Numerical solution of distributed order fractional differential equations by hybrid functions

Mashayekhi, Somayeh; Razzaghi, Mohsen

doi:10.1016/j.jcp.2016.01.041

Cited by 125 publications

(44 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and by replacing x by √ x in both sides of the second Sonine integral (Equation 30), relation 28 follows. We also note that Equation 29 is followed by replacing by − in Equation 28.…”

Section: Definitionmentioning

confidence: 99%

Generalized Bessel functions: Theory and their applications

Khosravian-Arab

Dehghan

Eslahchi

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper presents 2 new classes of the Bessel functions on a compact domain[0, T] as generalized-tempered Bessel functions of the first-and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders. KEYWORDS Bagley-Torvik equation, Bessel functions, fractional derivatives and integrals, Gaussian quadrature rules, relaxation-oscilation equation, self-adjoint operator, spectral Galerkin method, tempered fractional derivatives and integrals Math Meth Appl Sci. 2017;40:6389-6410.wileyonlinelibrary.com/journal/mma Esmaeili and Shamsi 30 were the first authors who introduced a new class of the Jacobi polynomials to solve some fractional differential equations. It is worthwhile noting that the new set, generally due to the nonpolynomial nature, is in fact, a set of orthogonal basis functions. These basis functions are also performed to solve some fractional optimal control problems and calculus of variations. [31][32][33] Later, Zayernouri and Karniadakis 34 introduced 2 new classes of Jacobi functions, which are orthogonal with respect to some suitable weight functions, as the eigenfunctions of some fractional Sturm-Liouville eigenvalue problems on [−1, 1]. They also used these basis functions to solve various fractional differential equations. (See for instance Zayernouri and Karniadakis 35-37 and other papers of the same authors. Also see Chen et al 38 and references therein.) After them, Khosravian-Arab et al 39 introduced 2 new classes of generalized Laguerre functions, which are also orthogonal with respect to some weight functions on [0, ∞).As we are aware, one of the most interesting and very practical functions which arises frequently in mathematical physics is the Bessel function(s) (BF(s)). From the historical point of view, the BFs arose in Daniel Bernoulli's investigation of the oscillations of a hanging chain, Euler's theory of the vibration of a circular membrane, and Bessel's studies of planetary motion. Recently, several applications of BFs have been discovered in physics and engineering such as propagation of waves, elasticity, fluid motion, and in many problems of potential theory and diffusion involving cylindrical symmetry. [40][41][42] Although, the BFs are usually known as the solutions of a second-order differential equation, so-called as Bessel's differential equation, which arises by the use of the separation of the variables for various problems in mathematical physics, but, in fact, there is a very close connection betw...

show abstract

“…and by replacing x by √ x in both sides of the second Sonine integral (Equation 30), relation 28 follows. We also note that Equation 29 is followed by replacing by − in Equation 28.…”

Section: Definitionmentioning

confidence: 99%

Generalized Bessel functions: Theory and their applications

Khosravian-Arab

Dehghan

Eslahchi

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The novel contributions for using the hybrid of block-pulse functions and Taylor polynomials together with their error analysis are given in [21]. In 2015, to the best of our knowledge, the Riemann-Liouville fractional integral operator for hybrid of block-pulse functions and Bernoulli polynomials was derived directly for the first time [22,23].…”

Section: Introductionmentioning

confidence: 99%

Hybrid approximations for fractional calculus

Razzaghi

2019

ITM Web Conf.

View full text Add to dashboard Cite

In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

show abstract

“…An implicit difference scheme for the time distributed-order and Riesz space FDE on bounded domains with Dirichlet boundary conditions was constructed by Ye et al [29]. Mashayekhi et al [30] introduced a new numerical method for solving the distributed-order FDEs, which is based upon hybrid functions approximation. Hu et al [31] provided an implicit numerical method of a new time distributed-order and two-sided space fractional advection-dispersion equation and proved the uniqueness, stability, and convergence of the method.…”

Section: Introductionmentioning

confidence: 99%

A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

et al. 2018

View full text Add to dashboard Cite

Recently, several problems in mathematics, physics, and engineering have been modeled via distributed-order fractional diffusion equations. In this paper, a new class of time distributed-order and space fractional diffusion equations with variable coefficients on bounded domains and Dirichlet boundary conditions is considered. By performing numerical integration we transform the time distributed-order fractional diffusion equations into multiterm time-space fractional diffusion equations. An implicit difference scheme for the multiterm time-space fractional diffusion equations is proposed along with a discussion about the unconditional stability and convergence. Then, the fast Krylov subspace methods with suitable circulant preconditioners are developed to solve the resultant linear system in light of their Toeplitz-like structures. The aforementioned methods are proved to acquire the capability to reduce the memory storage of the proposed implicit difference scheme from O(M 2) to O(M) and the computational cost from O(M 3) to O(M log M) during iteration procedures, where M is the number of grid nodes. Finally, numerical experiments are employed to support the theoretical findings and show the efficiency of the proposed methods.

show abstract

Numerical solution of distributed order fractional differential equations by hybrid functions

Cited by 125 publications

References 42 publications

Generalized Bessel functions: Theory and their applications

Generalized Bessel functions: Theory and their applications

Hybrid approximations for fractional calculus

A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

Contact Info

Product

Resources

About