scholarly journals EXISTENCE THEOREM FOR AN INITIAL VALUE PROBLEM OF FRACTIONAL ORDER IN L_1(0,1)

2014 ◽  
Vol 93 (3) ◽  
Author(s):  
I. Abouelfarag ◽  
A.M. Awad
Keyword(s):  
Existence Theorem ◽  
Fractional Order ◽  
Initial Value Problem ◽  
Initial Value

A Numerical Algorithm to Initial Value Problem of Caputo Fractional-Order Differential Equation

2015 ◽  
Author(s):  
Lu Bai ◽  
Dingyü Xue
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Algorithm ◽  
Initial Value Problem ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Initial Value

A numerical algorithm is presented to solve the initial value problem of linear and nonlinear Caputo fractional-order differential equations. Firstly, nonzero initial value problem should be transformed into zero initial value problem. Error analysis has been done to polynomial algorithm, the reason has been found why the calculation error of the algorithm is large. A new algorithm called exponential function algorithm is proposed based on the analysis. The obtained fractional-order differential equation is transformed into difference equation. If the differential equation is linear, the obtained difference equation is explicit, the numerical solution can be solved directly; otherwise, the obtained difference equation is implicit, the predictor of the numerical solution can be obtained with extrapolation algorithm, substitute the predictor into the equation, the corrector can be solved. Error analysis has been done to the new algorithm, the algorithm is of first order.

scholarly journals An existence theorem for solutions of an integro-differential equation in Banach spaces

2012 ◽  
Vol 45 (4) ◽  
Author(s):  
Aldona Dutkiewicz
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Initial Value Problem ◽  
Initial Value ◽  
Measures Of Noncompactness ◽  
Integro Differential Equation ◽  
Local Solutions

AbstractThe paper contains an existence theorem for local solutions of an initial value problem for a nonlinear integro-differential equation in Banach spaces. The assumptions and proofs are expressed in terms of measures of noncompactness.

scholarly journals Initial value problem of fractional order

2015 ◽  
Vol 2 (1) ◽  
pp. 1004797 ◽  
Author(s):  
A. Guezane-Lakoud
Keyword(s):  
Fractional Order ◽  
Initial Value Problem ◽  
Initial Value

scholarly journals Solution of linear fractional order systems with variable coefficients

2020 ◽  
Vol 23 (3) ◽  
pp. 753-763
Author(s):  
Ivan Matychyn ◽  
Viktoriia Onyshchenko
Keyword(s):  
Linear Systems ◽  
Fractional Order ◽  
Initial Value Problem ◽  
Transition Matrix ◽  
State Transition ◽  
Variable Coefficients ◽  
Explicit Solutions ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Theoretical Results

AbstractThe paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville derivatives. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by an example.

Numerical solutions for fractional initial value problems of distributed-order

2021 ◽  
pp. 2150096
Author(s):  
M. A. Abdelkawy
Keyword(s):  
Fractional Order ◽  
Initial Value Problem ◽  
Numerical Solutions ◽  
Jacobi Polynomials ◽  
Algebraic Equations ◽  
Initial Value ◽  
Orthogonal Functions ◽  
Jacobi Functions ◽  
Distributed Order ◽  
Numerical Approaches

This paper addresses spectral collocation techniques to treat with the fractional initial value problem of distributed-order. We introduce three algorithms based on shifted fractional order Jacobi orthogonal functions outputted by Jacobi polynomials. The shifted fractional order Jacobi–Gauss–Radau collocation method is developed for approximating the fractional initial value problem of distributed-order. The principal target in our techniques is to transform the fractional initial value problem of distributed-order to a system of algebraic equations. Some numerical examples are given to test the accuracy and applicability of our technique. It is known that the accuracy of numerical approaches for nonsmooth solution is deteriorated. Employing fractional order Jacobi functions instead of the classical Jacobi stopped this deterioration.

HOPF SOLUTIONS TO A FLUID-ELASTIC INTERACTION MODEL

2008 ◽  
Vol 18 (02) ◽  
pp. 215-269 ◽  
Author(s):  
M. GUIDORZI ◽  
M. PADULA ◽  
P. I. PLOTNIKOV
Keyword(s):  
Global Existence ◽  
Existence Theorem ◽  
Initial Value Problem ◽  
Interaction Model ◽  
Elastic Interaction ◽  
Initial Condition ◽  
Two Dimensional ◽  
Initial Value ◽  
Global Existence Theorem ◽  
Model Equations

In this paper, we give a global existence theorem of weak solutions to model equations governing interaction fluid structure in a two-dimensional layer, cf. Refs. 8 and 14. To our knowledge this is the first existence theorem of global in time solutions for such model. The interest of our result is double because, first, we change the original initial value problem by deleting one initial condition, second, we construct a solution through the classical Galerkin method for which several computing codes have been constructed.

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