10 research outputs found
OBLICZENIA NUMERYCZNE POCHODNEJ UŁAMKOWEGO RZĘDU W ZAGADNIENIACH POCZĄTKOWYCH, PRZYKŁADY W PROGRAMACH MATLAB I MATHEMATICA
The paper concerns a numerical method that deals with the computations of the fractional derivative in Caputo and Riemann-Liouville definitions. The method can be applied in time stepping processes of initial value problems. The name of the method is SubIval, which is an acronym of its previous name – the subinterval-based method. Its application in solving systems of fractional order state equations is presented. The method has been implemented into an ActiveX DLL. Exemplary MATLAB and Mathematica codes are given, which provide guidance on how the DLL can be used.Artykuł dotyczy numerycznej metody, którą wykorzystać można do obliczeń pochodnej ułamkowego rzędu w definicji Caputo i Riemanna-Liouville’a. Metoda ta może być wykorzystana przy rozwiązywaniu zagadnień początkowych. Metoda nosi nazwę SubIval, co jest akronimem jej poprzedniej, anglojęzycznej nazwy „subinterval-based method” (metoda podprzedziałów). Przedstawiono jej zastosowanie w rozwiązywaniu równań stanu ułamkowego rzędu. Metoda została zaimplementowana w bibliotece DLL z obsługą ActiveX. Przedstawiono przykładowe kody obliczeniowe (w oprogramowaniach MATLAB i Mathematica), które zawierają wskazówki dotyczące zastosowania biblioteki
Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions
This paper presents an efficient spectral method for solving the fractional
Fredholm integro-differential equations. The non-smoothness of the solutions to
such problems leads to the performance of spectral methods based on the
classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low
order of convergence. For this reason, the development of classic numerical
methods to solve such problems becomes a challenging issue. Since the
non-smooth solutions have the same asymptotic behavior with polynomials of
fractional powers, therefore, fractional basis functions are the best candidate
to overcome the drawbacks of the accuracy of the spectral methods. On the other
hand, the fractional integration of the fractional polynomials functions is in
the class of fractional polynomials and this is one of the main advantages of
using the fractional basis functions. In this paper, an implicit spectral
collocation method based on the fractional Chelyshkov basis functions is
introduced. The framework of the method is to reduce the problem into a
nonlinear system of equations utilizing the spectral collocation method along
with the fractional operational integration matrix. The obtained algebraic
system is solved using Newton's iterative method. Convergence analysis of the
method is studied. The numerical examples show the efficiency of the method on
the problems with smooth and non-smooth solutions in comparison with other
existing methods
Generalized Taylor Matrix Method for Solving Multi-Higher Nonlinear Integro-Fractional Differential Equations of Fredholm Type
In this study, generalized Taylor expansion approach formula is developed for solving approximately a Fredholm-Hammerstein type of multi-higher order nonlinear integro- fractional differential equations with variable coefficients under given mixed conditions. The fractional derivative is described in the Caputo sense. Using the collocation points, this new technique depends mainly on transform the nonlinear equation and conditions into the matrix equations which leads to solve a system of nonlinear algebraic equations with unknown generalized Taylor coefficients. A best algorithm for solving our equation numerically by applying this process has been developed in order to express these solution, programs are written in MatLab. In addition, the truth and reliability of this method is tested by several illustrative numerical examples are presented to show effectiveness and accuracy of this algorithm
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Using hybrid of block-pulse functions and bernoulli polynomials to solve fractional fredholm-volterra integro-differential equations
Fractional integro-differential equations have been the subject of significant interest in science and engineering problems. This paper deals with the numerical solution of classes of fractional Fredholm-Volterra integro-differential equations. The fractional derivative is described in the Caputo sense. We consider a hybrid of block-pulse functions and Bernoulli polynomials to approximate functions. The fractional integral operator for these hybrid functions together with the Legendre-Gauss quadrature is used to reduce the computation of the solution of the problem to a system of algebraic equations. Several examples are given to show the validity and applicability of the proposed computational procedure
New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations
It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function ( ). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0
New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations
It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function f(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods