Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications in Cell Biology

In this paper the multi-dimensional random walk models governed by distributed fractional order differential equations and multi-term fractional order differential equations are constructed. The scaling limits of these random walks to a diffusion process in the sense of distributions is proved. Simulations based upon multi-term fractional order differential equations are performed

Solution of Fractional Order Differential Equation Problems by Triangular Functions for Biomedical Applications

Abstract—Fractional Order Differential equations are used for modelling of a wide variety of biological systems but the solution process of such equations are quite complex. In this paper Orthogonal Triangular functions and their operational matrices have been used for finding an approximate solution of Fractional Order Differential Equations. This technique has been found to be more powerful in solving Fractional Order Differential Equations owing to the fact that the differential equations are reduced to systems of algebraic equations which are easy to solve numerically and the percentage error is lower compared to other methods of solutions (like: Laplace Transform Method). Also due to the recursive nature of this method, it can also be concluded that this method is less complex and more efficient in solving varieties of the Fractional Order Differential Equations

On simple iterative fractional order differential equations

In this paper, the simple fractional iterative differential equation will be the focus of study Dβv(s)=vn(s), v(s0)=a. where s0, v0 ∈ I = [0, b], and 0 < β < 1. One class of the iterative fractional differential equations is the simple iterative fractional differential equation. In this paper, the local existence and uniqueness results for the fractional order of degree n of simple iterative fractional differential equation are proven and the solutions found by using power series

Linear fractional order differential equations and their solution

Chapters 1, 2, 3, and 4 provide background material. Chapter 5 describes new results on the behaviour of solutions to (0.0.0)

Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications to Cell Biology

Mathematics Subject Classification: 65C05, 60G50, 39A10, 92C37In this paper the multi-dimensional Monte-Carlo random walk simulation models governed by distributed fractional order differential equations (DODEs) and multi-term fractional order differential equations are constructed. The construction is based on the discretization leading to a generalized difference scheme (containing a finite number of terms in the time step and infinite number of terms in the space step) of the Cauchy problem for DODE. The scaling limits of the constructed random walks to a diffusion process in the sense of distributions is proved

Numerical Solution of Fractional Order Differential Equations by Extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples

Fractional differential equations and related exact mechanical models

The aim of the paper is the description of fractional-order differential equations in terms of exact mechanical models. This result will be archived, in the paper, for the case of linear multiphase fractional hereditariness involving linear combinations of power-laws in relaxation/creep functions. The mechanical model corresponding to fractional-order differential equations is the extension of a recently introduced exact mechanical representation (Di Paola and Zingales (2012) [33] and Di Paola et al. (2012) [34]) of fractional-order integrals and derivatives. Some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots