scholarly journals Lucas polynomials semi-analytic solution for fractional multi-term initial value problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mahmoud M. Mokhtar ◽  
Amany S. Mohamed
Keyword(s):  
Initial Value Problems ◽  
Analytic Solution ◽  
Collocation Methods ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Initial Value ◽  
Lucas Polynomials ◽  
Error Analyses ◽  
Generalized Lucas Polynomials

AbstractHerein, we use the generalized Lucas polynomials to find an approximate numerical solution for fractional initial value problems (FIVPs). The method depends on the operational matrices for fractional differentiation and integration of generalized Lucas polynomials in the Caputo sense. We obtain these solutions using tau and collocation methods. We apply these methods by transforming the FIVP into systems of algebraic equations. The convergence and error analyses are discussed in detail. The applicability and efficiency of the method are tested and verified through numerical examples.

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 Error Upper Bounds for a Computational Method for Nonlinear Boundary and Initial-Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Osvaldo Guimarães ◽  
José Roberto C. Piqueira
Keyword(s):  
Hilbert Space ◽  
Initial Value Problems ◽  
Nonlinear Boundary ◽  
Upper Bounds ◽  
Computational Approach ◽  
Computational Method ◽  
Operational Matrices ◽  
Initial Value ◽  
Accuracy Measures

This work develops a computational approach for boundary and initial-value problems by using operational matrices, in order to run an evolutive process in a Hilbert space. Besides, upper bounds for errors in the solutions and in their derivatives can be estimated providing accuracy measures.

scholarly journals A Nonclassical Radau Collocation Method for Nonlinear Initial-Value Problems with Applications to Lane-Emden Type Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Maleki ◽  
M. Tavassoli Kajani ◽  
I. Hashim ◽  
A. Kilicman ◽  
K. A. M. Atan
Keyword(s):  
Orthogonal Polynomials ◽  
Numerical Method ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Weighted Interpolation ◽  
Collocation Points ◽  
Present Solution ◽  
Nonlinear Algebraic Equations

We propose a numerical method for solving nonlinear initial-value problems of Lane-Emden type. The method is based upon nonclassical Gauss-Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial-value problems to a system of nonlinear algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is very accurate.

scholarly journals A Numerical Method for Lane-Emden Equations Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Changqing Yang ◽  
Jianhua Hou
Keyword(s):  
Differential Equation ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions

A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

scholarly journals Analytical solution of a class of coupled second order differential-difference equations

1993 ◽  
Vol 16 (2) ◽  
pp. 385-396 ◽  
Author(s):  
L. Jódar ◽  
J. A. Martin Alustiza
Keyword(s):  
Analytical Solution ◽  
Difference Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Initial Value

In this paper coupled systems of second order differential-difference equations are considered. By means of the concept of co-solution of certain algebraic equations associated to the problem, an analytical solution of initial value problems for coupled systems of second order differential-difference equations is constructed.

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>

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