scholarly journals On stability of a class of second alpha-order fractal differential equations

2020 ◽  
Vol 5 (3) ◽  
pp. 2126-2142 ◽  
Author(s):  
Cemil Tunç ◽  
◽  
Alireza Khalili Golmankhaneh ◽  
Keyword(s):  
Differential Equations ◽  
Fractal Differential Equations

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.

Fractal differential equations on the Sierpinski gasket

1999 ◽  
Vol 5 (2-3) ◽  
pp. 203-284 ◽  
Author(s):  
Kyallee Dalrymple ◽  
Robert S. Strichartz ◽  
Jade P. Vinson
Keyword(s):  
Differential Equations ◽  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Fractal Differential Equations

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 A ROBUST OPERATIONAL MATRIX OF NONSINGULAR DERIVATIVE TO SOLVE FRACTIONAL VARIABLE-ORDER DIFFERENTIAL EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
MAYS BASIM ◽  
NORAZAK SENU ◽  
ZARINA BIBI IBRAHIM ◽  
ALI AHMADIAN ◽  
SOHEIL SALAHSHOUR
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Variable Order ◽  
Lagrange Polynomial ◽  
New Class ◽  
Fractal Differential Equations ◽  
Linear And Nonlinear ◽  
Predictor Corrector

Currently, a study has come out with a novel class of differential operators using fractional-order and variable-order fractal Atangana–Baleanu derivative, which in turn, became the source of inspiration for new class of differential equations. The aim of this paper is to apply the operation matrix to get numerical solutions to this new class of differential equations and help us help us to simplify the problem and transform it into a system of an algebraic equation. This method is applied to solve two types, linear and nonlinear of fractal differential equations. Some numerical examples are given to display the simplicity and accuracy of the proposed technique and compare it with the predictor–corrector and mixture two-step Lagrange polynomial and the fundamental theorem of fractional calculus methods.

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