Related papers: Wave Equation for Fractal Solid String
We use the fractional integrals to describe fractal media. We consider the fractal media as special ("fractional") continuous media. We discuss the possible experimental testing of the continuous medium model for fractal media that is…
This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with…
It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we…
Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of…
The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical…
It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable…
The Chapman-Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using…
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$…
We will give some regularity results about fractional diffusion-wave equations.
We develop an effective model to describe the dynamics of a system of particle moving in circular configurations around a central mass, by considering the continuum limit of the angular distribution, to obtain the stable configurations for…
We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the…
In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy…
We introduce fractal liquids by generalizing classical liquids of integer dimensions $d = 1, 2, 3$ to a fractal dimension $d_f$. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the…
Fractional calculus has been used to describe physical systems with complexity. Here, we show that a fractional calculus approach can restore or include complexity in any physical systems that can be described by partial differential…
This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a…
A fractional variational principle was derived in order to be used with lagrangians containing fractional derivatives of order 1/2. By forcing the action associated to this type of lagrangian to be stationary, a modified fractional…
A p.c.f. fractal with a regular harmonic structure admits an associated Dirichlet form, which is itself associated with a Laplacian. This Laplacian enables us to give an analogue of the damped stochastic wave equation on the fractal. We…
Classical wave equation is generalized for the case of viscoelastic materials obeying fractional Zener model instead of Hooke's law. Cauchy problem for such an equation is studied: existence and uniqueness of the fundamental solution is…
A fractional generalization of variations is used to define a stability of non-integer order. Fractional variational derivatives are suggested to describe the properties of dynamical systems at fractional perturbations. We formulate…
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms…