scholarly journals Solution of Euler’s Differential Equation and AC-Laplace Transform of Inverse Power Functions and Their Pseudofunctions, in Nonstandard Analysis

2021 ◽  
pp. 47-60
Author(s):  
Tohru Morita
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Complex Number ◽  
Nonstandard Analysis ◽  
Laplace Transforms ◽  
Power Functions ◽  
Liouville Fractional Derivative ◽  
Positive Integers

It is shown that the index law of the Riemann-Liouville fractional derivative is recovered when nonstandard analysis is applied, and then the solutions of Euler’s differential equation are obtained in nonstandard analysis, where infinitesimal number appears. They are given in the form, from which the solutions in distribution theory are obtained. In the derivation, the AC-Laplace transforms of functions tν and tν(loge t) m for complex number ν and positive integer m, are used. By using these formulas, the AC-Laplace transforms of functions t− n + andt− n +(loge t) m for positive integers n and m, and their pseudofunctions are obtained with the aid of nonstandard analysis.

scholarly journals Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Chen Yang ◽  
Jieming Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Point Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:D0+αu(t)+f(t,u,u',…,u(n-2))+g(t)=0, 0<t<1, n-1<α≤n, n≥2,u(0)=u'(0)=⋯=u(n-2)(0)=u(n-2)(1)=0, whereD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.

scholarly journals Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2117
Author(s):  
Tohru Morita ◽  
Ken-ichi Sato
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Nonstandard Analysis ◽  
Distribution Theory ◽  
Nonregular Distributions

For Euler’s differential equation of order n, a theorem is presented to give n solutions, by modifying a theorem given in a recent paper of the present authors in J. Adv. Math. Comput. Sci. 2018; 28(3): 1–15, and then the corresponding theorem in distribution theory is given. The latter theorem is compared with recent studies on Euler’s differential equation in distribution theory. A supplementary argument is provided on the solutions expressed by nonregular distributions, on the basis of nonstandard analysis and Laplace transform.

scholarly journals On the sulutions of a fractional differemtial equation

2016 ◽  
Vol 12 (2) ◽  
pp. 5925-5927
Author(s):  
Runqing Cui
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Complex Transform ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We have showed the results obtained in [1] are incorrect and the fractional complex transform is invalid to the fractional differential equation which contain modified Riemann-Liouville fractional derivative.

Some solutions of the nonhomogeneous Bagley–Torvik equation

2019 ◽  
Vol 10 (01) ◽  
pp. 1941002 ◽  
Author(s):  
Temirkhan S. Aleroev ◽  
Sergey Erokhin
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Numerical Solutions ◽  
Experimental Information ◽  
Polymer Concrete ◽  
The Laplace Transform ◽  
Nonhomogeneous Differential Equation ◽  
The Right ◽  
Derivative Order

In this study, nonhomogeneous differential equation of the second order is considered, which contains fractional derivative (Bagley–Torvik equation), where the derivative order ranges within 1 and 2. This equation is applied in mechanics of oscillation processes. To study the equation, we use the Laplace transform, which allows us to obtain an image of the solution in an explicit form. Two typical kinds of functions of the right-hand side of the equation are considered. Numerical solutions are constructed for each of them. The solutions obtained are compared with experimental information on polymer concrete samples. The comparison allows for the conclusion about the adequacy of the numerical and analytical solutions to the nonhomogeneous Bagley–Torvik equation.

scholarly journals Extended Jacobi Functions via Riemann-Liouville Fractional Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Bayram Çekim ◽  
Esra Erkuş-Duman
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Jacobi Polynomials ◽  
Liouville Fractional Derivative ◽  
Jacobi Functions ◽  
Fractional Differential

By means of the Riemann-Liouville fractional calculus, extended Jacobi functions are de…fined and some of their properties are obtained. Then, we compare some properties of the extended Jacobi functions extended Jacobi polynomials. Also, we derive fractional differential equation of generalized extended Jacobi functions.

scholarly journals Oscillation Behavior for a Class of Differential Equation with Fractional-Order Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shouxian Xiang ◽  
Zhenlai Han ◽  
Ping Zhao ◽  
Ying Sun
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Generalized Riccati Transformation ◽  
Fractional Differential ◽  
Oscillation Theorems ◽  
Positive Integers

By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation[atpt+qtD-αxt)γ′ − b(t)f∫t∞‍(s-t)-αx(s)ds = 0, fort⩾t0>0, whereD-αxis the Liouville right-sided fractional derivative of orderα∈(0,1)ofxandγis a quotient of odd positive integers. The results in this paper extend and improve the results given in the literatures (Chen, 2012).

A boundary value problem for a partial differential equation with fractional derivative

2021 ◽  
Vol 24 (2) ◽  
pp. 509-517
Author(s):  
Menglibay Ruziev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Moisture Transfer ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Generalized Equation ◽  
Nonlocal Boundary Value Problem ◽  
Liouville Fractional Derivative

Abstract In this paper, we investigate a nonlocal boundary value problem for an equation of special type. For y > 0 it is a fractional diffusion equation, which contains the Riemann-Liouville fractional derivative. For y < 0 it is a generalized equation of moisture transfer. A unique solvability of the considered problem is proved.

scholarly journals Analytic Solution for theRLElectric Circuit Model in Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
P. V. Shah ◽  
A. D. Patel ◽  
I. A. Salehbhai ◽  
A. K. Shukla
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Analytic Solution ◽  
Electrical Circuit ◽  
Special Cases ◽  
The Laplace Transform ◽  
Fractional Differential

This paper provides an analytic solution ofRLelectrical circuit described by a fractional differential equation of the order0<α≤1. We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.

Sign in / Sign up

Export Citation Format

Share Document