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A shifted Legendre spectral method for fractional-order multi-point boundary value problems Advances in Difference Equations 2012, 2012:8 doi:10.1186/1687-1847-2012-8 Ali H Bhrawy alibhr

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A shifted Legendre spectral method for fractional-order multi-point boundary

value problems

Advances in Difference Equations 2012, 2012:8 doi:10.1186/1687-1847-2012-8

Ali H Bhrawy (alibhrawy@yahoo.co.uk) Mohammed M Al-Shomrani (malshomrani@hotmail.com)

Article type Research

Acceptance date 9 February 2012

Publication date 9 February 2012

Article URL http://www.advancesindifferenceequations.com/content/2012/1/8

This peer-reviewed article was published immediately upon acceptance It can be downloaded,

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Ali H Bhrawy

A shifted Legendre spectral method for fractional-order multi-point boundary value problems

∗1,2 and Mohammed M Al-Shomrani1,3

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589 Saudi Arabia

2 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

3 Faculty Of Computer Science and Information Technology, Northern Border University, Saudi Arabia

∗Corresponding author: alibhrawy@yahoo.co.uk

solu-of fractional-order with variable coefficients Here the approximation is based onshifted Legendre polynomials and the quadrature rule is treated on shifted LegendreGauss-Lobatto points We also present a Gauss-Lobatto shifted Legendre collo-

cation method for solving nonlinear multi-order FDEs with multi-point boundaryconditions The main characteristic behind this approach is that it reduces suchproblem to those of solving a system of algebraic equations Thus we can finddirectly the spectral solution of the proposed problem Through several numericalexamples, we evaluate the accuracy and performance of the proposed algorithms.

Keywords: multi-term FDEs; multi-point boundary conditions; tau method;collocation method; direct method; shifted Legendre polynomials; Gauss-Lobattoquadrature

Fractional calculus, as generalization of integer order integration and tion to its non-integer (fractional) order counterpart, has proved to be a valuabletool in the modeling of many phenomena in the fields of physics, chemistry, engi-neering, aerodynamics, electrodynamics of complex medium, polymer rheology,etc [1–9] This mathematical phenomenon allows to describe a real object moreaccurately than the classical integer methods The most important advantage

differentia-of using FDEs in these and other applications is their non-local property It iswell known that the integer order differential operator is a local operator, butthe fractional-order differential operator is non-local This means that the nextstate of a system depends not only upon its current state but also upon all of itshistorical states This makes studying fractional order systems an active area

of research

Spectral methods are a widely used tool in the solution of differential tions, function approximation, and variational problems (see, e.g., [10, 11] and

equa-the references equa-therein) They involve representing equa-the solution to a problem interms of truncated series of smooth global functions They give very accurateapproximations for a smooth solution with relatively few degrees of freedom.

This accuracy comes about because the spectral coefficients, a n, typically tend

to zero faster than any algebraic power of their index n According to different

test functions in a variational formulation, there are three most common spectralschemes, namely, the collocation, Galerkin and tau methods Spectral methodshave been applied successfully to numerical simulations of many problems inscience and engineering, see [12–15]

Spectral tau method is similar to Galerkin methods in the way that the ferential equation is enforced However, none of the test functions need to satisfythe boundary conditions Hence, a supplementary set of equations are needed toapply the boundary conditions (see, e.g., [10] and the references therein) In thecollocation methods [16, 17], there are basically two steps to obtain a numericalapproximation to a solution of differential equation First, an appropriate finite

dif-or discrete representation of the solution must be chosen This may be done bypolynomials interpolation of the solution based on some suitable nodes such asthe well known Gauss or Gauss-Lobatto nodes The second step is to obtain asystem of algebraic equations from discretization of the original equation.Doha et al [18] proposed an efficient spectral tau and collocation methodsbased on Chebyshev polynomials for solving multi-term linear and nonlinearFDEs subject to initial conditions Furthermore, Bhrawy et al [19] proved

a new formula expressing explicitly any fractional-order derivatives of shiftedLegendre polynomials of any degree in terms of shifted Legendre polynomialsthemselves, and the multi-order fractional differential equation with variable co-efficients is treated using the shifted Legendre Gauss-Lobatto quadrature Saa-datmandi and Dehghan [20] and Doha et al [21] derived the shifted Legendreand shifted Chebyshev operational matrices of fractional derivatives and used

together spectral methods for solving FDEs with initial and boundary tions respectively In [18,22,23], the authors have presented spectral tau methodfor numerical solution of some FDEs Recently, Esmaeili and Shamsi [24] intro-duced a direct solution technique for obtaining the spectral solution of a specialfamily of fractional initial value problems using a pseudo-spectral method, andPedas and Tamme [25] developed the spline collocation method for solving FDEssubject to initial conditions.

condi-Multi-point boundary value problems appear in wave propagation and inelastic stability For examples, the vibrations of a guy wire of a uniform cross-

section, composed of m sections of different densities can be molded as a

multi-point boundary value problem The multi-multi-point boundary conditions can beunderstood in the sense that the controllers at the end points dissipate or add en-ergy according to censors located at intermediate points Rehman and Khan [26]introduced a numerical scheme, based on the Haar wavelet operational matrices

of integration for solving linear multi-point boundary value problems for tional differential equations with constant and variable coefficients Moreover,Rehman and Khan [27] derived a Legendre wavelet operational matrix of frac-tional order integration and applied it to solve FDEs with initial and boundaryvalue conditions In fact, the numerical solutions of multi-point boundary valueproblems for FDEs have received much less attention In this study, we focus onproviding a numerical scheme, based on spectral methods, to solve multi-pointboundary conditions for linear and nonlinear FDEs

frac-In this article, we are concerned with the direct solution technique for ing the multi-term FDEs subject to multi-point boundary conditions, using theshifted Legendre tau (SLT) approximation This technique requires a formulafor fractional-order derivatives of shifted Legendre polynomials of any degree interms of shifted Legendre polynomials themselves which is proved in Bhrawy et

solv-al [19] Another aim of this article is to propose a suitable way to approximate

the multi-term FDEs with variable coefficients subject to multi-point boundaryconditions, using a quadrature shifted Legendre tau (Q-SLT) approximation,this approach extended the tau method for variable coefficients FDEs by ap-proximating the weighted inner products in the tau method by using the shiftedLegendre-Gauss-Lobatto quadrature.

Moreover the treatment of the nonlinear multi-order fractional multi-point

value problems; with leading fractional-differential operator of order ν (m − 1 <

ν ≤ m), on the interval [0, t] is described, by shifted Legendre collocation (SLC)

method to find the solution u N (x) More precisely, such a technique is performed

in two successive steps, the first one to collocate the nonlinear FDE specified

at (N − m + 1) points; we use the (N − m + 1) nodes of the shifted

Legendre-Gauss-Lobatto interpolation on the interval [0, t], these equations together with

m equations comes form m multi-point boundary conditions generate (N + 1)

nonlinear algebraic equations, in general this step is cumbersome, and the ond one to solve these nonlinear algebraic equations using Newton’s iterativemethod The structure of this technique is similar to that of the two-step pro-cedure proposed in [18,20] for the initial boundary value problem and in [21] forthe two-point boundary value problem To the best of the our knowledge, suchapproaches have not been employed for solving fractional differential equationswith multi-point boundary conditions Finally, the accuracy of the proposedalgorithms are demonstrated by test problems

sec-The remainder of the article is organized as follows In the following section,

we introduce some notations and summarize a few mathematical facts used

in the remainder of the article In Section 3, we consider the SLT methodfor the multi-term FDEs subject to multi-point boundary conditions, and inSection 4, we construct an algorithm for solving linear multi-order FDEs withvariable coefficients subject to multi-point boundary conditions by using theQ-SLT method In Section 5, we study the general nonlinear FDEs subject to

multi-point boundary conditions by SLC method In Section 6, we present somenumerical results Finally, some concluding remarks are given in Section 7.

2.1 The fractional derivative in the Caputo sense

In this section, we first review the basic definitions and properties of fractionalintegral and derivative for the purpose of acquainting with sufficient fractionalcalculus theory Many definitions and studies of fractional calculus have beenproposed in the past two centuries (see, e.g., [8]) The two most commonly useddefinitions are the Riemann-Liouville operator and the Caputo operator Wegive some definitions and properties of the fractional calculus

Definition 2.1 The Riemann-Liouville fractional integral operator of order

For the Caputo derivative we have

We use the ceiling function⌈µ⌉ to denote the smallest integer greater than or

equal to µ, and the floor function ⌊µ⌋ to denote the largest integer less than or

equal to µ Also N = {1, 2, } and N0 ={0, 1, 2, } Recall that for µ ∈ N,

the Caputo differential operator coincides with the usual differential operator

of an integer order

2.2 Properties of shifted Legendre polynomials

Let L i (x) be the standard Legendre polynomial of degree i, then we have that

L i(−x) = (−1) i L i (x), L i(−1) = (−1) i , L i (1) = 1. (4)

Let w(x) = 1, then we define the weighted space L2w(−1, 1) ≡ L2(−1, 1) as

usual, equipped with the following inner product and norm

(u, v) =

1

∫

−1

u(x)v(x)w(x)dx, ∥u∥ = (u, u) 1/2

The set of Legendre polynomials forms a complete L2(−1, 1)-orthogonal system,

and

∥L i (x) ∥2= h i= 2

If we define the shifted Legendre polynomial of degree i by L t,i (x) =

L i(2x t − 1), t > 0, then the analytic form of the shifted Legendre

polynomi-als L t,i (x) of degree i is given by

The set of shifted Legendre polynomials forms a complete L w t [0,

t]-orthogonal system According to (5), we have

In practice, only the first (N + 1)-terms shifted Legendre polynomials are

con-sidered Hence we can write

Proof This lemma can be easily proved by using (6).

Next, the fractional derivative of order µ in the Caputo sense for the shifted

Legendre polynomials expanded in terms of shifted Legendre polynomials can

be represented formally in the following theorem

Theorem 2.2 The fractional derivative of order µ in the Caputo sense for the

shifted Legendre polynomials is given by

(For the proof, see, [19].)

Prompted by the application of multi-point boundary value problems to appliedmathematics and physics, these problems have provoked a great deal of attention

by many authors (see, for instance, [28–34] and references therein) In pursuit ofthis, we use the shifted Legendre tau method to solve numerically the followingFDE:

u (q0 )(0) =s0, u (q i)(x i ) = s i , u (q m−1)(t) = s m −1 ,

x i ∈]0, t[, i = 1, 2, , m − 2,

0≤ q0, q1, , q m −1 ≤ m − 1,

(16)

where 0 < β1< β2< · · · < β r −1 < ν, m − 1 < ν ≤ m are constants Moreover,

D ν u(x) ≡ u (ν) (x) denotes the Caputo fractional derivative of order ν for u(x),

γ i , i = 1, 2, , r are constant coefficients, s0, , s m −1 are given constants and

g(x) is a given source function.

The existence and uniqueness of solutions of FDEs have been studied by theauthors of [33–36]

Let us first introduce some basic notation that will be used in the sequel.

where the nonzero elements of matrices A, B i , i = 1, 2, , r − 1, C, and D are

given explicitly in the following theorem

elements of d kj are given by

Proof The square matrix A is defined from the bilinear form:

where Πν (j, l) is given in (14) By the orthogonality of the shifted Legendre

poly-nomials (8), we immediately with direct calculation observe that the nonzero

elements of a kj can be given as (23) The matrix B i for i = 1, 2, , r − 1 and

C defined by the bilinear forms:

then due to (7) and making use of the orthogonality relation of shifted Legendre

polynomials (8), and after some manipulation, one can show that the nonzero

elements of b i

kj ; i = 1, 2, , m −1, and c kjare given explicitly as (24) and (25),

respectively

The matrix D corresponding to the treatment of multi-point boundary

con-ditions (21), its elements can be written as

If we use the analytical form of shifted Legendre polynomial of degree i (6) and

in virtue of (4), then it can be easily shown that

4 A quadrature shifted Legendre tau method

In this section, we use the Q-SLT method to solve numerically the followingFDE with variable coefficients

D ν u(x) +

r −1

∑

i=1

γ i (x)D β i u(x) + γ r (x)u(x) = g(x), xin I = [0, t], (31)

subject to the multi-point boundary conditions (16)

It is worthy to mention that the pure spectral-tau technique is rarely used inpractice, since for variable coefficient terms and a general right-hand side func-

tion g one is unable to compute exactly its representation by Legendre

polynomi-als In fact, the so-called pseudospectral-tau (quadrature-tau) method is used

to treat the variable coefficient terms and right-hand side, (see for instance,Funaro [17] In fact, Doha et al [37] used a quadrature Jacobi dual-Petrov-Galerkin method for solving some ordinary differential equations with variablecoefficients but by considering their integrated forms Moreover, Bhrawy et

al [19] introduced a quadrature shifted Legendre tau method for developing adirect solution technique for solving multi-order fractional differential equationswith variable coefficients with respect to initial conditions

If we denote by x N,j (x t,N,j ), 0 6 j 6 N, and ϖ N,j (ϖ t,N,j ), (0 ≤ j ≤ N), the

nodes and Christoffel numbers of the standard (respectively shifted) Gauss-Lobatto quadratures on the intervals [−1, 1] and [0, t], respectively.

Legendre-Then one can easily show that

follows that for any ϕ ∈ S 2N +1 [0, t],

Associating with this quadrature rule, we denote by I L t

N the shifted Gauss-Lobatto interpolation,

Let us denote

g k = (g, L t,k)w t ,N , k = 0, 1, , N − m,

g = (g0, g1, , g N −m , s0, , s m −1)T ,

E i = (e i kj)0<k,j<N ; i=1,2, ,r −1 , F = (f kj)0<k,j<N ,