scholarly journals Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

2019 ◽  
Vol 25 (1) ◽  
Author(s):  
Kunio Hidano ◽  
Chengbo Wang
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Fractional Derivatives ◽  
Composite Functions ◽  
Half Wave ◽  
The Cauchy Problem ◽  
Derivatives Of

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

2021 ◽  
pp. 15-24
Author(s):  
Tatiana F. Dolgikh
Keyword(s):  
Cauchy Problem ◽  
Green Function ◽  
Differential Equations ◽  
Shallow Water ◽  
Ordinary Differential Equations ◽  
Riemann Invariants ◽  
Hodograph Method ◽  
First Order ◽  
The Cauchy Problem ◽  
Derivatives Of

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

scholarly journals ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

2020 ◽  
Vol 8 (2) ◽  
pp. 24-39
Author(s):  
V. Gorodetskiy ◽  
R. Kolisnyk ◽  
O. Martynyuk
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Initial Condition ◽  
Partial Derivatives ◽  
Time Problem ◽  
The Cauchy Problem ◽  
Space Of Generalized Functions ◽  
Derivatives Of

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

Wave equation involving fractional derivatives of real and complex fractional order

2019 ◽  
pp. 327-352
Author(s):  
Teodor M. Atanacković ◽  
Sanja Konjik ◽  
Stevan Pilipović
Keyword(s):  
Wave Equation ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Derivatives Of

scholarly journals Mathematical Modeling of Linear Fractional Oscillators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1879 ◽  
Author(s):  
Roman Parovik
Keyword(s):  
Cauchy Problem ◽  
Fractional Derivatives ◽  
Model Equation ◽  
Initial Conditions ◽  
Memory Function ◽  
Analytical Formula ◽  
Forced Oscillations ◽  
Caputo Fractional Derivatives ◽  
Specific Test ◽  
The Cauchy Problem

In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out.

scholarly journals On the Cauchy Problem for the n-Dimensional Wave Equation

1966 ◽  
Vol 15 (1) ◽  
pp. 29-32
Author(s):  
R. M. Young
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
The Cauchy Problem ◽  
Image Position ◽  
Dimensional Wave

Consider the n-dimensional wave equation

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