAVIKUMAR K
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Local Conditions ◽  
Non Local ◽  
Inequality Techniques ◽  
Stability Results

The paper is concerned with stochastic random impulsive integro-differential equations with non-local conditions. The sufficient conditions guarantees uniqueness of mild solution derived using Banach fixed point theorem. Stability of the solution is derived by incorporating Banach fixed point theorem with certain inequality techniques.

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