Abstract
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.
Highlights
Fractal geometry includes shapes which are scale invariant and have fractional dimensions and self-similar properties [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
We present that which corresponds to a given fractal difference equation fractal differential equation
We provide analogues of the numerical method for finding the solutions of the fractal differential equations such as the fractal logistic equation
Summary
Fractal geometry includes shapes which are scale invariant and have fractional dimensions and self-similar properties [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. In the seminal papers [57,58], generalized standard calculus was adopted to include functions with support on totally disconnected fractal sets and self-avoiding curves. We present that which corresponds to a given fractal difference equation fractal differential equation. We provide analogues of the numerical method for finding the solutions of the fractal differential equations such as the fractal logistic equation.
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