scholarly journals A Fractional Equation with Left-Sided Fractional Bessel Derivatives of Gerasimov–Caputo Type

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1216 ◽  
Author(s):  
Elina Shishkina ◽  
Sergey Sitnik
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Transform ◽  
Bessel Operator ◽  
Explicit Solutions ◽  
Solutions To Equations ◽  
Fractional Powers ◽  
Derivatives Of ◽  
Fractional Equation

In this article we propose and study a method to solve ordinary differential equations with left-sided fractional Bessel derivatives on semi-axes of Gerasimov–Caputo type. We derive explicit solutions to equations with fractional powers of the Bessel operator using the Meijer integral transform.

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 the solution of two-sided fractional ordinary differential equations of Caputo type

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Ma. Elena Hernández-Hernández ◽  
Vassili N. Kolokoltsov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Well Posedness ◽  
Solutions To Equations ◽  
Fractional Differential ◽  
The Right ◽  
Stochastic Representations ◽  
Exit Problems ◽  
Special Case

AbstractThis paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain Lévy processes.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

scholarly journals Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application

2021 ◽  
Vol 5 (3) ◽  
pp. 130
Author(s):  
Ahmed Ibrahim El-Mesady ◽  
Yaser Salah Hamed ◽  
Abdullah M. Alsharif
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Transform ◽  
Inverse Operator ◽  
Integral Transforms ◽  
Delta Function ◽  
Medical Application ◽  
Heaviside Function ◽  
Algebraic Equations ◽  
General Integral

Integral transformations are essential for solving complex problems in business, engineering, natural sciences, computers, optical science, and modern mathematics. In this paper, we apply a general integral transform, called the Jafari transform, for solving a system of ordinary differential equations. After applying the Jafari transform, ordinary differential equations are converted to a simple system of algebraic equations that can be solved easily. Then, by using the inverse operator of the Jafari transform, we can solve the main system of ordinary differential equations. Jafari transform belongs to the class of Laplace transform and is considered a generalization to integral transforms such as Laplace, Elzaki, Sumudu, G\_transforms, Aboodh, Pourreza, etc. Jafari transform does not need a large computational work as the previous integral transforms. For the Jafari transform, we have studied some valuable properties and theories that have not been studied before. Such as the linearity property, scaling property, first and second shift properties, the transformation of periodic functions, Heaviside function, and the transformation of Dirac’s delta function, and so on. There is a mathematical model that describes the cell population dynamics in the colonic crypt and colorectal cancer. We have applied the Jafari transform for solving this model.

scholarly journals On the Mohand Transform and Ordinary Differential Equations with Variable Coefficients

2020 ◽  
Vol 35 (1) ◽  
pp. 01-06
Author(s):  
Mohamed E. Attaweel ◽  
Haneen Almassry
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Integral Transform ◽  
Variable Coefficients ◽  
Sumudu Transforms ◽  
Differential And Integral Equations

The Mohand transform is a new integral transform introduced by Mohand M. Abdelrahim Mahgoub to facilitate the solution of differential and integral equations. In this article, a new integral transform, namely Mohand transform was applied to solve ordinary differential equations with variable coefficients by using the modified version of Laplace and Sumudu transforms.

scholarly journals A diagonal recurrence relation for the Stirling numbers of the first kind

2018 ◽  
Vol 12 (1) ◽  
pp. 153-165 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recurrence Relation ◽  
Stirling Numbers ◽  
Bernstein Function ◽  
Linear Ordinary Differential Equations ◽  
The Family ◽  
Significant Expression ◽  
Lerch Transcendent ◽  
Derivatives Of

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

Computer Programming

SIMULATION ◽  
1965 ◽  
Vol 4 (5) ◽  
pp. 317-323 ◽  
Author(s):  
Joseph L. Hammond
Keyword(s):  
Differential Equations ◽  
Computer Program ◽  
Ordinary Differential Equations ◽  
Computer Programming ◽  
Characteristic Root ◽  
Analog Computer ◽  
State Variables ◽  
State Variable ◽  
Forcing Function ◽  
Derivatives Of

State variable techniques are reviewed and applied to analog computer programming. The concise rep resentation for ordinary differential equations made possible by this technique is used to formulate a gen eral program for all such equations. It is shown that an analog computer program based on state variables will not have redundant integrators. The fact that the use of state variables facilitates the choice of variables internal to an analog com puter program is illustrated by two techniques, namely, (1) a technique for avoiding derivatives of the forcing function in programming a large class of ordinary differential equations, and (2) a technique for simulating certain systems in such a way that the effect of each characteristic root is placed in evi dence.

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