Exact solutions for some of the fractional integro-diﬀerential equations with the nonlocal boundary conditions by using the modiﬁcation of He’s variational iteration method

Irandoust‐pakchin, Safar; Abdi-Mazraeh, Somayeh

doi:10.14419/ijams.v1i3.983

Cited by 5 publications

(5 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The derivatives in these equations can be replaced with suitable finite difference approximations on a discretized domain. The accuracy of the solution depends on the number of mesh points such that if the number of mesh points is increased, then the solution will be more accurate [12]. However, Taylor Series can be used to derive some of finite difference approximations.…”

Section: Numerical Differentiation and Integrationmentioning

confidence: 99%

Numerical Solutions for Linear Fractional Differential Equations of Order 1 <Alpha<2 Using Finite Difference Method (ffdm)

Albadarneh¹,

Batiha²,

Zurigat³

2016

J. Math. Computer Sci.

View full text Add to dashboard Cite

The major goal of this paper is to find accurate solutions for linear fractional differential equations of order 1 < α < 2 . Hence, it is necessary to carry out this goal by preparing a new method called Fractional Finite Difference Method (FFDM). However, this method depends on several important topics and definitions such as Caputo's definition as a definition of fractional derivative, Finite Difference Formulas in three types (Forward, Central and Backward) for approximating the second and third derivatives and Composite Trapezoidal Rule for approximating the integral term in the Caputo's definition. In this paper, the numerical solutions of linear fractional differential equations using FFDM will be discussed and illustrated. The purposed problem is to construct a method to find accurate approximate solutions for linear fractional differential equations. The efficiency of FFDM will be illustrated by solving some problems of linear fractional differential equations of order 1 < α < 2.

show abstract

Section: Numerical Differentiation and Integrationmentioning

confidence: 99%

Numerical Solutions for Linear Fractional Differential Equations of Order 1 <Alpha<2 Using Finite Difference Method (ffdm)

Albadarneh¹,

Batiha²,

Zurigat³

2016

J. Math. Computer Sci.

View full text Add to dashboard Cite

show abstract

“…Te existence and uniqueness of the solution of such equations have been widely studied in many papers, as can be seen in [11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

2023

View full text Add to dashboard Cite

In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.

show abstract

“…Also many attempts have been made to find analytical and numerical solutions for the fractional problems. These attempts include introducing finite difference [5,20], collocation-shooting [6], spline and B-spline collocation [18], Adomian decomposition [11], variational iteration [17], operational matrix [22] and many other methods. Some authors present spectral or pseudospectral integration methods proven successful in the numerical solutions of many problems (see for example [2,3,4,12]).…”

Section: Introductionmentioning

confidence: 99%

Solving fractional differential equations by the ultraspherical integration method

Ali

Panahi

2021

bspm

View full text Add to dashboard Cite

In this paper, we present a numerical method to solve a linear fractional differential equations. This new investigation is based on ultraspherical integration matrix to approximate the highest order derivatives to the lower order derivatives. By this approximation the problem is reduced to a constrained optimization problem which can be solved by using the penalty quadratic interpolation method. Numerical examples are included to confirm the efficiency and accuracy of the proposed method.

show abstract

Exact solutions for some of the fractional integro-diﬀerential equations with the nonlocal boundary conditions by using the modiﬁcation of He’s variational iteration method

Cited by 5 publications

References 16 publications

Numerical Solutions for Linear Fractional Differential Equations of Order 1 <Alpha<2 Using Finite Difference Method (ffdm)

Numerical Solutions for Linear Fractional Differential Equations of Order 1 <Alpha<2 Using Finite Difference Method (ffdm)

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

Solving fractional differential equations by the ultraspherical integration method

Contact Info

Product

Resources

About