scholarly journals Finding symmetries by incorporating initial conditions as side conditions

2008 ◽  
Vol 19 (6) ◽  
pp. 701-715 ◽  
Author(s):  
JOANNA GOARD
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Initial Condition ◽  
Initial Value ◽  
Side Condition ◽  
Symmetry Generator ◽  
Infinitesimal Symmetry

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

scholarly journals A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra’s Population Growth Model

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Majid Tavassoli Kajani ◽  
Mohammad Maleki ◽  
Adem Kılıçman
Keyword(s):  
Population Growth ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Initial Value ◽  
Step Size ◽  
Integral Term ◽  
Multiple Step

A new shifted Legendre-Gauss collocation method is proposed for the solution of Volterra’s model for population growth of a species in a closed system. Volterra’s model is a nonlinear integrodifferential equation on a semi-infinite domain, where the integral term represents the effects of toxin. In this method, by choosing a step size, the original problem is replaced with a sequence of initial value problems in subintervals. The obtained initial value problems are then step by step reduced to systems of algebraic equations using collocation. The initial conditions for each step are obtained from the approximated solution at its previous step. It is shown that the accuracy can be improved by either increasing the collocation points or decreasing the step size. The method seems easy to implement and computationally attractive. Numerical findings demonstrate the applicability and high accuracy of the proposed method.

scholarly journals Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

scholarly journals Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

2021 ◽  
Vol 24 (4) ◽  
pp. 32-39
Author(s):  
Hussein M. Sagban ◽  
◽  
Fadhel S. Fadhel ◽  
Keyword(s):  
Approximate Solution ◽  
Rate Of Convergence ◽  
Initial Value Problem ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

scholarly journals Initial value problems should not be associated to fractional model descriptions whatever the derivative definition used

2021 ◽  
Vol 6 (10) ◽  
pp. 11318-11329
Author(s):  
Jocelyn SABATIER ◽  
◽  
Christophe FARGES
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Model Analysis ◽  
Fractional Differentiation ◽  
Initial Value ◽  
Validity Of Results ◽  
Fractional Model ◽  
Definition Of

<abstract> <p>The paper shows that the Caputo definition of fractional differentiation is problematic if it is used in the definition of a time fractional model and if initial conditions are taken into account. The demonstration is done using simple examples (or counterexamples). The analysis is extended to the Riemann-Liouville and Grünwald-Letnikov definitions. These results thus question the validity of results produced in the field of time fractional model analysis in which initial conditions are involved.</p> </abstract>

On generalized fractional operators and a gronwall type inequality with applications

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5457-5473 ◽  
Author(s):  
Yassine Adjabi ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equations ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Type Inequality ◽  
Weighted Spaces ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Derivatives

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

scholarly journals The Order Completion Method for Systems of Nonlinear PDEs: Solutions of Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Jan Harm van der Walt
Keyword(s):  
Initial Value Problems ◽  
Lower Semicontinuous ◽  
Nonlinear Pdes ◽  
Generalized Solutions ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Initial Condition ◽  
Initial Value ◽  
Extended Sense ◽  
Order Completion

We present an existence result for generalized solutions of initial value problems obtained through the order completion method. The solutions we obtain satisfy the initial condition in a suitable extended sense, and each such solution may be represented in a canonical way through its generalized partial derivatives as nearly finite normal lower semicontinuous function.

scholarly journals Explicit Solutions of Initial Value Problems for Linear Scalar Riemann-Liouville Fractional Differential Equations With a Constant Delay

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 32 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Initial Condition ◽  
Constant Delay ◽  
Initial Value ◽  
Fractional Differential ◽  
Set Up ◽  
Ordinary Derivative ◽  
Linear Scalar

In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.

scholarly journals Addendum to stability analysis of neutral linear fractional system with distributed delays (Filomat 30:3 (2016), 841-851)

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 5013-5017
Author(s):  
Magdalena Veselinova ◽  
Hristo Kiskinov ◽  
Andrey Zahariev
Keyword(s):  
Fractional Integral ◽  
Initial Conditions ◽  
Initial Condition ◽  
Initial Value ◽  
Short Article ◽  
Delayed Argument ◽  
Fractional Differential ◽  
Fractional System ◽  
The Right ◽  
Equations With Delay

In this short article we discuss the initial condition of the initial value problem for fractional differential equations with delayed argument and derivatives in Riemann-Liouville sense. We provide also a new lemma - a ?mirror? analogue of the Kilbas Lemma, concerning the right side Riemann-Liouville fractional integral, which is important for the correct setting of the initial conditions, especially in the case of equations with delay.

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