scholarly journals On the Solutions of Okubo-Type Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Galina Filipuk ◽  
Alberto Lastra
Keyword(s):  
Integral Equation ◽  
Power Series ◽  
Differential Equations ◽  
Recurrence Relation ◽  
Volterra Integral Equation ◽  
Type System ◽  
Type Systems ◽  
Solution Vector

We show that for certain systems of Okubo-type, we can find a solution vector, all components of which are expressed in terms of the first one. This first component can be expressed in two ways. It solves a Volterra integral equation with the kernel expressed in terms of the solutions of a reduced Okubo-type system of smaller dimension. It is also expressed as a power series about the origin with coefficients satisfying certain recurrence relation. This extends the results in (W. Balser, C. Röscheisen, J. Differential Equations, 2009).

scholarly journals Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

2021 ◽  
Vol 45 (4) ◽  
pp. 571-585
Author(s):  
AMIRAHMAD KHAJEHNASIRI ◽  
◽  
M. AFSHAR KERMANI ◽  
REZZA EZZATI ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Volterra Integral Equation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Basis Functions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Operational Matrix Method

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

Numerical solution of fractional differential equations via a Volterra integral equation approach

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Shahrokh Esmaeili ◽  
Mostafa Shamsi ◽  
Mehdi Dehghan
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Volterra Integral Equation ◽  
Equation Approach ◽  
Algebraic Equations ◽  
Integral Equation Approach ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
The Given

AbstractThe main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.

Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations

2018 ◽  
Vol 68 (1) ◽  
pp. 77-88
Author(s):  
Marcin Borkowski ◽  
Daria Bugajewska
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Integral Equations ◽  
Bounded Variation ◽  
Volterra Integral Equation ◽  
Functions Of Bounded Variation ◽  
Hammerstein Integral Equation ◽  
Linear Differential ◽  
Unbounded Intervals ◽  
Differential And Integral Equations

Abstract In this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

Solving Volterra integral equation by using a new transformation

2021 ◽  
Vol 24 (3) ◽  
pp. 735-741
Author(s):  
Sahar Muhsen Jaabar ◽  
Ahmed Hadi Hussain
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation

On approximate solutions of the generalized Volterra integral equation

2014 ◽  
Vol 20 ◽  
pp. 59-66 ◽  
Author(s):  
Anna Bahyrycz ◽  
Janusz Brzdȩk ◽  
Zbigniew Leśniak
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Approximate Solutions

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

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