scholarly journals On the Fractional Diffusion-Advection Equation for Fluids and Plasmas

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 62 ◽  
Author(s):  
Gaetano Zimbardo ◽  
Silvia Perri
Keyword(s):  
Practical Interest ◽  
Fractional Diffusion ◽  
Advection Equation ◽  
Astrophysical Plasmas ◽  
Caputo Fractional Derivatives ◽  
Fractional Contribution ◽  
Liouville Derivative ◽  
Superdiffusive Transport ◽  
The Right ◽  
State Case

The problem of studying anomalous superdiffusive transport by means of fractional transport equations is considered. We concentrate on the case when an advection flow is present (since this corresponds to many actual plasma configurations), as well as on the case when a boundary is also present. We propose that the presence of a boundary can be taken into account by adopting the Caputo fractional derivatives for the side of the boundary (here, the left side), while the Riemann-Liouville derivative is used for the unbounded side (here, the right side). These derivatives are used to write the fractional diffusion–advection equation. We look for solutions in the steady-state case, as such solutions are of practical interest for comparison with observations both in laboratory and astrophysical plasmas. It is shown that the solutions in the completely asymmetric cases have the form of Mittag-Leffler functions in the case of the left fractional contribution, and the form of an exponential decay in the case of the right fractional contribution. Possible applications to space plasmas are discussed.

Numerical Method for Solving Fractional Optimal Control Problems

2009 ◽  
Author(s):  
Raj Kumar Biswas ◽  
Siddhartha Sen
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Numerical Technique ◽  
Control Problems ◽  
Caputo Fractional Derivatives ◽  
Fractional Optimal Control ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Fractional Optimal Control Problems ◽  
The Right

A numerical technique for the solution of a class of fractional optimal control problems has been proposed in this paper. The technique can used for problems defined both in terms of Riemann-Liouville and Caputo fractional derivatives. In this technique a Reflection Operator is used to convert the right Riemann-Liouville derivative into an equivalent left Riemann-Liouville derivative, and then the two point boundary value problem is solved numerically. The proposed method is straightforward and it uses an available numerical technique to solve fractional differential equations resulting from the formulation. Examples considered here show that the numerical results obtained using this and other techniques agree very well.

scholarly journals On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 41
Author(s):  
Ravshan Ashurov ◽  
Yusuf Fayziev
Keyword(s):  
Fractional Derivatives ◽  
Separable Hilbert Space ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Nonlocal Problems ◽  
Existence Of A Solution ◽  
Left Hand ◽  
Liouville Derivative ◽  
The Right

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

A REACTION–DIFFUSION–ADVECTION EQUATION WITH COMBUSTION NONLINEARITY ON THE HALF-LINE

2018 ◽  
Vol 98 (2) ◽  
pp. 277-285
Author(s):  
FANG LI ◽  
QI LI ◽  
YUFEI LIU
Keyword(s):  
Boundary Condition ◽  
Wave Speed ◽  
Reaction Diffusion ◽  
Robin Boundary Condition ◽  
Travelling Wave ◽  
Half Line ◽  
Advection Equation ◽  
The Right ◽  
Combustion Nonlinearity ◽  
Robin Boundary

We study the dynamics of a reaction–diffusion–advection equation $u_{t}=u_{xx}-au_{x}+f(u)$ on the right half-line with Robin boundary condition $u_{x}=au$ at $x=0$, where $f(u)$ is a combustion nonlinearity. We show that, when $0<a<c$ (where $c$ is the travelling wave speed of $u_{t}=u_{xx}+f(u)$), $u$ converges in the $L_{loc}^{\infty }([0,\infty ))$ topology either to $0$ or to a positive steady state; when $a\geq c$, a solution $u$ starting from a small initial datum tends to $0$ in the $L^{\infty }([0,\infty ))$ topology, but this is not true for a solution starting from a large initial datum; when $a>c$, such a solution converges to $0$ in $L_{loc}^{\infty }([0,\infty ))$ but not in $L^{\infty }([0,\infty ))$ topology.

scholarly journals On the Kolmogorov forward equations within Caputo and Riemann-Liouville fractions derivatives

2016 ◽  
Vol 24 (3) ◽  
pp. 5-19
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Laplace Transforms ◽  
Caputo Fractional Derivatives ◽  
The Right ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

AbstractIn this work, we focus on the fractional versions of the well-known Kolmogorov forward equations. We consider the problem in two cases. In case 1, we apply the left Caputo fractional derivatives for α ∈ (0, 1] and in case 2, we use the right Riemann-Liouville fractional derivatives on R+, for α ∈ (1, +∞). The exact solutions are obtained for the both cases by Laplace transforms and stable subordinators.

scholarly journals Analytical solutions for the fractional diffusion-advection equation describing super-diffusion

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 668-675 ◽  
Author(s):  
Francisco Gómez ◽  
Enrique Escalante ◽  
Celia Calderón ◽  
Luis Morales ◽  
Mario González ◽  
...  
Keyword(s):  
Quantum Optics ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Turbulent Diffusion ◽  
Fractional Diffusion ◽  
Mathematical Representation ◽  
Advection Equation ◽  
Physical Systems ◽  
Complex Processes ◽  
Alternative Construction

AbstractThis paper presents the alternative construction of the diffusion-advection equation in the range (1; 2). The fractional derivative of the Liouville-Caputo type is applied. Analytical solutions are obtained in terms of Mittag-Leffler functions. In the range (1; 2) the concentration exhibits the superdiffusion phenomena and when the order of the derivative is equal to 2 ballistic diffusion can be observed, these behaviors occur in many physical systems such as semiconductors, quantum optics, or turbulent diffusion. This mathematical representation can be applied in the description of anomalous complex processes.

scholarly journals X. A method of measuring the pressure produced in the detonation of high, explosives or by the impact of bullets

1914 ◽  
Vol 213 (497-508) ◽  
pp. 437-456 ◽  
Keyword(s):  
Practical Interest ◽  
Direct Determination ◽  
High Explosives ◽  
Time Curve ◽  
Considerable Difficulty ◽  
Cylindrical Steel ◽  
Set Up ◽  
The Right ◽  
The Impact

The determination of the actual pressures produced by a blow such as that of a rifle bullet or by the detonation of high explosives is a problem of much scientific and practical interest but of considerable difficulty. It is easy to measure the transfer of momentum associated with the blow, which is equal to the average pressure developed, multiplied by the time during which it acts, but the separation of these two factors has not hitherto been effected. The direct determination of a force acting for a few hundred-thousandths of a second presents difficulties which may perhaps be called insuperable, but the measurement of the other factor, the duration of the blow, is more feasible. In the case of impacts such as those of spheres or rods moving at moderate velocities the time of contact can be determined electrically with considerable accuracy.* The present paper contains an account of a method of analysing experimentally more violent blows and of measuring their duration and the pressures developed. If a rifle bullet be fired against the end of a cylindrical steel rod there is a definite pressure applied on the end of the rod at each instant of time during the period of impact and the pressure can be plotted as a function of the time. The pressure-time curve is a perfectly definite thing, though the ordinates are expressed in tons and the abscissae in millionths of a second; the pressure starts when the nose of the bullet first strikes the end of the rod and it continues until the bullet has been completely set up or stopped by the impact. Subject to qualifications, which will be considered later, the result of applying this varying pressure to the end is to send along the rod a wave of pressure which, so long as the elasticity is perfect, travels without change of type. If the pressure in different sections of the rod be plotted at any instant (fig. l) then at a later time the same curve shifted to the right by a distance proportional to the time will represent the then distribution of pressure. The velocity with which the wave travels in steel is approximately 17,000 feet per second. As the wave travels over any section of the rod, that section successively experiences pressures represented by the successive ordinates of the curve as they pass over it. Thus the curve also represents the relation between the pressure at any point of the rod and the time, the scale being such that one inch represents the time taken by the wave to travel that distance which is very nearly 1/200,000 of a second. In particular the curve giving the distribution of pressure in the rod along its length is, assuming perfect elasticity, the same as the curve connecting the pressure applied at the end and the time, the scale of time being that just given.

scholarly journals Topography of lymph nodes around the right main bronchus

2020 ◽  
pp. 5-9
Author(s):  
O. I. Fedulov ◽  
V. V. Maslyakov
Keyword(s):  
Lymph Nodes ◽  
Main Bronchus ◽  
Age Groups ◽  
Practical Interest ◽  
Forensic Autopsy ◽  
Lung Pathology ◽  
Age Group ◽  
Thoracic Cavity ◽  
Topographic Features ◽  
The Right

The study is based on the study of 30 organ complexes from the human thoracic cavity, distributed into 5 age groups with 10-year interval, 6 observations in each group: 1st age group – 20–30 years, 2nd age group – 31–40 years, 3rd age group – 41–50 years, 4th age group – 51–60 years, 5th age group – 61–70 years. All medications were taken from people without heart and lung pathology at forensic autopsy according to the generally accepted method. Based on the results obtained, topographic features of lymph nodes location around the right main bronchus are given, which is of practical interest.

scholarly journals PLANNING OF THE LEGISLATIVE ACTIVITY AND ITS IMPORTANCE IN THE IMPROVEMENT OF LEGISLATION AND THE LEGISLATIVE PROCESS OF THE REPUBLIC OF KAZAKHSTAN

2017 ◽  
pp. 28-35
Author(s):  
Жанна Тлембаева ◽  
Zhanna Tlembaeva
Keyword(s):  
Practical Interest ◽  
Legislative Process ◽  
Information Transparency ◽  
Legislative Initiative ◽  
Legislative Activity ◽  
The Republic ◽  
The Government ◽  
The Right ◽  
Legislative Activities

Some issues of lawmaking activity planning in the Republic of Kazakhstan as one of the important components of legislative activity are discussed, and its importance in improving legislation is analyzed in the article. The author pays special attention to the types and stages of the legislative process In the Republic of Kazakhstan. The main problems of planning the legislative activity of the Government and of other subjects of lawmaking are considered. Also the ways to improve the planning of lawmaking activity taking into account the current realities of the development of the legislative process in the Republic of Kazakhstan are proposed. Planning of legislative activities in Kazakhstan needs to be improved and, first of all, by means of increasing the information transparency of planning, the development of forecasting, improving the coordination of planning of subjects of the right of legislative initiative and the development of regulatory support for planning. The issues of application of technologies of legislative forecasting as an obligatory element of lawmaking are separately considered. The conclusion about the role of planning of lawmaking activity in counteraction to the processes of «shadow lobbying» is substantiated. It seems that the implementation of these proposals will ensure an increased role for planning in the country’s legislative process. In the context of the problems studied, the question of the legislative activity of the subjects of the legislative initiative and the subjects of lawmaking has considerable scientific and practical interest. The author reveals a tendency to reduce the lawmaking activity of the deputies of the Parliament against the backdrop of the growing legislative activity of the Government.

scholarly journals Solving a Higher-Dimensional Time-Fractional Diffusion Equation via the Fractional Reduced Differential Transform Method

2021 ◽  
Vol 5 (4) ◽  
pp. 168
Author(s):  
Salah Abuasad ◽  
Saleh Alshammari ◽  
Adil Al-rabtah ◽  
Ishak Hashim
Keyword(s):  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Diffusion Equations ◽  
Caputo Fractional Derivatives ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
Higher Dimensional ◽  
Differential Transform ◽  
Reduced Differential Transform Method

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions.

scholarly journals Continuous action right knife unit with ground interaction analysis

2020 ◽  
Vol 17 (4) ◽  
pp. 452-463
Author(s):  
V. A. Nikolayev ◽  
D. I. Troshin
Keyword(s):  
Total Energy ◽  
Soil Layer ◽  
Practical Interest ◽  
Longitudinal Force ◽  
Energy Costs ◽  
Continuous Action ◽  
Action Unit ◽  
The Face ◽  
The Right ◽  
The Impact

Introduction. To solve the problem of accelerating the construction of roads, improving their quality, it is advisable to use a continuous action unit to form a underlying layer. The main working bodies of this unit are buckets, which cut off the soil layer from below and on the side. At the same time, the bottom knife cuts off the ground layer from below, the right knife on the side, and the console knife partially cuts the top layer of soil from below for the next bucket. In particular, the analysis of interaction with the soil of the right knife of the continuous action unit is of theoretical and practical interest. To do this, the right knife is divided into elements and analyzed the interaction of these elements with the ground. The consistent impact on the soil of many right knives, within the width of the grip of the unit, is replaced by the impact on the ground of one conventional right knife at a distance necessary for the development of one cubic meter of soil. The forces of interaction of the conventional right knife with the ground are called conditional forces.The method of research. The method for calculating the energy costs during punching the right knife into the ground is shown: on separating the formation of the ground from its body, on overcoming the ground friction on the edge of the blade, on overcoming the ground pressure on the edge of the blade, on accelerating the ground of the blade by means of the axle, on overcoming the ground friction on the shelf, to overcome the ground friction against the outside surface.The total energy costs of interacting with a soil of one cubic meter are derived from the addition of private energy costs. The method of calculating the horizontal longitudinal force needed to move the right knife is given.Results. On the basis of the methodology developed, energy costs are calculated when introducing the right knife into the ground: on separating the soil from its body, on overcoming the friction of the ground on the edge of the blade, on overcoming the pressure of the ground on the face of the blade, on the acceleration of the ground with a fascia blade, on overcoming the ground friction on the face. The total energy costs of the right knife interact with the soil of one cubic meter. The horizontal long-lived force needed to move the right knife has been determined.Conclusion. As a result of the calculations: the energy needed to cut the ground with the right knives, more than 71 J/cube. The horizontal longitudinal force needed to move the right knife is 730 N. To determine the total energy spent on cutting the ground by buckets of the unit to remove the top layer of soil from the underlying layer of the road, it is necessary to analyze the interaction with the soil of other elements of the bucket.

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