scholarly journals On self-similar solutions of time and space fractional sub-diffusion equations

2021 ◽  
Vol 11 (3) ◽  
pp. 16-27
Author(s):  
Fatma Al-Musalhi ◽  
Erkinjon Karimov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivatives ◽  
Essential Role ◽  
Differential Operators ◽  
Diffusion Equations ◽  
Fractional Partial Differential Equation ◽  
Successive Iteration ◽  
Self Similar ◽  
Similar Solutions

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.

Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

2018 ◽  
Vol 2020 (23) ◽  
pp. 9440-9470
Author(s):  
Jian Lu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Smooth Function ◽  
Existence Result ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

Abstract In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}$$is studied, where $f$ is a positive $2\pi $-periodic smooth function. By virtue of a new generalized Blaschke–Santaló inequality, we obtain a new existence result of solutions.

scholarly journals Alternative Form of Ordinary Differential Equation of Electroencephalography Signals During an Epileptic Seizure

2021 ◽  
Vol 17 (2) ◽  
pp. 109-113
Author(s):  
Ameen Omar Barja
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Epileptic Seizure ◽  
Essential Role ◽  
Clinical Neurophysiology ◽  
Alternative Form ◽  
Eeg Signals ◽  
Seizure Disorders ◽  
Electroencephalogram Eeg ◽  
The Brain

One of the most important fields in clinical neurophysiology is an electroencephalogram (EEG). It is a test used to detect problems related to the brain electrical activity, and it can track and records patterns of brain waves. EEG continues to play an essential role in diagnosis and management of patients with epileptic seizure disorders. Nevertheless, the outcome of EEG as a tool for evaluating epileptic seizure is often interpreted as a noise rather than an ordered pattern. The mathematical modelling of EEG signals provides valuable data to neurologists, and is heavily utilized in the diagnosis and treatment of epilepsy. EEG signals during the seizure can be modeled as ordinary differential equation (ODE). In this study we will present an alternative form of ODE of EEG signals through the seizure.

Integrable-square solutions of a singular ordinary differential equation

1981 ◽  
Vol 89 (3-4) ◽  
pp. 267-279 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Characteristic Polynomial ◽  
Characteristic Equation ◽  
Complex Plane ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Singular Ordinary Differential Equation ◽  
Ordinary Differential Operators

SynopsisAbsolutely square integrable solutions are determined for the equation= λywhere the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions asxapproaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

scholarly journals Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

2021 ◽  
Vol 5 (2) ◽  
pp. 48
Author(s):  
Alessandro De Gregorio ◽  
Roberto Garra
Keyword(s):  
Differential Equation ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Diffusion Equations ◽  
Heat Equations ◽  
Abstract Differential Equation ◽  
Fractional Heat Equation ◽  
Random Models ◽  
Particular Solution ◽  
Fractional Cauchy Problem

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

On Linear Ordinary Differential Equation With Orthogonal Polynomial Solutions

2018 ◽  
Vol 1 (20) ◽  
pp. 559-572
Author(s):  
ولدان وليد محمود
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Differential Operators ◽  
Linear Ordinary Differential Equation ◽  
Large Piece ◽  
Polynomial Solutions ◽  
The Subject

: The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. In this paper we give a  survey of the orthogonal polynomial solutions of second-order and fourth-order linear ordinary differential equation, on the generated self-adjoint differential operators .

scholarly journals An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions

2021 ◽  
pp. 2327-2333
Author(s):  
Nabaa N. Hasan ◽  
Omar H. Salim
Keyword(s):  
Differential Equation ◽  
Heat Equation ◽  
Fractional Derivatives ◽  
Polynomial Spline ◽  
Spline Functions ◽  
Two Dimensional ◽  
Fractional Partial Differential Equation ◽  
Numerical Examples ◽  
One Dimensional ◽  
The One

     The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear non-polynomial spline to a two-dimensional spline  to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.

scholarly journals On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function

2021 ◽  
Vol 24 (5) ◽  
pp. 1559-1570
Author(s):  
Riccardo Droghei
Keyword(s):  
Differential Equation ◽  
Fractional Derivatives ◽  
Special Function ◽  
Special Functions ◽  
Wright Function ◽  
Fractional Partial Differential Equation ◽  
Caputo Fractional Derivatives ◽  
Multi Index ◽  
Multiple Parameters ◽  
Bessel Differential Equation

Abstract In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation.

Sign in / Sign up

Export Citation Format

Share Document