Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions

Nemati, Somayeh; Lima, Pedro M.

doi:10.1016/j.amc.2018.01.030

Cited by 25 publications

(31 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, n − 1, with h = t f n . Then the generalized modified hat functions consist of a set of n + 1 linearly independent functions in L 2 [0, t f ] that are defined as follows [15,16]:…”

Section: Properties Of the Modified Hat Functionsmentioning

confidence: 99%

“…where P (α) is a matrix of dimension (n + 1) × (n + 1) called the operational matrix of fractional integration of order α. This matrix is given (see [15,16]) as…”

Section: Otherwise;mentioning

confidence: 99%

“…The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems.…”

mentioning

confidence: 99%

“…Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems.…”

mentioning

confidence: 99%

See 3 more Smart Citations

A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati

Lima

Torres

2019

Communications in Nonlinear Science and Numerical Simulation

Self Cite

View full text Add to dashboard Cite

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems. More precisely, in this work we begin by considering the following fractional optimal control problem (FOCP):

show abstract

“…, n − 1, with h = t f n . Then the generalized modified hat functions consist of a set of n + 1 linearly independent functions in L 2 [0, t f ] that are defined as follows [15,16]:…”

Section: Properties Of the Modified Hat Functionsmentioning

confidence: 99%

“…where P (α) is a matrix of dimension (n + 1) × (n + 1) called the operational matrix of fractional integration of order α. This matrix is given (see [15,16]) as…”

Section: Otherwise;mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati

Lima

Torres

2019

Communications in Nonlinear Science and Numerical Simulation

Self Cite

View full text Add to dashboard Cite

show abstract

“…This kind of equations appears in the field of elasticity and fracture mechanics, 20 radiative equilibrium, 21 heat conduction problem, 22 and so on. During the recent years, a few methods have been used to solve the fractional integro-differential equations with weakly singular kernels numerically, such as the second kind Chebyshev polynomials, 23 spline collocation, 24 second kind Chebyshev wavelets (SCW) method, 25 Legendre wavelets (LW), 26 an operational Jacobi Tau method, 27 hat functions, 28 and the shifted Jacobi polynomials. 29 The various applications of fractional integro-differential equations with weakly singular kernels is our main motivation from the present work.…”

Section: Introductionmentioning

confidence: 99%

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

Keshavarz

Ordokhani

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, a fast numerical algorithm based on the Taylor wavelets is proposed for finding the numerical solutions of the fractional integro-differential equations with weakly singular kernels. The properties of Taylor wavelets are given, and the operational matrix of fractional integration is constructed. These wavelets are utilized to reduce the solution of the given fractional integro-differential equation to the solution of a linear system of algebraic equations. Also, convergence of the proposed method is studied. Illustrative examples are included to demonstrate the validity and applicability of the technique. KEYWORDS convergence, numerical solution, operational matrix of fractional integration, Taylor wavelets, weakly singular integro-differential equations Math Meth Appl Sci. 2019;42:4427-4443.wileyonlinelibrary.com/journal/mma

show abstract

Wavelet‐based collocation technique for fractional integro‐differential equation with weakly singular kernel

Mouley

Mandal

2021

Comp and Math Methods

View full text Add to dashboard Cite

Fractional integro-differential equation (FIDE) with weakly singular kernel is an important topic in mathematics and engineering dealing with mathematical modeling and simulation of numerous systems and processes. A wavelet-based collocation technique has been developed here to obtain approximate numerical solution of a FIDE with weakly singular kernel. The present method avoids complicated integrations and elaborate numerical calculations. The multiscale error approximation associated with this method has also been explained. The efficiency of the proposed method has been demonstrated by including some illustrative examples. K E Y W O R D Scollocation method, fractional derivative, multiscale basis of Daubechies family, weakly singular fractional integro-differential equation Here D y(x) represents the fractional derivative of order of the unknown function y(x) and the fractional derivative is understood in Caputo sense in this paper. The explicit forms of V(x, t) and f (x) are known and they are considered sufficiently smooth to confirm the existence and uniqueness of solution. 4

show abstract

Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions

Cited by 25 publications

References 15 publications

A numerical approach for solving fractional optimal control problems using modified hat functions

A numerical approach for solving fractional optimal control problems using modified hat functions

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

Wavelet‐based collocation technique for fractional integro‐differential equation with weakly singular kernel

Contact Info

Product

Resources

About