Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.


Introduction
It is necessary to determine the maxima and minima of certain functionals in study problems in analysis, mechanics, and geometry.These problems are known as variational problems in calculus of variations.Variational problems have many applications in various fields like physics [1], engineering [2], and areas in which energy principles are applicable [3][4][5].
Nowadays, fractional calculus is a very interesting branch of mathematics.Fractional calculus has many real applications in science and engineering, such as fluid dynamics [6], biology [7], chemistry [8], viscoelasticity [9,10], signal processing [11], bioengineering [12], control theory [13], and physics [14].Due to the importance of the fractional derivatives established through real-life applications, several authors have considered problems in calculus of variations by replacing the integer-order derivative with fractional orders in objective functionals, and this is thus known as fractional calculus of variations.Some of these studies are of a fractionally damped system [15], energy control for a fractional linear control system [16], a fractional model of a vibrating string [17], and an optimal control problem [18].In this paper, our aim is to minimize non-linear fractional variational problems (NLFVPs) [19] of the following form: under the constraints y(0) = a, I 1−α y(1) = , where g and h are two functions of class C 1 with g(x) = 0 on [0, 1], α and are real numbers with α ∈ (0, 1), and a is a constant.The pioneer approach for solving the fractional variational problems originates in reference [20] where Agrawal derived the formulation of the Euler-Langrage equation for fractional variational problems.Further, in reference [4], he gave a general formulation for fractional variational problems.
In reference [5], the authors used an analytical algorithm based on the Adomian decomposition method (ADM) for solving problems in calculus of variations.In [21,22], Legendre orthonormal polynomials and Jacobi orthonormal polynomials, respectively, were used to obtain an approximate numerical solution of fractional optimum control problems.In [23], the Haar wavelet method was used to obtain numerical solution of these problems.Some other numerical methods for the approximate solution of fractional variational problems are given in [24][25][26][27][28][29][30][31][32][33][34].Recently, in [19], the authors gave a new class of fractional variational problems and solved this using a decomposition formula based on Jacobi polynomials.The operational matrix methods (see [35][36][37][38][39][40][41]) have been found to be useful for solving problems in fractional calculus.
In present paper, we extend the Rayleigh-Ritz method together with operational matrices of different orthogonal polynomials such as Shifted Legendre polynomials, Shifted Chebyshev polynomials of the first kind, Shifted Chebyshev polynomials of the third kind, Shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials to solve a special class of NLFVPs.The Rayleigh-Ritz methods have been discussed by many researchers in the literature for different kinds of variational problems, i.e., fractional optimal control problems [18,21,22,32,33]; here we cite only few, and many more can be found in the literature.In this method, first we take a finite-dimensional approximation of the unknown function.Further, using an operational matrix of integration and the Rayleigh-Ritz method in the variational problem, we obtain a system of non-linear algebraic equations whose solution gives an approximate solution for the non-linear variational problem.Error analysis of the method for different orthogonal polynomials is given, and convergence of the approximate numerical solution to the exact solution is shown.A comparative study using absolute error and root-mean-square error tables for all five kinds of polynomials is analyzed.Numerical results are discussed in terms of the different values of fractional order involved in the problem and are shown through tables and figures.

Basic Preliminaries
The definition of fractional order integration in the Riemann-Liouville sense is defined as follows.
Definition 1.The Riemann-Liouville fractional order integral operator is given by The analytical form of the shifted Jacobi polynomial of degree i on [0, 1] is given as where a and b are certain constants.Jacobi polynomials are orthogonal in the interval [0, 1] with respect to the weight function w (a,b) (x) = (1 − x) a x b and have the orthogonality property where δ mn is the Kronecker delta function and For certain values of the constants a and b, the Jacobi polynomials take the form of some well-known polynomials, defined as follows.Case 1: Legendre polynomials (S1) For a = 0, b = 0 in Equation (3), we get Legendre polynomials.
Case 2: Chebyshev polynomials of the first kind (S2) For a = 1 2 , b = 1 2 in Equation ( 3), we get Chebyshev polynomials of the first kind.
where c i = f (t), Ψ i (t) and −, − is the usual inner product space.
Theorem 1.Let H be a Hilbert space and Z be a closed subspace of H with dim Z < ∞; let {z 1 , z 2 , . . . ,z N } be any basis for Z. Suppose that y is an arbitrary element in H and z 0 is the unique best approximation to y out of Z. Then where Proof .Please see references [42,43].
, and suppose Proof .Please see Appendix A.
Mathematics 2019, 7, 224 Now, in particular cases, the operational matrix of integration for various polynomials is given as follows.
For Shifted Legendre polynomials (S1), the (i, j)th entry of the operational matrix of integration is given as For Shifted Chebyshev polynomials of the first kind (S2), the (i, j)th entry of the operational matrix of integration is given as: For Shifted Chebyshev polynomials of the third kind (S3), the (i, j)th entry of the operational matrix of integration is given as For Shifted Chebyshev polynomials of the fourth kind (S4), the (i, j)th entry of the operational matrix of integration is given as For Shifted Gegenbauer polynomials (S5), the (i, j)th entry of the operational matrix of integration is given as . ( 18)

Method of Solution
Approximating the unknown function in terms of orthogonal polynomials has been practiced in several papers in recent years [18,21,22,32,33] for different types of problems.Here, for solving the problem in Equation (1), we approximate We are approximating the derivative first because we want to use the initial condition.Taking the integral of order α on both sides of Equation ( 19), we get Using the operational matrix of integration, Equation ( 20) can be written as where y(0) = a ∼ = A T Φ n (x) and I (α) is the operational matrix of integration of order α.
Using Equation ( 19), we can write Using Equations ( 19) and ( 22) in Equation ( 1), we obtain Equation ( 23) can then be written as We further take the following approximations: where and −, − is the usual inner product space.
Using Equations ( 25) and ( 26) we can write where and From Equations ( 24) and ( 27)-( 29), we get From Equations ( 31) and ( 32), we get where P is a square matrix given by P = 1 0 Φ n (x)Φ n (x) T dx.Using Equation ( 22), the boundary condition can be written as Using the Lagrange multiplier method [18,[20][21][22]32,33], the necessary extremal condition for the functional in Equation (33) becomes From Equations ( 34) and ( 35), we get a set of n + 1 equations.Solving these n + 1 equations, we get unknown parameters c 0 , c 1 , . . ., c n .Using these unknown parameters in Equation ( 21), we get the unknown function's extreme values of the non-linear fractional functional.

Error Analysis
The upper bound of error for the operational matrix of fractional integration of a Jacobi polynomial of the ith degree is given as From Equation ( 36), we can write Taking the integral operator of order α on both sides of Equation ( 3), we get From the construction of the operational matrix we can write Using Theorem 1 we can write From Equations ( 37)-(39), we get Using Equation (40) in Equation (41), we obtain the error bound for the operational matrix of integration of an ith-degree polynomial, which is given as Mathematics 2019, 7, 224 Now, in particular cases, the error bounds for different orthogonal polynomials are given as follows.Case 1: For Legendre polynomials (S1) the error bound is given as Case 2: For Chebyshev polynomials of the first kind (S2) the error bound is given as 2 )Γ(i+k+2) 2 )Γ(i+2)Γ(α+k+1) Case 3: For Chebyshev polynomials of the third kind (S3) the error bound is given as 2 )Γ(i+k+1) Case 4: For Chebyshev polynomials (S4) the error bound is given as 2 )Γ(i+k1) 2 )Γ(i+1)Γ(α+k+1) Case 5: For Gegenbauer polynomials (S5) the error bound is given as Let e α,w I,n denote the error vector for the operational matrix of integration of order α obtained by using (n + 1) orthogonal polynomials in L 2 w [0, 1]; then e α,w From Theorems 1 and 2 and from Equations ( 43)-(47), it is clear that as n → ∞ the error vector in Equation (48) tends to zero.

Convergence Analysis
A set of orthogonal polynomials on [0, 1] forms a basis for L 2 w [0, 1].Let S n be the n-dimensional subspace of L 2 w [0, 1] generated by (Φ i ) 0≤i≤n .Thus, every functional on S n can be written as a linear combination of orthogonal polynomials (Φ i ) 0≤i≤n .The scalars in the linear combinations can be chosen in such a way that the functional minimizes.Let the minimum value of a functional on space S n be denoted by m n .From the construction of S n and m n , it is clear that S n ⊂ S n+1 and m n+1 ≥ m n .Theorem 4. Consider the functional J, then Proof .Using Equation (48) in Equation ( 23), we have Taking n → ∞ and using Equations ( 25)-( 27) and (48) in Equation (49), we get where , and J e is the error term of the functional.
Using Equations ( 30) and (32) in Equation ( 50), we get where Solving Equation (51) similarly to the original functional, Equation (51) reduces to the following form: Using Equation (48) in Equation ( 34), we get Similar to above, by using the Rayleigh-Ritz method on Equation (53) with the boundary condition in Equation (54) we obtain the extreme value of the functional defined in Equation (53).Let this extreme value be denoted by m * n (t).Now, from Equation (48), it is obvious that e w E i 1 ,n , e w E i 2 ,n , e w E 3 ,n → 0 as n → ∞, which implies that e w n (J e ) → 0 as n → ∞.So, it is clear that as n → ∞ , the functional J e in Equation (53) comes close to the functional J in Equation ( 23) and the boundary condition in Equation (54) comes close to Equation (34).
So, for large values of n, m * n (t) → m n (t).
From Theorem 4 and Equation (55), we conclude that Proof completed.

Numerical Results and Discussions
In this section, we investigate the accuracy of the method by testing it on some numerical examples.We apply the numerical algorithm to two test problems using different orthogonal polynomials as a basis.The results for the test problems are shown through the figures and tables.
The exact solution of the above equation is given as We discuss this example for different values of α = 0.5, 0.6, 0.7, 0.8, 0.9, or 1, β = 5, and = 1.
In Figures 1-5, it is shown that the solutions for the two different values of α = 0.8 and α = 1 coincide with the exact solutions for different orthogonal polynomials at n = 5.In Figures 6-10, it is shown that the solution varies continuously for Shifted Legendre polynomials, Shifted Chebyshev polynomials of the second kind, Shifted Chebyshev polynomials of the third kind, Shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials, respectively, with different values of fractional order.In Table 1, we have listed the maximum absolute errors (MAE) and root-mean-square errors (RMSE) for Example 1 for the two different n values of 2 and 6.In Table 1, we have listed the maximum absolute errors (MAE) and root-mean-square errors (RMSE) for Example 1 for the two different n values of 2 and 6.In Table 1, we have compared results for different polynomials, and it is observed that the results for Shifted Legendre polynomials and Gegenbauer polynomials are better than those for the other polynomials.It is also observed that the MAE and RMSE decrease with increasing n.
The exact solution of the above equation is given as where E a,b (x) is the Mittag-Leffler function of order a and b and is defined as                        In Table 2, we have listed the maximum absolute errors (MAE) and root-mean-square errors (RMSE) for Example 2 for the two n values 2 and 6.In Table 2, we have listed the maximum absolute errors (MAE) and root-mean-square errors (RMSE) for Example 2 for the two n values 2 and 6.In Table 2, we have listed the maximum absolute errors (MAE) and root-mean-square errors (RMSE) for Example 2 for the two n values 2 and 6.In Table 2, we have compared results for different polynomials, and it is observed that the results for the Shifted Legendre polynomial are better than those for the other polynomials.It is also observed that the MAE and RMSE decrease as n increases.

Conclusions
We extended the Ritz method [18,[20][21][22]32,33] for solving a class of NLFVPs using different orthogonal polynomials such as shifted Legendre polynomials, shifted Chebyshev polynomials of the first kind, shifted Chebyshev polynomials of the third kind, shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials.These polynomials were used to approximate the unknown function in the NLFVP.The advantage of the method is that it converts the given NLFVPs into a set of non-linear algebraic equations which are then solved numerically.The error bound of the approximation method for NLFVP was established.It was also shown that the approximate numerical solution converges to the exact solution as we increase the number of basis functions in the approximation.At the end, numerical results were provided by applying the method to two test examples, and it was observed that the results showed good agreement with the exact solution.Numerical results obtained using different orthogonal polynomials were compared.A comparative study showed that the shifted Legendre polynomials were more accurate in approximating the numerical solution.
The upper bound of the error of the Taylor polynomial is given as

Figure 5 .
Figure 5.Comparison of exact and numerical solutions using S5 for α = 0.8 and α = 1, Example 1.In Figures6-10, it is shown that the solution varies continuously for Shifted Legendre polynomials, Shifted Chebyshev polynomials of the second kind, Shifted Chebyshev polynomials of the third kind, Shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials, respectively, with different values of fractional order.

Figures 16 -
Figures 16-20 reflect that the approximate solution varies continuously for Shifted Legendre polynomials, Shifted Chebyshev polynomials of the second kind, Shifted Chebyshev polynomials of the third kind, Shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials, respectively, with different values of fractional order.

Figures 16 - 24 Figure 15 .
Figures 16-20 reflect that the approximate solution varies continuously for Shifted Legendre polynomials, Shifted Chebyshev polynomials of the second kind, Shifted Chebyshev polynomials of the third kind, Shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials, respectively, with different values of fractional order.

Figures 16 -
Figures 16-20 reflect that the approximate solution varies continuously for Shifted Legendre polynomials, Shifted Chebyshev polynomials of the second kind, Shifted Chebyshev polynomials of the third kind, Shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials, respectively, with different values of fractional order.

Table 1 .
Result comparison of Example 1 for different orthogonal polynomials at different values of n.

Table 1 .
Result comparison of Example 1 for different orthogonal polynomials at different values of n.

Table 2 .
Result comparison of Example 2 for different orthogonal polynomials at different values of n.

Table 2 .
Result comparison of Example 2 for different orthogonal polynomials at different values of n.