Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

In this paper, we present a numerical scheme for ﬁnding numerical solution of a class of weakly singular nonlinear fractional integro-diﬀerential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-diﬀerential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the eﬀectiveness and accuracy of the proposed method.


Introduction
In this study, the operational alternative Legendre method is introduced and employed to solve a class of nonlinear fractional integro-differential equations with weakly singular kernel: where F: C([0, 1]) ⟶ R and G: C([0, 1]) ⟶ R (sufficiently smooth operators) are considered to be nonlinear. D α t are the Caputo fractional derivative operators, ⌈α⌉ is the ceiling function of α, f(t) is a continuous function, and y(t) is an unknown function to be determined. ese equations come from the mathematical modeling of various phenomena, such as radiative equilibrium, heat conduction problems, elasticity, and fracture mechanics (see [1][2][3][4][5]). Since the numerical solution of the nonlinear fractional equations is almost a new subject and because of having a singular kernel, there are many schemes for solving this kind of equations. Heydari et al. have utilized the Chebyshev wavelet method to solve systems of the linear and nonlinear singular fractional Volterra integro-differential equations (see [6]). Mohammadi has applied the block pulse functions for the linear and nonlinear singular fractional integro-differential equations (see [7]). Zhao et al. have proposed the piecewise polynomial collocation method for solving the fractional integro-differential equations with weakly singular kernels (see [8]). In [9,10], the operational Tau method has been used to solve this kind of equations. A spectral method based on the Chebyshev polynomials of the second kind has been applied in [11]. Yi et al. have dealt with the CAS wavelets and Legendre wavelets method for solving the linear and nonlinear fractional weakly singular integrodifferential equations (see [12,13]), and so on (see [14][15][16]).
In this paper, application of the alternative Legendre polynomials is extended to solve the nonlinear weakly singular fractional order integro-differential equations. For this purpose, the fractional operational matrices of integration and product are derived. Also, an operational matrix is derived to approximate the integral part with the singular kernel in equation (1). e matrices and approximations resulted are substituted into the given equation to convert it into the nonlinear system of algebraic equations. e nonlinear systems can be solved by the well-known Newton iteration method. Also, error analysis and convergence analysis of the proposed method are presented.

Alternative Legendre Polynomials and Their
Operational Matrices e set P n � [P nk (t)] n k�0 of alternative Legendre polynomials of degree n in [0, 1] (see [19]) is given by ey are orthogonal over the interval [0, 1] with respect to the weight function ω(t) � 1 and satisfy the property as follows: Equation (6) may be rewritten as Rodrigues's type: Integrating equation (8) from 0 to 1, we have Suppose the alternative Legendre polynomials vector as Equation (10) can be rewritten as the form where T(t) � 1, t, t 2 , . . . , t n T , (12) and Q is the upper triangular matrix defined by A square integrable function y(t) ∈ L 2 (0, 1) can be expressed in terms of the alternative Legendre polynomials basis as follows: In practice, only the first (n + 1) term of alternative Legendre polynomials is considered. Hence, one has 2 Journal of Mathematics where C � [c 0 , c 1 , . . . , c n ] T , c k � 〈y, P nk 〉/〈P nk , P nk 〉 � (2k + 1)〈y, P nk 〉, k � 0, 1, . . . , n are called alternative Legendre polynomials coefficients, and 〈y, y〉 � 1 0 y 2 (t)dt. In implementing the operations on the alternative Legendre basis, we frequently encounter the integration and the product of the vector Φ(t) and also it is necessary to evaluate the integration, the product of the vector Φ(t). For this purpose, some operational matrices will be derived. To pursue, we need the following lemmas. e proofs of these lemmas are quite easy by using the definition. Lemma 1. Let P nk (t) � n r�0 p (k) r t r and P nj (t) � n r�0 p (j) r t r be the k th and j th alternative Legendre polynomials, respectively. en, the product of P nk (t) and P nj (t) can be written as where q Lemma 2. Let r be a positive integer, then we have Lemma 3. Let P ni (t), P nj (t), and P nk (t) be i th , j th , and k th alternative Legendre polynomials, respectively. en, we have e alternative Legendre polynomials operational matrix of fractional integration is shown by the following theorem. Theorem 1. Let Φ(t) be the alternative Legendre polynomials vector obtained by equation (10), then we have the following: where J � [p kr ] n k,r�0 is the alternative Legendre polynomials operational matrix of fractional integration of order (n + 1) × (n + 1) in which e following theorem presents a general formula for finding the operational matrix of the product.

c n ] T be an arbitrary vector, then we have
where C � [c ik ] n i,k�0 is the alternative Legendre polynomials operational matrix of the product in which Journal of Mathematics e values of ξ ijk are calculated by using Lemma 3. Proof (see [20]).
where k�0 is a (n + 1) × (n + 1) matrix and its elements are as follows: Proof. According to the definition of the vector Φ(t), one has Using equation (6) leads to Using equation (15), we can approximate t k+j− β+1 in terms of alternative Legendre polynomials as follows: where, by Lemma 2, one can obtain b kjr � (2r + 1) kr P nr (t). (29) So, equation (25) can be written as follows: is proof is completed.

Solution of Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel
To explain the implementation process of the proposed method, the following case of equation (1) is considered: where m 1 and m 2 are the positive integers. By applying the introduced matrices and the basis vector in the previous section, the terms of the equations under investigation may be approximated in terms of the alternative Legendre polynomials as follows.
D α t y(t) can be approximated as By applying the fractional integral operator of order α to equation (28), an approximation of unknown function y(t) results as follows: where X T is a known n + 1 vector and Y T � C T J + X T . Using eorem 2, we can obtain where C m 1 − 1 and C m 2 − 1 are m 1 − 1 and m 2 − 1 order, the alternative Legendre polynomials operational matrix of the product.
Using eorem 3 and equation (35), the integral parts in equation (31) are approximated as follows: After substituting approximations equations (32)-(36) into equation (31), the main equation is converted into the following nonlinear algebraic equation: Equation (37) is collocated at the n + 1 Gauss-Chelyshkov nodes (see [21]). By solving the nonlinear systems of the generated algebraic equations, the unknown vector C T can be determined. e resultant nonlinear systems can be solved by the Newton iteration method.

Error Analysis and Convergence
for all i such that 0 ≤ i ≤ n + 1, then we have the following theorem.
where C is a constant such that Proof. Assuming that q n (t) are the interpolating polynomials to y(t) at points t l (t l are the roots of the shifted Chebyshev polynomials of degree n + 1), one has e estimates for Chebyshev interpolation nodes is that (see [22]) As discussed in [20], y n (t) is the unique best approximation of y(t), then we get is proof is completed.

Convergence Analysis.
Before the convergence analysis, we firstly transform equation (1) into an equivalent nonlinear Volterra integral equation with the same initial condition (2). (1) and (2) are equivalent nonlinear Volterra integral equations as follows:

Theorem 5. If the right-hand side of equation (1) is continuous, then equations
where Proof. Applying the Riemann-Liouville integral operator I α t to both sides of equation (1), we have Journal of Mathematics where g(t) is defined by equation (46).
Using the Dirichlet's formula to the last part of the righthand side of equation (48), we get where B is the beta function. is proof is completed. Let g n (t) be the approximation of g(t) and f n (t) be the alternative Legendre approximation of f(t) and suppose the nonlinear terms F(u) and G(u) are satisfied in Lipschitz condition such that □ Theorem 6. e solution of the nonlinear fractional order integro-differential equation (45) by using alternative Legendre polynomials approximations is convergent when n ⟶ + ∞. Proof.
is proof is completed. Note: the conditions of eorem 6 are the same as eorems 4 and 5. By combining eorems 5 and 6, we can easily obtain the following theorem. e solution of the nonlinear fractional order integro-differential equation (1) by using alternative Legendre polynomials approximations is convergent when n ⟶ + ∞.

Numerical Examples
In this section, some numerical examples are presented to illustrate the proposed alternative Legendre polynomials method. In order to demonstrate the error of the method, the notation is shown as where y(t) and y n (t) are the exact solution and the numerical solution, respectively, t i ∈ [0, 1]. All algorithms are performed by Mathematica 10.0.

Example 1.
Consider the following nonlinear fractional order integro-differential equation with weakly singular kernel (see [23]): where with initial value y(0) � 0. e exact solution of this equation is y(t) � t 3/2 . e absolute errors between the numerical solutions and the exact solution with different values of n are displayed in Table 1 and compared with the results obtained by MHFs (modification of hat functions) (see [23]). From Table 1, it can be seen that the absolute errors become smaller and smaller with n increasing. Satisfactory results would be acquired by a small number of alternative Legendre polynomials. Figure 1 shows the comparison of e n obtained by our method and MHFs for different n on logarithmic scale. From Figure 1, we find that e n are smaller than that, and the accuracy of our method is higher.
Example 2. Consider the nonlinear fractional integro-differential equation with weakly singular kernel as follows (see [23]): where g(t) � 3t 2 − ( � � π √ Γ(7)/(Γ(15/2)))t 23/2 , with y(0) � 0. e exact solution of the nonlinear equation for α � 1 is e numerical solutions and comparisons are given in Table 2 and Figure 2. e absolute errors for α � 1 are listed in Table 2. It can be seen that the values of the absolute errors decay as n increases from 5 to 8. e comparisons of e n obtained by our method and MHFs for different n on logarithmic scale are plotted in Figure 2. is figure shows good coincidence between the numerical and exact solutions.
According to Figure 2, it can be deduced that the results obtained by the alternative Legendre polynomials method are more precise than those obtained by MHFs. Figure 3 displays the numerical results for α � 0.7, 0.8, 0.9, 1 and the exact result for α � 1. It shows that the numerical solutions converge to the exact solution as α ⟶ 1. Example 4. Consider the following nonlinear fractional integro-differential equation with weakly singular kernel:

Example 3. Consider this equation:
where Γ(2m+ 1)/(Γ(2m + 1.5)))t 2m+0.5 , such that y(0) � 0. e exact solution is y(t) � t m if α � 1, and m is a positive integer. e graphs of e n for different m on logarithmic scale are depicted in Figures 8-10. From Figures 8-10, one can find that for large n, the error norm becomes smaller and their logarithm behaves approximately as a linear function of n, which means that the error norm decreases linearly.

Conclusion
e alternative Legendre polynomials operational matrix method has been developed for solving the weakly singular nonlinear integro-differential equations with fractional derivatives. e method exploits the operational matrices of the fractional integration and the product of alternative Legendre polynomials as the primary underlying tool. A given nonlinear equation is converted into a nonlinear system of algebraic equations. e method has also been analyzed for error and convergence. A new operational matrix is obtained to approximate the integral parts with the singular kernels. To illustrate the reliability and efficiency of the suggested approach, the alternative Legendre operational matrices and the relevant approximations are employed to solve several examples. In the examples in which the order of the fractional derivatives is known, the numerical results show good agreement between the numerical and exact solutions. When the order of the fractional derivative α is uncertain, the numerical solutions for the various values of α approach to the exact solutions as α ⟶ 1. e results obtained by our method are also compared with those obtained by some existing methods. From the tables and figures, it can be concluded that the results obtained by the suggested method are more precise than those. e results confirm the ability of the alternative Legendre polynomials method to solve the nonlinear fractional integro-differential equations with weakly singular kernels.

Data Availability
No data were used to support the study.