On shifted Jacobi spectral approximations for solving fractional differential equations

doi:10.1016/j.amc.2013.01.051

Applied Mathematics and Computation

Volume 219, Issue 15, 1 April 2013, Pages 8042-8056

https://doi.org/10.1016/j.amc.2013.01.051 Get rights and content

Abstract

In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

Highlights

► A new formula of fractional-order derivatives of shifted Jacobi polynomials is proved. ► A Jacobi spectral tau approximation for solving linear FDEs with constant coefficients is proposed. ► A quadrature tau approximation is shown for linear FDEs with variable coefficients. ► A Jacobi collocation method for nonlinear multi-order FDEs is introduced. ► The advantages of using the proposed algorithms are discussed.

Introduction

It has proved that many phenomena in science and engineering can be described accurately by models based on mathematical tools from fractional calculus (see, for instance, refs. [5], [22], [28], [35], [36], [38], [39], [47] and the references therein). There are several analytical techniques for solving FDEs such as Adomian decomposition method [44], variational iteration method [46], homotopy-perturbation method [19], [37], perturbative Laplace method [27] and homotopy analysis method [20], [23]. For some recent developments on the subject the reader can see for example the articles [1], [3], [4], [31], [41], [47], [48], [49] and the references therein.

As it is known the spectral method is one of the versatile methods of discretization for most types of differential equations. We notice that the most used spectral versions are the Galerkin, collocation and tau methods (see, for more details, [9], [16], [40] and the references therein). Recently, many numerical methods have been proposed to address the initial and the boundary value problems of fractional order. Doha et al. [13] suggested the shifted Chebyshev tau and collocation spectral approximations for the numerical solution of multi-term FDEs including linear and nonlinear terms. Furthermore, in [6] it was proved a new formula of any fractional-order derivatives of shifted Legendre polynomials of any degree, this formula is used to extend the application of tau method based on shifted Legendre Gauss–Lobatto quadrature for solving multi-order FDE with variable coefficients. The authors of [15] implemented Jacobi operational matrix of fractional derivatives in combination with Jacobi collocation approximation for treating nonlinear multi-term FDEs. It is noted that the shifted Chebyshev operational matrix [14] and shifted Legendre operational matrix [43] have been obtained as special cases from the shifted Jacobi operational matrix.

In [26], Kadem and Baleanu introduced Chebyshev spectral approach for fractional radiative transfer equation, in which the multidimensional problem has been transformed into a system of fractional differential equations. The pseudo-spectral method [7], [8] has been developed to solve accurately a class of fractional nonlinear problems with approximations converging rapidly to exact solutions. Liu et al. [33] used an integral formula of a truncated Jacobi polynomials for estimating the fractional derivative of an unknown signal, they applied modulating functions method to find the unknown coefficients of the truncated series expansion of the solution by solving a linear system issued from the approximated fractional differential equation. The Wavelet collocation method for the numerical solution of a class of FDEs is investigated in [21]. Very recently, Waveform relaxation method is used in [24] to solve fractional differential equations under linear and nonlinear conditions for the right-hand side of equations. The algorithms in the present article are related to the ideas introduced in [13], [6] in developing accurate approximations for various purposes.

In this paper we are concerned with the direct solution techniques for solving the linear multi-order fractional initial value problems with constant and variable coefficients using shifted Jacobi tau (SJT) method and quadrature shifted Jacobi tau (Q-SJT) method respectively. We note that the two shifted Chebyshev tau and quadrature shifted Legendre tau approximations developed by Doha et al. [13] and Bhrawy et al. [6] respectively, and some other very interesting cases, can be obtained directly as special cases from our proposed SJT and Q-SJT approximations. As a result we motivated our interest in shifted Jacobi-spectral approximations. Moreover, we present a generalized spectral method for tackling nonlinear fractional initial value problems in $(0, L)$ by using shifted Jacobi collocation (SJC) method with general parameters α and β. The implementation of this collocation method for the problem depends on $(N - m + 1)$ nodes of the shifted Jacobi–Gauss interpolation on $(0, L)$ as a collocation points, and then the problem with m initial conditions reduce to $(N + 1)$ nonlinear algebraic equations which can be solved by using any standard iterative method.

The paper is organized as follows. In Section 2 we present some mathematical preliminaries. In Section 3 we introduce the main theorem of the paper which gives explicitly a formula that expresses the fractional-order derivatives of the Jacobi polynomials of any degree. In Sections 4 Linear multi-order FDE with constant coefficients, 5 Linear multi-order FDE with variable coefficients, we construct and develop two algorithms for solving linear multi-order FDEs with constant and variable coefficients by using tau and quadrature tau spectral method respectively, based on the shifted Jacobi polynomials. In Section 6 we presente SJC method for solving nonlinear initial value problems of fractional-order. Some illustrative numerical experiments were given in Section 7. Finally, the concluding remarks are depicted in Section 8.

Section snippets

Mathematical tools

In this section, we are going to state the definition and preliminaries of fractional calculus [42] and some properties of shifted Jacobi polynomials.

Let m be the smallest integer that exceeds ν, then Caputo’s fractional derivative operator of order $ν > 0$ is defined as: $D^{ν} f (x) = \{\begin{matrix} J^{m - ν} D^{m} f (x), & if m - 1 < ν < m, \\ D^{m} f (x), & if ν = m, m \in N, \end{matrix}$ where $J^{ν} f (x) = \frac{1}{Γ (ν)} \int_{0}^{x} (x - t)^{ν - 1} f (t) dt, ν > 0, x > 0 .$ For the Caputo derivative we have $D^{ν} x^{β} = \{\begin{matrix} 0, & for β \in N_{0} and β < ⌈ ν ⌉, \\ \frac{Γ (β + 1)}{Γ (β + 1 - ν)} x^{β - ν}, & for β \in N_{0} and β ⩾ ⌈ ν ⌉ or β \notin N and β > ⌊ ν ⌋ . \end{matrix}$ The Caputo’s fractional differentiation

Jacobi polynomials and their fractional derivatives

The main goal of this section is to find explicitly the Caputo derivatives of any fractional order of the shifted Jacobi polynomials of any degree which will be used in the sequel.

Lemma 3.1

Let $P_{L, i}^{(α, β)} (x)$ be a shifted Jacobi polynomial of degree i, then $D^{ν} P_{L, i}^{(α, β)} (x) = 0, i = 0, 1, \dots, ⌈ ν ⌉ - 1, ν > 0 .$

Theorem 3.2

The Caputo fractional derivative of order ν of the shifted Jacobi polynomials of degree i is obtained from $D^{ν} P_{L, i}^{(α, β)} (x) = \sum_{j = 0}^{\infty} S_{ν} (i, j, α, β) P_{L, j}^{(α, β)} (x), i = ⌈ ν ⌉, ⌈ ν ⌉ + 1, \dots,$ where $S_{ν} (i, j, α, β) = \sum_{k = ⌈ ν ⌉}^{i} (- 1)^{i - k} L^{α + β - ν + 1} Γ (j + β + 1) Γ (i + β + 1) Γ$

Linear multi-order FDE with constant coefficients

In the following we are interested in using the SJT method to solve the linear multi-order FDE $\begin{matrix} D^{ν} u (x) + \sum_{i = 1}^{r - 1} γ_{i} D^{δ_{i}} u (x) + γ_{r} u (x) = f (x), in I = (0, L), \\ u^{(i)} (0) = d_{i}, i = 0, \dots, m - 1, \end{matrix}$ such that $γ_{i} (i = 1, \dots, r)$ and $0 < δ_{1} < δ_{2} < \dots < δ_{r - 1} < ν, m - 1 < ν ⩽ m$ are constants. Moreover, $D^{ν} u (x) \equiv u^{(ν)} (x)$ denotes the Caputo fractional derivative of order ν for $u (x)$ and the values of $d_{i} (i = 0, \dots, m - 1)$ describe the initial data of $u (x)$ and $f (x)$ is a given source function.

As it is known, the existence and uniqueness of the FDEs have been introduced by

Linear multi-order FDE with variable coefficients

In this section, we use the Q-SJT method to solve numerically the following FDE (in the Caputo sense) with variable coefficients $\begin{matrix} D^{ν} u (x) + \sum_{i = 1}^{r - 1} γ_{i} (x) D^{δ_{i}} u (x) + γ_{r} (x) u (x) = f (x), in I = (0, L), \\ u^{(i)} (0) = d_{i}, i = 0, \dots, m - 1, \end{matrix}$ where $0 < δ_{1} < δ_{2} < \dots < δ_{r - 1} < ν, m - 1 < ν ⩽ m$ are constants.

The existence and uniqueness of solutions of FDEs have been studied by [10].

Let $x_{N, j}^{(α, β)}, 0 ⩽ j ⩽ N$ , be the zeros of $(1 - x^{2}) \partial_{x} P_{N}^{(α, β)}$ . Denote by $ϖ_{N, j}^{(α, β)}, 0 ⩽ j ⩽ N$ , the weights of the corresponding Gauss–Lobatto quadrature formula. These values are arranged in

Nonlinear multi-order initial value problems

In this section, we develop the SJC method for the numerical solution of the nonlinear multi-order initial value problems $D^{ν} u (x) = F (x, u (x), D^{δ_{1}} u (x), \dots, D^{δ_{k}} u (x)), x \in I,$ $u^{(i)} (0) = d_{i}, i = 0, 1, \dots, m - 1,$ where $m - 1 < ν ⩽ m, 0 < δ_{1} < δ_{2} < \dots < δ_{k} < ν$ . If we consider $u_{N} (x) = \sum_{j = 0}^{N} a_{j} P_{L, j}^{(α, β)} (x),$ then, by making use of (3.2) we write explicitly the derivatives $D^{ν} u (x), D^{δ_{1}} u (x), \dots, D^{δ_{k}} u (x)$ in terms of the expansion coefficients $a_{j}$ . The implementation of spectral shifted Jacobi collocation approximation for solving numerically (6.1), (6.2) is to

Numerical results

By using the algorithms presented in the previous sections in the following we give some numerical results. Comparisons of our results with those obtained by other methods reveal that our methods are very effective and convenient.

In the following we present the following examples.

Example 1

Consider the equation, see [13] $D^{2} u (x) + D^{\frac{3}{2}} u (x) + u (x) = g (x), u (0) = 0, u^{'} (0) = γ, x \in [0, 1],$ having an exact solution given by $u (x) = \sin (γ x)$ .

In Table 1 we introduce MAE by using SJT with various choices of $N, γ, α, and β$ .

In case of $α = β = - 1$

Concluding remarks

In this paper, we have derived an explicit formula for any fractional-order derivatives in the Caputo type of the shifted Jacobi polynomial of any degree in terms of the shifted Jacobi polynomials themselves. We have developed direct solvers techniques for the spectral solutions of the general linear multi-term FDEs with constant and variable coefficients by using standard tau and quadrature tau approximations based on shifted Jacobi polynomials. A Jacobi–Gauss–Lobatto quadrature has been used

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On shifted Jacobi spectral approximations for solving fractional differential equations

Abstract

Highlights

Introduction

Section snippets

Mathematical tools

Jacobi polynomials and their fractional derivatives

Linear multi-order FDE with constant coefficients

Linear multi-order FDE with variable coefficients

Nonlinear multi-order initial value problems

Numerical results

Concluding remarks

Appl. Math. Lett.

Appl. Math. Lett.

Commun. Nonlinear Sci. Numer. Simul.

Boundary Value Prob.

Appl. Math. Lett.

Appl. Math. Comput.

Appl. Math. Modell.

Comput. Math. Appl.

Appl. Math. Modell.

J. Approximation Theory

Comput. Math. Appl.

Commun. Nonlinear Sci. Numer. Simul.

J. Comput. Appl. Math.

J. Math. Anal. Appl.

Comput. Math. Appl.

Appl. Math. Comput.

Chaos Solitons Fract.

Comput. Math. Appl.

Appl. Math. Comput.

Appl. Math. Comput.

Solution to system of partial fractional differential equations using the fractional exponential operators

Math. Methods Appl. Sci.

On the appearance of the fractional derivative in the behaviour of real materials

J. Appl. Mech.

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