scholarly journals Spectral approximations to the fractional integral and derivative

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Changpin Li ◽  
Fanhai Zeng ◽  
Fawang Liu
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Caputo Derivative ◽  
Initial Value Problems ◽  
Fractional Integral ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Initial Value ◽  
Numerical Examples ◽  
Spectral Approximations

AbstractIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

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.

Existence and uniqueness for two-point boundary value problems

1981 ◽  
Vol 88 (3-4) ◽  
pp. 219-234 ◽  
Author(s):  
John V. Baxley ◽  
Sarah E. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Point Boundary ◽  
Initial Value

SynopsisBoundary value problems associated with y″ = f(x, y, y′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f(x, y,y′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g(y(0), y′(0)) = 0, h(y(0), y′(0), y(1), y′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.

High Order Numerical Methods for Fractional Terminal Value Problems

2014 ◽  
Vol 14 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Neville J. Ford ◽  
Maria L. Morgado ◽  
Magda Rebelo
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Scheme ◽  
Initial Value ◽  
Shooting Algorithm ◽  
Effective Approximation

Abstract. In this paper we present a shooting algorithm to solve fractional terminal (or boundary) value problems. We provide a convergence analysis of the numerical method, derived based upon properties of the equation being solved and without the need to impose smoothness conditions on the solution. The work is a sequel to our recent investigation where we constructed a nonpolynomial collocation method for the approximation of the solution to fractional initial value problems. Here we show that the method can be adapted for the effective approximation of the solution of terminal value problems. Moreover, we compare the efficiency of this numerical scheme against other existing methods.

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