Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2021.015310

ARTICLE

A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

1Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt

2Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

3Department of Applied Mathematics, Physics Division, National Research Centre, Dokki, Cairo, 12622, Egypt

4Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, 620002, Russia

5Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

*Corresponding Author: M. A. Zaky. Email: ma.zaky@yahoo.com; zaky.nrc@gmail.com

Received: 08 December 2020; Accepted: 04 March 2021

Abstract: We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives. The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations. Then, a Legendre-based spectral collocation method is developed for solving the transformed system. Therefore, we can make good use of the advantages of the Gauss quadrature rule. We present the construction and analysis of the collocation method. These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations. Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.

Keywords: Spectral collocation method; weakly singular integral equations; two-point boundary value problems; convergence analysis

Fractional-order differential operators have recently risen to prominence in the modelling of several processes. The mathematical models involving these operators have also attracted much attention, a survey of recent activity is given in [1–3]. The issue we address in this paper is to construct and analyse a spectral collocation method to solve the following nonlinear system of Caputo fractional two-point boundary value problems:

where fR and

Because the fractional-order differential operators are nonlocal with weakly singular kernels, the numerical discretization of the fractional models is more change than the classical schemes. There are several analytical schemes to solve fractional differential equations, such as the Green’s function method, the Mellin transform method, the Laplace transform method, the Fourier transform method, and so on [4–7]. However, analytical methods are rare for most of fractional differential equations, e.g., with non linearities or linear equations with time-dependent coefficients. Hence, constructing efficient numerical approaches is of great importance in practical applications.

Many numerical schemes have been developed to solve the fractional differential equations, mostly with the finite element methods (e.g., [8–11] and the references therein) and the finite difference methods (e.g., [12–18] and the references therein). Since spectral methods are capable of providing high-order accurate numerical approximations with less degrees of freedoms [19–23], they have been widely used for numerical approximations of fractional differential equations [24–29] or its related integral equations [30–37]. In particular, well designed spectral methods appear to be particularly attractive to tackle the difficulties associated with the weakly singular kernels of the fractional differential operators and the integral equations [38,39].

The system of fractional two-point boundary value problems (1) can be converted to an equivalent weakly singular nonlinear system of Volterra integral equations (15). The key idea of the presented approach is to solve (15) using the Legendre spectral collocation scheme. The aim of that convert to (15) is to approximate the related integral terms by the Gauss quadrature formula. The presented method has spectral convergence. This theoretical estimate is confirmed by two numerical test examples. Specifically, our strategies and contributions are highlighted as follows:

i) The system of fractional two-point boundary value problems is recast into an equivalent weakly singular nonlinear system of Volterra integral equations.

ii) The Legendre spectral collocation method is applied to the transformed equation.

iii) The convergence analysis of the Legendre collocation method under the L2-norms is derived.

The structure of the paper is as follows. In Section 2, we introduce some necessary definitions, notations and lemmas. The Legendre spectral collocation scheme is presented in Section 3. The convergence analysis is provided in Section 4. In Section 5, numerical examples are performed to confirm the efficiency of the numerical method. A brief conclusion is highlighted in Section 6.

In this section, we provide some notations, definitions, and some useful lemmas about the fractional differential and integral operators [4] and the Jacobi polynomials.

Definition 1. Let

where

Definition 2. The left- and right-sided Caputo fractional derivatives are defined as:

where ,

Definition 3. The left- and right-sided Riemann-Liouville fractional derivatives are defined as:

It is worthy to mention here that the left- and right-sided Caputo fractional derivatives satisfy the following fundamental properties

Theorem 1. There hold [4]

where

The following formulas introduce the relationship between the Riemann-Liouville and the Caputo fractional derivatives [5].

Let

and for

Now, we give some basic properties of the Jacobi polynomials and related Jacobi–Gauss interpolation. For

where,

Denote

The formula (11) is exact for any

For any

The above condition indicates that

The Legendere polynomials can be obtained directly from the properties of the Jacobi polynomials by setting

We consider the system of two-point fractional boundary value problems (1) with homogeneous boundary conditions. There is no loss of generality since this can always be accomplished by a simple change of variables.

The above system is equivalent to the following system of weakly singular integral equations:

The variable transformations

The Legendre spectral collocation scheme for (17) and (18) is to seek uN(x) and

We now describe the implementation procedure of (19), (20) in detail. We consider the following Legendre approximations

and

Then, by (21) and (9) direct computations lead to

Applying (12) to (21), one can verify readily that

Similarly

Hence, by using (17)–(26) we deduce that

Finally, using (9) we obtain

This system of equations can be solved for

In this section, we introduce some functional spaces. We denote

Definition 4. Let

with the inner product and norm

Definition 5. For a non-negative integer s. The weighted Jacobi non-uniformly Sobolev space

with the inner product, norm, and semi-norm

In particular,

The space

Lemma 1. (cf. [40]) Let

where

Lemma 2. (cf. [40]) For any

Lemma 3. (cf. [41]) Let

Let

Hence

and

It is not difficult to obtain the following results

Similarly

We denote

Let

Hence,

and

We can also derive the following results

Similarly

and

In this section, we analyze the numerical errors of the systems (19) and (20). Let E = |Eu|+|Ev|, where Eu = u − uN, Ev = v − vN and denote by

Lemma 4. The following inequality holds

where

Proof. It follows from (16) and (46) that

Subtracting (57) from (56) yields

Similarly, from (16) and (52) we deduce that

Subtracting (60) from (59) yields

and adding (58) and (61) yields

The above equation can be rewritten as

The desired result follows immediately from the above.

Throughout this section, we suppose that fL and fR fulfil the Lipschitz conditions

with the Lipschitz positive constants L11 and L22, are chosen such that

Theorem 2. Let uN(x) and vN(x) be the solutions of the systems of Eqs. (17) and (18), respectively,

Proof. Using Lemma (4), we get

and

We now estimate

using the Cauchy-Schwarz inequality, we further get

making use of (48) leads to

Using similar arguments leads to

We now estimate

Using (47), we obtain

Using (46) and the Lipschitz condition, we get

For

hence,

Similarly, we can deduce that

Using the condition (65) and knowing that

In this section, two numerical examples are provided to illustrate the efficiency and applicability of the proposed method.

Example 1. We consider the following system of fractional differential equations:

The boundary conditions are chosen such that the exact solution is given by u(z) = (1+z)3.5 and v(z) = (1 − z)2.5. In Figs. 1 and 2 we list the L2-error of the approximate solutions uN and vN, respectively, in log scale against various N and

Example 2. We consider the following system of fractional differential equations: [28]:

In Tab. 2, we compare our results with those reported in [42]. These results indicate that the proposed spectral method is more accurate than the Ritz method [42]. In Tab. 3, we list the maximum absolute errors for different values of N and

Numerically solving a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives was fulfilled as a target of that work. The way to achieve that was done indirectly by recasting the considered system into a weakly singular integral system for the sake of the possibility of applying the Gauss quadrature rule on the transformed integral system. The construction and analysis of the used collocation method is proposed. The obtained results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations [43,44]. A numerical example was given to confirm the convergence analysis and robustness of the scheme. Our future work is related to spectral methods for systems of nonlinear fractional differential equations and system of integral equations with non-smooth solutions.

Funding Statement: The Russian Foundation for Basic Research (RFBR) Grant No. 19-01-00019.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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