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Shifted ultraspherical pseudoGalerkin method for approximating the solutions of some types of ordinary fractional problems
Advances in Difference Equations volume 2021, Article number: 110 (2021)
Abstract
In this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integrodifferential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integrodifferential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.
Introduction
OFDEs play a role in many branches and applications. These applications can be found in risk theory [1], physics [2], biological phenomena, and diseases [3–6]. Also, the applications of FIDEs have been presented for electromagnetic [7], microelectronics [8], and hematopoietic stem cell modeling [9]. Moreover, many models have been constructed using FOCPs [10, 11]. For that reason, the importance of fractional calculus has emerged.
Most of these problems and models have no exact solutions. Consequently, the researchers focus on investigating and developing new numerical methods. For OFDEs, in [12] the authors presented a spectral method using shifted Chebyshev polynomials (SCHPs) of the second kind. While in [13], the same authors used the same technique but with the sixth kind of SCHPs. A different technique by using a nonpolynomial spline function was employed in [14, 15]. The second kind Wright function with Erdélyi–Kober fractional derivatives was used in [16]. Similar techniques were used for approximating FIDEs. Alternative Legendre functions in [17] were used to set up an operational matrix by the collocation method. In [9], the authors used smoothed pseudosplines refinable functions to construct Riesz wavelets. And as a direct development of the mentioned methods and algorithms, several methods have been formulated to approximate FOCPs. For details about those methods, refer to [10, 18, 19].
The spirit in using the spectral method is the choice of the basis function [20, 21]. The smoothness properties of the basis functions control the decay rate of the coefficients of expansion. The boundary conditions have no effect on the rate. Unlike the other orthogonal polynomials, ultraspherical polynomials (UPs) are not nearly as popular in use as basis functions [22–24]. However, UPs motivate our interest because they include Chebyshev and Legendre polynomials and some other polynomials as subclasses of them [25].
The importance of the Galerkin method (GM) is that, for the presented technique, it can be used for the solution of a wide class of ordinary fractional problems (OFPs). Throughout this work, OFPs are OFDEs, FIDEs, or OCFPs. One of the advantages of GM is that it is not a difficult method. On the other hand, every equation has its own algorithm. The presented technique is a mix of two wellknown spectral methods: collocation method [26–28] and Galerkin method [29–31]. We call that method pseudoGalerkin method (pseudoGM).
The work is coordinated as follows: in Sect. 2, we introduce Caputo’s fractional derivative. Then, UPs (SUPs) and some of their properties are presented. Finally, the notion of integration matrix (Bmatrix) is presented. In Sect. 3, the main results of the paper are stated and proved. Those results give formulae that assert the derivatives of SUPs and the spectral expansion. In Sect. 4, the approximate of OFPs is stated by employing shifted ultraspherical pseudoGM (SUpseudoGM). Then, the error analysis and upper bound for the expansion are estimated in Sect. 5. The correctness and effectiveness of SUpseudoGM are proved by different types of OFPs in Sect. 6. The last section “7” contains the conclusion and remarks.
Preliminaries and notations
This section presents some core definitions and concepts needed throughout this paper. We shall begin with the wellknown Caputo fractional derivative for the function \(\phi (x)\) of order δ denoted by \(D^{\delta } ( \phi (x) )\):
where \(r1\le \delta < r\), \(r\in \mathbb{N}\).
As a direct result of Eq. (1), we have
Besides, one of the advantages of Caputo’s fractional derivative is the linearity property. For any two arbitrary constants \(C_{1}\) and \(C_{2}\),
For more details about fraction derivatives, refer to the review [32].
Proceeding with the fundamental concepts, some properties of UPs \((\mathcal{U}_{j}^{\nu }(x) )\) and SUPs \((\mathcal{U}_{j}^{*\nu }(x) )\), of degree j and real parameter \(\nu >\frac{1}{2}\), will be presented. UPs are the special case of Jacobi polynomials with analytic form [25]:
As special cases: \(L_{j}(x)=\mathcal{U}^{0.5}_{j}(x)\) and \(\mathcal{T}_{j}(x)=\frac{j}{2} \lim_{\nu \rightarrow 0} \frac{\mathcal{U}^{\nu }_{j}(x)}{\nu }, j \ge 1\), where \(L_{j}(x)\) and \(\mathcal{T}_{j}(x)\) are Legendre and Chebyshev polynomials, respectively.
Also, UPs can be generated from Rodrigueś formula [33]
while SUPs, defined on the interval \([0,1 ]\), can be obtained from [34]
where \(\mathcal{U}_{0}^{*\nu }(x)=1\) and \(\mathcal{U}_{1}^{*\nu }(x)=2\nu ( 2 x1 ) \).
Or in terms of its derivatives [35]
SUPs at the boundaries are as follows:
The set \(\lbrace \mathcal{U}_{j}^{*\nu }(x) \rbrace _{j} \) forms an orthogonal set w.r.t. the weight function \(w^{\nu }(x)= (xx^{2} )^{\nu \frac{1}{2}}\):
where \(\psi _{r}= \frac{\pi 2^{14\nu }\Gamma (r+2\nu )}{(r+\nu ) \Gamma (r+1) ( \Gamma (\nu ) ) ^{2}}\).
Finally, the expression Bmatrix will be presented. This matrix will be used for solving FIDEs and FOCPs. According to the spectral method, let \(\phi (x)\) be a differential function over the interval \([0,1]\). Then \(\phi (x)\) can be approximated using \(M+1\) points as follows [36]:
where
such that: \(\mathcal{T}^{*}_{m}(x)\) is the SCHP of degree m, \(c_{0}=c_{M}=0.5\) and \(c_{m}=1\) for \(m=1,2,\ldots,M1\), \(x_{r}=0.5* (1\cos \frac{r \pi }{M} ) \) are the shifted Chebyshev points (SCHpoints).
From Eq. (7) and Eq. (8), we get
According to ElGendi [37], from Eq. (9),
where
are the elements of the Bmatrix. Equation (10) and Eq. (11) can be written in the matrix form:
such that
In the next section, some important theorems and lemmas are investigated. These theorems are needed in the process of SUpseudoGM. As mentioned before, this method is used to solve several types of fractional problems.
Shifted ultraspherical pseudoGalerkin method (SUpseudoGM)
The first step, some rules and forms of UPs need to be shifted. Rodrigueś (Eq. (4)) formula is generalized in the shifted form
Thus, SUPs can be obtained from Eq. (5) or Eq. (12). The next lemma will be used to introduce a general form for the fractional derivative and the integer order integration of SUPs.
Lemma 1
The analytic form of SUPs is as follows:
Proof
Straightforward using Eq. (3) and the binomial theorem. □
The following corollary proves the fractional derivatives of SUPs in the Caputo sense.
Corollary 2
The fractional derivatives of SUPs in the Caputo sense of order δ are as follows:
where \(G_{rk}=(1)^{jrk} \frac{2^{j2r+k} \Gamma (jr+\nu )}{\Gamma (\nu )\Gamma (r+1)\Gamma (k\delta +1)\Gamma (j2rk+1)}x^{k \delta }\).
Proof
Using Lemma 1 and the results from Eq. (2). □
The next theorem will establish the general form of the SUPs approximation of a function \(\phi (x)\). This form is used in SUpseudoGM.
Theorem 3
Consider \(\phi (x)\) to be an “\(s+1\)” differentiable function. Then \(\phi (x)\) can be approximated as
where \(\lbrace \mathcal{A}_{m} \rbrace \) are unknown constants to be determined later and
Proof
Let \(\phi (x)\) be approximated as follows:
So, by the assumption in the theorem,
and
Substituting by \(\mathcal{U}^{*\nu }_{0}(x)\) and \(\mathcal{U}^{*\nu }_{1}(x)\) and using Eq. (6), we obtain
By differentiating Eq. (15) w.r.t. x, we have
Both of the equations (16) and (17) represent the \(s+1\) derivative of \(\phi (x)\). Thus the following equation must be satisfied:
and choose a set of constants \(\lbrace \theta _{m} \rbrace \) as defended in (14). □
SUpseudoGM for solving OFPs
This section is divided into three subsections. Each section concerns solving a different type of OFP. Also, an algorithm for each problem has been added “Algorithm 1, Algorithm 2, and Algorithm 3”.
Before proceeding to the techniques of the solution, the “M+1” collection points must be chosen. Throughout this paper three spaces of points are used. The set of equally spaced points “\(S_{1}\)”:
while the second is Gauss quadrate points “\(S_{2}\)”, i.e., the zeros of SUPs:
The last set is the SCHpoints “\(S_{3}\)”:
“\(S_{3}\)” is used specially for solving FIDEs.
Solving OFDEs by SUpseudoGM
Consider OFDEs as follows:
such that
where \(r,k\in \mathbb{Z}^{+}\), \(\lbrace \delta _{i} \rbrace _{1}^{r} \in \mathbb{R}^{+}\), \(\lbrace \eta _{i}(x) \rbrace _{0}^{k}\) are functions of x, the values of \(d_{j}(i=0, \dots,r1)\) and \(h_{q}(q=0, \dots,r1)\) describe the boundary or the initial state of \(\phi (x)\). The linearity and the nonlinearity of OFDE (21) depend on the function F.
Now, by using SUPs as the base function in the spectral approximation of \(\phi (x)\) as in Eq. (13), we have
Using the straightforward differentiation in the sense of Corollary 2 leads to
Also, the initial and boundary conditions (22) have series expansion of the form
Substitute from Eqs. (23) and (24) into (21) by choosing \(S_{1}\) or \(S_{2}\) as nodes to get a system of linear/nonlinear algebraic equations together with Eq. (25). The unknowns of this system are \(\lbrace \mathcal{A}_{m} \rbrace _{0}^{M}\). The system can be easily solved by any method.
Solving FIDEs by SUpseudoGM
Herein, the SUpseudoGM for solving OFDEs is extended to some types of FIDEs. Consider the following FIDE:
under a sufficient number of initial conditions according to δ. By applying the expansion described at Eq. (23), we obtain
where \(s=0,1,\ldots,M\). Replacing the integration in Eq. (27) by the Bmatrix (10) gives
As in the previous section, the obtained Eq. (28) is an algebraic system. The linearity of that system depends on the linearity of the function F.
Solving FOCPs by SUpseudoGM
Finally, we proceed to approximating the FOCP that takes the form
under the conditions
where \(c_{1},c_{2} \in \mathbb{R}\). By taking both techniques that are included in the last two subsections, the FOCP is transformed into a regular optimization problem.
Error analysis
The convergence and the error analysis for using SUPs are inevitable. Theorems and concepts for these aspects are investigated in [34, 35].
Lemma 4
Let \(\phi (x) \in L^{2}_{w^{\nu }}[0,1]\), \(\vert \phi ''(x) \vert \le B\), and it can be expanded according to expansion (13). Then
where \(0< \nu <1\).
Proof
For the proof, see [38]. □
Theorem 5
Let \(\phi _{M}(x)=\sum_{m=0}^{M}\theta _{m} \mathcal{A}_{m} \mathcal{U}^{*\nu }_{m}(x)\) and \(\phi (x)\), ν satisfies Lemma 4. Then
Proof
See [38]. □
The next section proves the effectiveness and correctness of SUpseudoGM by solving different types of OFPs.
Numerical examples
In this section, numerical examples of SUpseudoGM are presented. The results to those obtained are compared with other methods and the exact solution.
Example 1
Consider the linear OFDE
subject to \(\phi (0)=1\), \(\phi '(0)=c\), and \(\phi ''(0)=c^{2}\) with the exact solution \(\phi (x)=e^{c x}\) and \(u(x)= \frac{c^{2/3} e^{c x} [3 \Gamma (1/3,c x ) + ( c^{11/6} erf ( \sqrt{c x} )3 ) \Gamma (1/3 ) ] }{\Gamma (1/3 )}\).
According to Sect. 4.1, the maximum absolute error (MAE) for SUpseudoGM and MAE for others are presented in Table 1 and Table 2. These tables show the demonstration of SUpseudoGM over the methods in [39] and [40] for all values of c and the parameter ν for the two spaces \(S_{1}\) and \(S_{2}\) at different values of M, while Fig. 1 represents the convergence at an exponential rate using \(S_{3}\).
Example 2
Consider the nonlinear OFDE
subject to \(\phi (0)=0\) with the exact solution \(\phi (x)=x^{2.5}\).
Applying the technique discussed in Sect. 4.1, we obtained the pointwise absolute error (PWAE) shown in Table 3 for \(\delta =0.5\) and Table 4 for \(\delta =0.7\). Comparisons between SUpseudoGM “at different values of ν”, the neural network method (NNM) in [41], the differentiated radial basis function method (DRBF) in [42], and the integrated radial basis function method (IRBF) in [42] have been done. From both tables we recognize that:

SUpseudoGM is more accurate and efficient than NNM in [41] for the presented values of the parameter ν, \(\delta =0.5,0.7\) and both spaces \(S_{1}\) and \(S_{2}\).

Using the equidistance space \(S_{1}\), SUpseudoGM is of almost the same accuracy as DRBF and IRBF methods in [42] for the presented values of the parameter ν and \(\delta =0.5,0.7\).

For \(\delta =0.5\), SUpseudoGM got the same accuracy as DRBF and IRBF methods in [42] at \(\nu =0.49\) using the \(S_{2}\) space.

For \(\delta =0.5\), SUpseudoGM is more accurate than DRBF and IRBF methods in [42] for \(\nu =1\) and \(\nu =0.49\) using the \(S_{2}\) space.

For \(\delta =0.7\), SUpseudoGM is more accurate and efficient than DRBF and IRBF methods in [42] for the presented values of the parameter ν using the \(S_{2}\) space.

The values of the parameter ν do not affect the results while using \(S_{1}\).
During the process of approximation, we recognized that the results were not affected by changing the values of parameters ν at the small values of M.
Example 3
Consider the Bagley–Torvik fractional BVP
subject to \(\phi (0)=0\) and \(\phi (1)=0\) with the exact solution \(\phi (x)=x+x^{3}\).
SUpseudoGM reaches the doubleprecision (e16) as MAE only at \(M=3\) using any space. While in [43], MAE varies from e04 to e05 by increasing M from 10 to 40.
On the other hand, the authors in [44] investigated four methods. They treated the same problem with the first and fourth methods. The authors had to increase M to 256. But they only reached e08 for the first method and e05 for the fourth method.
Example 4
Consider the linear FIDE (Volterra type)
subject to \(\phi (0)=0\) with the exact solution \(\phi (x)=x^{1.5}\).
In this example, we have to use the space \(S_{3}\) to meet the conditions of the Bmatrix. The method in [45] solved Eq. (32) and got 9.5e04 as MAE at \(M=5\). Also, [46] solved it at the same number of points but by three methods. MAE were 1.9e02, 9.8e0.3, and 9.8e0.3 for the three methods. While the MAE of SUpseudoGM is 7.0e04 using \(M=5\) too. The authors in [46] increased M to increase the accuracy. Their best MAE was 10^{−4} at \(M=40,80\). But by using SUpseudoGM, MAE is 8.7e05 at \(M=10\) and 9.4e06 at \(M=21\). Some results and comparisons are summarized in Table 5. The stability and convergence rate are shown in Fig. 2.
Example 5
Consider the nonlinear FIDE (Volterra–Fredholm type)
subject to the boundary conditions of mixed type \(\phi (0)+\phi '(0)=0\) and \(\phi (1)+\phi '(1)=3\) with the exact solution \(\phi (x)=x^{2}\).
The boundary conditions of mixed type do not affect the steps. They are just two extra equations added to the algebraic system. The PWAE of this example using SUpseudoGM and two other methods in [47] is written in Table 6. The first method in [47] is “Nystrom”, while the second method depends on the wavelet. This method used the two parameters m and k for the approximation. So, \(2^{k1}m\) is the equivalent number of iterations. The following observations have been noted from Table 6. This first method in [47] approximates the solution to 10^{−5} as PWAE at \(M=16\). But SUpseudoGM got the same PWAE at \(M=5\) only. On the other hand, PWAE decreases from 10^{−6} to 10^{−2} as x changes from 0.1 to 0.9 by using “Nystrom” at \(M=16\). The doubleprecision “10^{−16}” was reached using SUpseudoGM at \(M=16\) for all the domain points. Figure 3 ensures the accuracy, stability, and convergence rate for SUpseudoGM.
Example 6
Consider the nonlinear FOCP
subject to \(D^{1.1}\phi (x)=x^{2} \phi (x)+v(x)\) and \(\phi (0)=\phi '(0)=0\) with the exact solution \((\mathcal{J}(v), \phi (x), v(x) ) = ( 0 , x^{2}, \frac{20 x^{0.9}}{9 \Gamma (0.9)}x^{4} ) \).
According to the integration “from 0 to 1”, all spaces”\(S_{1},S_{2},S_{3}\)” may be used. The same technique is applied. In the past examples we got algebraic systems to solve. But in the case of FOCP, the problem is transformed into an optimization problem.
The best result is \(\mathcal{J}=4.1280e17\) at “\(M=2\)” when “\(\nu =0.49\)” using “\(S_{2}\)” with run time 0.019. The results of SUpseudoGM have been compared with the results from [18, 19, 48]. In [18], the authors used wavelet expansion. They got \(\mathcal{J}=4.7187e06\) using six iterations with run time 0.046. While in [19], \(\mathcal{J}=6.16e16\) but at \(M=4\) with run time 0.030 and in [48], \(\mathcal{J}=5.44e08\) at \(M=9\) as best results. Table 7 compares the results obtained by SUpseudoGM when \(\nu =0.49\) using \(S_{1}\) with results from [48].
These comparisons show the accuracy and the efficiency of SUpseudoGM for solving FCOPs.
Example 7
Consider the nonlinear FOCP
subject to \(D^{1.5}\phi (x)=\sin (\phi (x) )+v(x)\), \(\phi (0)=1\), and \(\phi '(0)=0\) with the exact solution \((\mathcal{J}(v), \phi (x), v(x) ) = ( 0 , 1x^{2}, 4\sqrt{\frac{x}{\pi }}+\sin (1+x^{2} ) ) \).
By applying the same steps, Table 8 summarizes the best results of \(\mathcal{J}\), MAE of the state function \(\phi (x)\), and MAE of the control function \(v(x)\). All those results are compared with the methods in [18, 19, 49, 50]. The obtained results prove the high correctness and effectiveness of SUpseudoGM. Figure 4 shows the stability of the optimality value.
Conclusion
A general method developed from GM has been introduced in this work. This method uses SUPs as trails functions. We call this method SUpseudoGM. New formulae for the spectral expansion and fractional derivatives of SUPs have been proved. This method can handle several types of OFPs. Finally, SUpseudoGM was used to solve several examples. As shown in those examples, the method was easy to apply in FODEs, IFDEs, and FOCPs. To show and prove the high correctness of that method, several graphs have been constructed. Also, comparisons with other methods have been done. Three spaces have been used in the examples. The results show that the quadrature points (\(S_{2}\)) are slightly more accurate than the equidistant Riemann points (\(S_{1}\)).
Availability of data and materials
All data are included within this paper.
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Acknowledgements
The authors are sincerely grateful to Dr. Youssri H. Youssri, Associate Professor at Cairo University in Egypt, for his valuable support during this research.
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The first, third, and fourth authors participated equally in developing the idea and derivations. The second author participated in the derivation part. All authors read and approved the final manuscript.
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Abdelhakem, M., Mahmoud, D., Baleanu, D. et al. Shifted ultraspherical pseudoGalerkin method for approximating the solutions of some types of ordinary fractional problems. Adv Differ Equ 2021, 110 (2021). https://doi.org/10.1186/s13662021032476
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Keywords
 Shifted ultraspherical polynomials
 Fractional differential equations
 Fractional integrodifferential equations
 Fractional optimal control problems
 Galerkin method
 Spectral method
 Error analysis