scholarly journals A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Yong Xu ◽  
Qixian Liu ◽  
Jike Liu ◽  
Yanmao Chen
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Separation Of Variables ◽  
Computational Cost ◽  
Diffusion Equations ◽  
Novel Method ◽  
High Computational Efficiency

We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

The Mixing of Two Parallel Streams of Dissimilar Fluids—Part I: Analytical Development

1971 ◽  
Vol 38 (2) ◽  
pp. 301-309 ◽  
Author(s):  
R. L. Baker ◽  
L. N. Tao ◽  
H. Weinstein
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Schmidt Number ◽  
Similarity Transformation ◽  
Velocity Ratio ◽  
Density Ratio ◽  
Large Density ◽  
Parallel Streams ◽  
Analytical Development

The mixing region between dissimilar fluids is investigated in the region where the similarity transformation is valid. The treatment is complete in that laminar and turbulent cases both with and without large density differences are considered. The Schmidt number is an arbitrary input to the problem and may be varied. The ordinary differential equation resulting from the similarity transformation is integrated numerically and some solutions are presented. The three boundary conditions are proper; the so-called arbitrary third boundary condition is treated as originally suggested by von Karman and extended to the case of large density difference. Illustrations of the effects of varying velocity ratio, density ratio, and Schmidt number are presented.

scholarly journals Additive Noise Effects on the Stabilization of Fractional-Space Diffusion Equation Solutions

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 130
Author(s):  
Wael W. Mohammed ◽  
Naveed Iqbal ◽  
Thongchai Botmart
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Additive Noise ◽  
Diffusion Equations ◽  
Critical Dynamics ◽  
Ginzburg Landau ◽  
Noise Effects ◽  
Special Cases ◽  
Ginzburg Landau Equations ◽  
Fractional Space

This paper considers a class of stochastic fractional-space diffusion equations with polynomials. We establish a limiting equation that specifies the critical dynamics in a rigorous way. After this, we use the limiting equation, which is an ordinary differential equation, to approximate the solution of the stochastic fractional-space diffusion equation. This equation has never been studied before using a combination of additive noise and fractional-space, therefore we generalize some previously obtained results as special cases. Furthermore, we use Fisher’s and Ginzburg–Landau equations to illustrate our results. Finally, we look at how additive noise affects the stabilization of the solutions.

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme

2017 ◽  
Vol 22 (4) ◽  
pp. 1028-1048 ◽  
Author(s):  
Yonggui Yan ◽  
Zhi-Zhong Sun ◽  
Jiwei Zhang
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fast Evaluation ◽  
Fractional Diffusion Equations ◽  
Long Time ◽  
Order Scheme ◽  
Speed Up

AbstractThe fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on theL2-1σformula proposed in [A. Alikhanov,J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

scholarly journals Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread

2020 ◽  
Vol 18 (04) ◽  
pp. 585-614 ◽  
Author(s):  
Yuan-Hang Su ◽  
Wan-Tong Li ◽  
Fei-Ying Yang
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Nonlocal Dispersal ◽  
Asymptotic Behaviors ◽  
Cost Parameter ◽  
The Cost

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator [Formula: see text] and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalized principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter [Formula: see text]. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter [Formula: see text] is in a different range. In particular, for the case [Formula: see text], we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition.

scholarly journals Gluon dynamics from an ordinary differential equation

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
A. C. Aguilar ◽  
M. N. Ferreira ◽  
J. Papavassiliou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Gluon Propagator ◽  
Dyson Equation ◽  
Kinetic Term ◽  
Practical Implementation ◽  
Gluon Vertex ◽  
Central Hypothesis ◽  
Approximate Form ◽  
Novel Method

AbstractWe present a novel method for computing the nonperturbative kinetic term of the gluon propagator from an ordinary differential equation, whose derivation hinges on the central hypothesis that the regular part of the three-gluon vertex and the aforementioned kinetic term are related by a partial Slavnov–Taylor identity. The main ingredients entering in the solution are projection of the three-gluon vertex and a particular derivative of the ghost-gluon kernel, whose approximate form is derived from a Schwinger–Dyson equation. Crucially, the requirement of a pole-free answer determines the initial condition, whose value is calculated from an integral containing the same ingredients as the solution itself. This feature fixes uniquely, at least in principle, the form of the kinetic term, once the ingredients have been accurately evaluated. In practice, however, due to substantial uncertainties in the computation of the necessary inputs, certain crucial components need be adjusted by hand, in order to obtain self-consistent results. Furthermore, if the gluon propagator has been independently accessed from the lattice, the solution for the kinetic term facilitates the extraction of the momentum-dependent effective gluon mass. The practical implementation of this method is carried out in detail, and the required approximations and theoretical assumptions are duly highlighted.

scholarly journals Similarity Solution for Fractional Diffusion Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Ai-Ping Guo ◽  
Wen-Zai Yun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Probability Density Function ◽  
Hausdorff Dimension ◽  
Diffusion Equation ◽  
Similarity Solution ◽  
Mellin Transform ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Fractal Media

Fractional diffusion equation in fractal media is an integropartial differential equation parametrized by fractal Hausdorff dimension and anomalous diffusion exponent. In this paper, the similarity solution of the fractional diffusion equation was considered. Through the invariants of the group of scaling transformations we derived the integro-ordinary differential equation for the similarity variable. Then by virtue of Mellin transform, the probability density functionp(r,t), which is just the fundamental solution of the fractional diffusion equation, was expressed in terms of Fox functions.

scholarly journals A new fractional derivative operator and its application to diffusion equation

2021 ◽  
Author(s):  
Ruchi Sharma ◽  
Pranay Goswami ◽  
RAVI DUBEY ◽  
Fethi Belgacem
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Derivative Operator ◽  
Fractional Diffusion Equations

In this paper, we introduced a new fractional derivative operator based on Lonezo Hartely function, which is called G-function. With the help of the operator, we solved a fractional diffusion equations. Some applications related to the operator is also discussed as form of corollaries.

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