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1.
We establish a relation between stable distributions in probability theory and the fractional integral. Moreover, it turns out that the parameter of the stable distribution coincides with the exponent of the fractional integral. It follows from an analysis of the obtained results that equations with the fractional time derivative describe the evolution of some physical system whose time degree of freedom becomes stochastic, i.e., presents a sum of random time intervals subject to a stable probability distribution. We discuss relations between the fractal Cantor set (Cantor strips) and the fractional integral. We show that the possibility to use this relation as an approximation of the fractional integral is rather limited. 相似文献
2.
Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous
and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal
subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate
framework. A new calculus calledF
α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF
α-integral andF
α-derivative respectively. TheF
α-integral is suitable for integrating functions with fractal support of dimension α, while theF
α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function
occur naturally as solutions ofF
α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems.
We discuss construction and solutions of some fractal differential equations of the form
whereh is a vector field andD
F,t
α
is a fractal differential operator of order α in timet. We also consider some equations of the form
whereL is an ordinary differential operator in the real variablex, and(t,x) ∈F × Rn whereF is a Cantor-like set of dimension α.
Further, we discuss a method of finding solutions toF
α-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed
back to get those of the former. This is illustrated with a couple of examples. 相似文献
3.
This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method. 相似文献
4.
Mokhtar Kirane Makhmud A. Sadybekov Abdisalam A. Sarsenbi 《Mathematical Methods in the Applied Sciences》2019,42(6):2043-2052
We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables. 相似文献
5.
《Mathematical Methods in the Applied Sciences》2018,41(13):4906-4922
In this paper, we propose an efficient numerical method for a distributed order fractional subdiffusion problem using nonpolynomial spline approach. The solvability, stability, and convergence of the scheme are established rigorously, and it is shown that the spatial convergence order improves some previous work done. Simulation is then conducted to verify the accuracy of the proposed scheme as well as to compare with earlier work. 相似文献
6.
KOU S.C. 《中国科学A辑(英文版)》2009,52(6):1181-1211
Advances in nanotechnology enable scientists for the first time to study biological processes on a nanoscale molecule-by-molecule
basis. They also raise challenges and opportunities for statisticians and applied probabilists. To exemplify the stochastic
inference and modeling problems in the field, this paper discusses a few selected cases, ranging from likelihood inference,
Bayesian data augmentation, and semi- and non-parametric inference of nanometric biochemical systems to the utilization of
stochastic integro-differential equations and stochastic networks to model single-molecule biophysical processes. We discuss
the statistical and probabilistic issues as well as the biophysical motivation and physical meaning behind the problems, emphasizing
the analysis and modeling of real experimental data.
This work was supported by the United States National Science Fundation Career Award (Grant No. DMS-0449204) 相似文献
7.
Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation
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We present the finite difference/element method for a
two-dimensional modified fractional diffusion equation. The analysis
is carried out first for the time semi-discrete scheme, and then for
the full discrete scheme. The time discretization is based on the
$L1$-approximation for the fractional derivative terms and the
second-order backward differentiation formula for the classical
first order derivative term. We use finite element method for the
spatial approximation in full discrete scheme. We show that both the
semi-discrete and full discrete schemes are unconditionally stable
and convergent. Moreover, the optimal convergence rate is obtained.
Finally, some numerical examples are tested in the case of one and
two space dimensions and the numerical results confirm our
theoretical analysis. 相似文献
8.
The finite‐difference method applied to the time‐fractional subdiffusion equation usually leads to a large‐scale linear system with a block lower triangular Toeplitz coefficient matrix. The approximate inversion method is employed to solve this system. A sufficient condition is proved to guarantee the high accuracy of the approximate inversion method for solving the block lower triangular Toeplitz systems, which are easy to verify in practice and have a wide range of applications. The applications of this sufficient condition to several existing finite‐difference schemes are investigated. Numerical experiments are presented to verify the validity of theoretical results. 相似文献
9.
Tadeusz Kosztołowicz Aldona Dutkiewicz 《Mathematical Methods in the Applied Sciences》2020,43(18):10500-10510
We consider three different diffusion processes in a system with a thin membrane: normal diffusion, classical subdiffusion, and slow subdiffusion. We conduct the considerations following the rule: If a diffusion equation is derived from a certain theoretical model, boundary conditions at a thin membrane should also be derived from this model with additional assumptions taking into account selective properties of the membrane. To derive diffusion equations and boundary conditions at a thin membrane, we use a particle random walk model in one-dimensional membrane system in which space and time variables are discrete. Then we move from discrete to continuous variables. We show that the boundary conditions depend on both selective properties of the membrane and a type of diffusion in the system. 相似文献
10.
Three-dimensional (3D) Fick's diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model (CTRW) to investigate the time-averaged mean square dis- placement (MSD) of a 3D single particle trajectory. Theoretical results show that the ensemble average of the time-averaged MSD can be expressed analytically by a Mittag-Leffler function. Our new expression is in agreement with previous formu- las in two limiting cases: (^-δ2) ~ △1 in short lag time and (^-δ2} ~ △1 -α in long lag time. We also simulate the experimental data of mRNA diffusion in living E. coli using a 3D CTRW model under confined and crowded conditions. The simulation results are well consistent with experimental results. The calculations of power spectral density (PSD) further indicate the subdiffsive behavior of an individual trajectory. 相似文献