A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

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.


Introduction
is paper is dedicated to the numerical solution of the multiorder fractional differential equation with Caputo fractional derivative based on Müntz-Legendre wavelets. Let α ∈ R + , N ∍ n ≔ [− α], for α ∉ N. Further, let α j ∈ R + (j � 1, . . . , σ ∈ N) which satisfies 0 � α 0 < α 1 < · · · < α σ < α. (1) We aim to compute the approximate solution of the linear or nonlinear fractional differential equation where c D α 0 is the Caputo fractional differential operator [1] and the function f[x, y, y 1 , . . . , y σ ] ∈ C[0, 1] with y 1 , . . . , y σ ∈ G (G ⊂ C is an open set) satisfies the Lipschitz condition f x, y, y 1 , . . . , y σ (x) − f x, u, u 1 , . . . , u σ (x) ≤ L σ σ j�0 y j − u j , y j , u j ∈ G, where L σ > 0 is independent of x. We also assume for simplicity that e existence and uniqueness of L 1 solution of the multiorder fractional differential equation with the Riemann-Liouville and Caputo fractional derivatives under the assumption that f(x, t) ∈ L 1 [0, 1] satisfies the Lipschitz condition with respect to the second variable are investigated by Kilbas et al. [1]. e 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.
ere 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 eir investigation is based on the collocation method using the Chebyshev-Gauss-Lobatto collocation points. e 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. e 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 fractionalorder 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][8][9][10].
is 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.

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][18][19][20][21][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 Ω is a bounded interval or is equal to R.
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 V J (J ∈ Z + ∪ 0 { }) is spanned by a set of bases which are called multiscaling functions or mother wavelets, i.e., where B ≔ 0, 1, . . . , 2 J − 1 and M ≔ 0, 1, . . . , r − 1 { }, r ∈ N. e parameter r is called the multiplicity parameter, and J is the refinement level. In the following, we introduce the functions ψ m J,b . Assume that λ k ≔ kμ where μ is a real constant. Denote by L m (x) the Müntz-Legendre polynomials [24] which are defined on Ω ≔ [0, 1] as where the coefficient l k,m is defined by [24] Among the properties of these functions, we can mention their orthogonality. ese polynomials form an orthogonal system that satisfies the following relation: where δ m′,m is used for the Kronecker symbol and is given by Now we are ready to introduce the functions ψ m J,b . e Müntz-Legendre wavelets on the interval [0, 1] are defined as follows [24]: 2 Complexity Let P J be an operator that projects any function f(x) ∈ L 2 [0, 1] onto the subspace V J as follows: where the coefficients f b,m are evaluated by Let F and Ψ(x) be vectors of dimension N � 2 J r whose (br + m + 1)-th element is f b,m and ψ m J,b (x), respectively. Hence, it follows from equation (15) that where the superscript T is used for the matrix transpose. It follows from [24] that there are some error estimates in the sense of Sobolev norms.
and for s ≥ 1, we have where H n (0, 1) is the Sobolev space and

Representation of the Caputo Fractional Derivative Operator in Müntz-Legendre Wavelets.
Recall that the Riemann-Liouville fractional integral operator I α 0 (α ∈ R + ) is determined by We know that there is an operator that satisfies the relation where D ≔ (d/dx) and RL D α a is called the Riemann-Liouville fractional derivative operator.
ere 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 I α 0 . en, applying the operational matrix of derivative D for Müntz-Legendre wavelets [25] and , we can find the operational matrix of fractional derivative for Müntz-Legendre wavelets.
Applying the projection operator P, the fractional integral operator I α 0 acting on the vector function Ψ(x) can be approximated by where I α (x) 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 fractionalorder Taylor functions. For a fixed J ∈ Z + ∪ 0 { }, these functions are constructed as Let Φ(x) be a vector of dimension N whose (br + m + 1)-th element is ϕ m J,b (x). To derive matrix I α (x), we first introduce a matrix of dimension N × N that is used to transfer the Müntz-Legendre wavelets Ψ(x) to the piecewise fractionalorder Taylor functions Φ(x). Assume that there is a matrix of dimension N × N such that where T − 1 J stands for the inverse of the matrix T. e matrix T is called the transformation matrix whose (i, j)-th element is evaluated by Let Λ be a vector of dimension r whose i-th element is x λ i , and thus it follows from equation (24) that It is easy to verify that the Riemann-Liouville fractional integration of the power functions x κ is equal to power functions of the same form, i.e., is gives rise to find the i-th element of us, there exists a diagonal matrix I Φ,α (x) of dimension N × N such that e matrix I Φ,α (x) elements are obtain as follows: where (x)) and G 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 is gives rise to Now using c D α a ≔ I n− α a D n , we can introduce the operational matrix for the Caputo fractional derivative

Pseudospectral Method
To derive the numerical solution of equation (2), the approximate solution can be approximated by Müntz-Legendre wavelets as follows: where Y is a vector of dimension N that should be determined. A similar expression is valid for To compute the elements of matrix F, let y J � P(y); then, using c D α a ≔ I n− α a D n , 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 r J (x i ) � 0 where x i are the collocation points. In this paper, we use the shifted Legendre and Chebyshev polynomial zeros. is gives rise to a system of linear or nonlinear algebraic equations that should be solved to find the unknown coefficients Y. To apply the initial conditions (2), we replace the first n equations of the obtained system of the pseudospectral method with them.

Numerical Implementation
In this section, we reported the numerical results for some examples to show the accuracy and efficiency of the method.
To this end, we have performed all numerical computations in Maple and Matlab simultaneously. Wherever collocation nodes have not been reported, we have used Legendre nodes. Example 1. Let us dedicate the first example to the following one.
e exact solution is reported in [4] as follows: To show the ability and efficiency of the proposed method, Tables 1 and 2 are reported. e absolute error for proposed method is compared with the fractional-Lucas optimization method [4] and Chebyshev wavelet method [26] in Table 1.
e results illustrate that the proposed method is flexible against other methods and gives a better approximation. We show L 2 -error, L ∞ -error, and CPU time for different values of α and α 1 taking μ � 0.5 in Table 2.
e second example is dedicated to the following equation: (55) e exact solution is reported in [5] as follows: In Table 3, we compare the maximum of absolute value error of our method with fractional-order generalized Laguerre functions (FGLFs) [5]. In this example, we set the value of μ equal to α. L 2 -errors taking different vales for r and absolute value of error for different values of α are plotted in Figure 1.
In Figure 2, we plot the exact solution along with the approximate solution. Also, in this figure, the absolute value of error is reported for μ � 0.5, r � 3, and J � 1. In Table 4, we compare the proposed method with the Bessel     Complexity collocation method [28]. We observe that our method gives a better result than [28]. In this example, we put μ � 0.5.

Conclusion
In this paper, we apply the pseudospectral method based on Müntz-Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. To this end, we represent the Caputo fractional derivative operator in the Müntz-Legendre wavelets. e results illustrate that by selecting the proper value for μ, the proposed method gives better results than others. e most important advantage of this method over other methods is its flexibility and ease of use. In most cases, the approximate solution is very close to the exact solution and we can almost say that the exact solution is obtained.

Data Availability
e data used to support this study are included within this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.