Abstract

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

1. Introduction

This paper is dedicated to the numerical solution of the multiorder fractional differential equation with Caputo fractional derivative based on Müntz–Legendre wavelets. Let , , for . Further, let which satisfies

We aim to compute the approximate solution of the linear or nonlinear fractional differential equationwhere is the Caputo fractional differential operator [1] and the function with ( is an open set) satisfies the Lipschitz conditionwhere is independent of . We also assume for simplicity that

The existence and uniqueness of solution of the multiorder fractional differential equation with the Riemann–Liouville and Caputo fractional derivatives under the assumption that satisfies the Lipschitz condition with respect to the second variable are investigated by Kilbas et al. [1]. The previous investigation is based on converting equation (2) into the equivalent Volterra integral equation and then solving it. But in this study, we solve the problem directly.

There exist some numerical methods that solve the desired problem. Laksetani et al. [2] introduced an operational method using B-spline functions to solve the multiorder fractional differential equation

In this paper, the operational matrix of the Caputo fractional derivative has been constructed directly, and then using the collocation method, the problem is solved. In [3], the authors applied the collocation method to solve the fractional differential equation

Their investigation is based on the collocation method using the Chebyshev–Gauss–Lobatto collocation points. The main advantage of this method is its superior accuracy. We can point out its exponential convergence too. But solving this problem is not a big challenge because it lacks multiorder fractional derivatives. Dehestani et al. [4] applied the fractional-Lucas optimization method to solve the multidimensional and multiorder fractional differential equation with Caputo fractional derivative. To this end, they used the operational matrix of fractional derivative for Lucas functions and reduced the problem into a linear or nonlinear system. The result shows the accuracy and efficiency of the method. A special type of equation (2) is considered by Bhrawy et al. [5] as

To solve this equation, they utilized the Laguerre tau technique. To this end, firstly fractional-order generalized Laguerre functions are introduced and Caputo fractional-order derivative is represented by these bases. In [6], after introducing the fractional-order Legendre functions and the operational matrix of Caputo derivative, the multiorder fractional differential equation is solved. For more details, we refer the readers to [7–10].

The fractional differential equations are applied to model various physical phenomena, such as heat conduction, viscoelasticity, dynamical behavior of quantum particles, and laxation and diffusion problems [11–16].

This paper is organized as follows. In Section 2, we introduce the Müntz–Legendre wavelets, and we construct the operational matrix of fractional integration and Caputo fractional derivative. In Section 3, the pseudospectral method is applied to solve the generalized Cauchy-type problems with Caputo fractional differentiation based on Müntz–Legendre wavelets, and then the error analysis is investigated. Section 4 is devoted to some numerical examples to illustrate the accuracy and efficiency of the proposed method.

2. Müntz–Legendre Wavelets

In the last decade, wavelets have been able to get a special place in numerical analysis and especially in the numerical solution of equations [17–22]. As you know, one of the ways to get wavelets is to use multiresolution analysis (MRA). MRA is a family of nested spaces that satisfies certain circumstances [23], namely,where is a bounded interval or is equal to .

In this paper, we apply Müntz–Legendre wavelets to solve the multiorder fractional differential equations. To this end, we give a brief introduction to Müntz–Legendre wavelets. Assume that the space () is spanned by a set of bases which are called multiscaling functions or mother wavelets, i.e.,where and , . The parameter is called the multiplicity parameter, and is the refinement level. In the following, we introduce the functions .

Assume that where is a real constant. Denote by the Müntz–Legendre polynomials [24] which are defined on aswhere the coefficient is defined by [24]

Among the properties of these functions, we can mention their orthogonality. These polynomials form an orthogonal system that satisfies the following relation:where is used for the Kronecker symbol and is given by

Now we are ready to introduce the functions . The Müntz–Legendre wavelets on the interval are defined as follows [24]:

Let be an operator that projects any function onto the subspace as follows:where the coefficients are evaluated by

Let and be vectors of dimension whose -th element is and , respectively. Hence, it follows from equation (15) thatwhere the superscript is used for the matrix transpose.

It follows from [24] that there are some error estimates in the sense of Sobolev norms.

Lemma 1 (see [24]). Let and . If , thenand for , we havewhere is the Sobolev space and

2.1. Representation of the Caputo Fractional Derivative Operator in Müntz–Legendre Wavelets

Recall that the Riemann–Liouville fractional integral operator () is determined by

Note that if , then the function . We know that there is an operator that satisfies the relationwhere and is called the Riemann–Liouville fractional derivative operator. There is also another fractional derivative operator that satisfies the relation and is called the Caputo fractional derivative.

In this section, we would like to represent the Caputo fractional derivative operator in Müntz–Legendre wavelets. To this end, we first construct the operational matrix for fractional integral operator . Then, applying the operational matrix of derivative for Müntz–Legendre wavelets [25] and relation , we can find the operational matrix of fractional derivative for Müntz–Legendre wavelets.

Applying the projection operator , the fractional integral operator acting on the vector function can be approximated bywhere is the operational matrix of integral for the Müntz–Legendre wavelets.

To facilitate the evaluation of the operational matrix elements of fractional integration for the Müntz–Legendre wavelets, it is necessary to introduce the piecewise fractional-order Taylor functions. For a fixed , these functions are constructed as

Let be a vector of dimension whose -th element is .

To derive matrix , we first introduce a matrix of dimension that is used to transfer the Müntz–Legendre wavelets to the piecewise fractional-order Taylor functions . Assume that there is a matrix of dimension such thatwhere stands for the inverse of the matrix . The matrix is called the transformation matrix whose -th element is evaluated by

Let be a vector of dimension whose -th element is , and thus it follows from equation (24) that

It is easy to verify that the Riemann–Liouville fractional integration of the power functions is equal to power functions of the same form, i.e.,

This gives rise to find the -th element of , via

Thus, there exists a diagonal matrix of dimension such that

The matrix elements are obtain as follows:where () and is a diagonal matrix of the form

To derive the operational matrix of integral for the Müntz–Legendre wavelets, it follows from equation (25) that

This gives rise to

Now using , we can introduce the operational matrix for the Caputo fractional derivative

3. Pseudospectral Method

To derive the numerical solution of equation (2), the approximate solution can be approximated by Müntz–Legendre wavelets as follows:where is a vector of dimension that should be determined. A similar expression is valid for where is the space of functions that satisfies for .

To compute the elements of matrix , let ; then, using ,

It follows from equations (36) and (38) that we can compute the residual in approximating equation (2) as follows:

We aim to reduce the residual to zero. One of the available methods is to use the pseudospectral method such that where are the collocation points. In this paper, we use the shifted Legendre and Chebyshev polynomial zeros. This gives rise to a system of linear or nonlinear algebraic equations that should be solved to find the unknown coefficients . To apply the initial conditions (2), we replace the first equations of the obtained system of the pseudospectral method with them.

3.1. Convergence Analysis

It follows from [1] that the fractional integration operators is bounded in (see [1] Lemma 1(a))

Also there is an optimal error estimate in term of error between the Müntz–Legendre polynomials derivative and the exact derivative , viawhere is a constant andis a seminorm.

Theorem 1. Let and . Assume that and satisfies the Lipschitz condition (3). Also, assume that and are the exact and the approximate solutions (39) of equation (2), respectively.
If is a sufficiently smooth function, then the overall errorsatisfies

Proof. If , then by Lemma 1, we can writeUsing the Lipschitz conditions (3) and (41), we can write the following bound:Now, applying equation (41) and Theorem 2 [24], we haveAlso, using Lemma 1 and equation (41), we haveSubtracting equation (2) from equation (39), one can writeTaking norms from both sides of equation (50) and using equations (46)–(49), we end up withSuppose that () and ; then, we can obtain the following from equation (51):It is easy to show that as or .