Densities of scaling limits of coupled continuous time random walks

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Marcin Magdziarz ◽  
Tomasz Zorawik
Keyword(s):  
Fractional Differential Equations ◽  
Continuous Time ◽  
Stability Index ◽  
Scaling Limits ◽  
Explicit Formulas ◽  
Material Derivative ◽  
Lévy Walks ◽  
Continuous Time Random Walks ◽  
Fractional Differential ◽  
The Stability

AbstractIn this paper we derive explicit formulas for the densities of Lévy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represented as an integral of Meijer

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

scholarly journals Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

scholarly journals Non-Linear Macroeconomic Models of Growth with Memory

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2078 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Economic Growth ◽  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Continuous Time ◽  
Non Linear ◽  
Standard Models ◽  
Fractional Differential ◽  
Derivatives Of ◽  
With Memory

In this article, two well-known standard models with continuous time, which are proposed by two Nobel laureates in economics, Robert M. Solow and Robert E. Lucas, are generalized. The continuous time standard models of economic growth do not account for memory effects. Mathematically, this is due to the fact that these models describe equations with derivatives of integer orders. These derivatives are determined by the properties of the function in an infinitely small neighborhood of the considered time. In this article, we proposed two non-linear models of economic growth with memory, for which equations are derived and solutions of these equations are obtained. In the differential equations of these models, instead of the derivative of integer order, fractional derivatives of non-integer order are used, which allow describing long memory with power-law fading. Exact solutions for these non-linear fractional differential equations are obtained. The purpose of this article is to study the influence of memory effects on the rate of economic growth using the proposed simple models with memory as examples. As the methods of this study, exact solutions of fractional differential equations of the proposed models are used. We prove that the effects of memory can significantly (several times) change the growth rate, when other parameters of the model are unchanged.

scholarly journals Qualitative Analysis of Implicit Dirichlet Boundary Value Problem for Caputo-Fabrizio Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Rozi Gul ◽  
Muhammad Sarwar ◽  
Kamal Shah ◽  
Thabet Abdeljawad ◽  
Fahd Jarad
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Fractional Differential Equations ◽  
Dirichlet Boundary ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Dirichlet Boundary Conditions ◽  
Fractional Differential ◽  
The Stability ◽  
Implicit Fractional Differential Equations

This article studies a class of implicit fractional differential equations involving a Caputo-Fabrizio fractional derivative under Dirichlet boundary conditions (DBCs). Using classical fixed-point theory techniques due to Banch’s and Krasnoselskii, a qualitative analysis of the concerned problem for the existence of solutions is established. Furthermore, some results about the stability of the Ulam type are also studied for the proposed problem. Some pertinent examples are given to justify the results.

Correlated continuous-time random walks—scaling limits and Langevin picture

2012 ◽  
Vol 2012 (04) ◽  
pp. P04010 ◽  
Author(s):  
Marcin Magdziarz ◽  
Ralf Metzler ◽  
Wladyslaw Szczotka ◽  
Piotr Zebrowski
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Scaling Limits ◽  
Continuous Time Random Walks
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