A new operational matrix for solving two-dimensional nonlinear integral equations of fractional order

Abstarct: In this paper, first, we derive the operational matrix of two-dimensional orthogonal triangular functions (2D-TFs) for two-dimensional fractional integrals. Then, we apply this operational matrix and properties of Two-dimensional orthogonal triangular functions to reduce two-dimensional fractional integral equations to a system of algebraic equations. Finally, in order to show the validity and efficiency, we present some numerical examples.


Introduction
As a branch of mathematics, fractional calculus provides an excellent tool for describing and modeling such complex engineering and scientific phenomena as fluid-dynamic traffic model (He, 1999), model frequency-dependent damping behavior of viscoelastic materials (Bagley & Torvik, 1983, 1985, economics (Baillie, 1996), continuum and statistical mechanics (Mainardi, 1997), solid

PUBLIC INTEREST STATEMENT
As we know, we can convert many initial and boundary value problems into problems of solving integral equations. So, it is important to develop numerical methods for solving integral equations. In this article, after deriving the operational matrix of two-dimensional orthogonal triangular functions for two-dimensional fractional integrals, we reduce two-dimensional fractional integral equations to a system of algebraic equations by applying this operational matrix. It is necessary to say that the introduced operational matrix in this paper can be applied for solving differential equations of fractional order, integro-differential equations of fractional order, etc.
mechanics (Rossikhin & Shitikova, 1997), and dynamics of interfaces between soft-nanoparticles and rough substrates (Chow, 2005). Several numerical methods to solve fractional differential equations and fractional integro-differential equations have been recently presented by many authors. In EI-Wakil, Elhanbaly, and Abdou (2006) authors used Adomian decomposition method for Fractional nonlinear differential equations. Saadatmandi and Dehghan in Saadatmandi and Dehghan (2010) used the Legendre operational matrix to solve fractional-order differential equations. In Saadatmandi (2014), Bernstein polynomials were used for solving partial differential equations. In Maleknejad and Asgari (2015) used triangular functions for multi-order fractional differential equations. In Chen, Liu, Turner, and Anh (2013), two-dimensional fractional percolation equation was solved. In Najafalizadeh and Ezzati (2016) we see two-dimensional block pulse operational matrix is used for two-dimensional nonlinear integral equations of fractional order.
Here, we try to extend the application of 2D-TFs to solve two-dimensional nonlinear integral equations of fractional order in. Our main aim is to obtain 2D-TFs operational matrix for two-dimensional fractional integral to reduce the original problem to a system of algebraic equations. In this paper first, we briefly review fractional calculus and one-dimensional triangular functions (1D-TFs). In Section 3, we present the approximation of function via 2D-TFs. Also, by using the properties of 2D-TFs, we derive the operational matrix of two-dimensional integration of fractional order. Section 4 is devoted to solving two-dimensional nonlinear fractional integral equations by applying the operational matrix of integration of fractional order introduced in previous section. In Section 5, we show the accuracy and the efficiency of the proposed method through several examples. Finally, a conclusion is given in Section 6.

Brief review for fractional calculus
The most commonly used definitions for fractional derivative and fractional integration are Caputo and Riemann-Liouville definitions, respectively. if p, g ∈ (−1, ∞), then x p+r 1 y q+r 2 .

One-dimensional triangular functions
Triangular functions are among orthogonal functions that are introduced by authors of Deb, Dasgupta, and Sarkar (2006), Deb, Sarkar, and Dasgupta (2007). Maleknejad and Asgari (2015) applied these functions to solve nonlinear integro-differential equations of fractional order. In , an m-set of 1D-TFs over interval [0, T) are defined as: Clearly, we can define m-set of 1D-TF vectors as the following: and The vector T(t) is called 1D-TFs vector.
The operational matrix for fractional integration can be obtained as Maleknejad and Asgari (2015): where and (2.5)

Two-dimensional triangular functions (2D-TFs)
In Babolian, Maleknejad, Roodaki, and Almasieh (2010), authors defined an m 1 × m 2 -set of 2D-TFs on [0, T 1 ) × [0, T 2 ) as follows: . m 1 and m 2 are arbitrary positive integers. Also, they defined the following vectors: With the following properties: By considering above vectors, authors of Babolian et al. (2010) defined 2D-TF vector as the following form: According to this fact , to construct the operational matrix of 2D-TFs for the fractional integration in Section 4, we need to derive T11(s, t), T12(s, t), T21(s, t), and T22(s, t) by Kronecker product of T1(t) and T1(s). So, using (3.1) and (3.2), we can write where ⊗ denotes the Kronecker product defined for two arbitrary matrices A and B as and also it has the following two basic properties (Zhang & Ding, 2013): In Babolian et al. (2010), it is proved that 2D-TFs are disjoint, orthogonal. Thus, for every where T(x, y) and T(s, t) are 2D-TFs vectors of dimension 4m 1 m 2 and 4m 3 m 4 , respectively, and K is a (4m 1 m 2 × 4m 3 m 4 ) 2D-TFs coefficient matrix.

Operational matrix of 2D-TFs for the fractional integration
In this section, we construct operational matrix of 2D-TFs for the fractional integration.

Numerical solution of two-dimensional nonlinear fractional integral equations
In this section, we present an effective method to solve two-dimensional nonlinear integral equations of fractional order. For this purpose, we apply two-dimensional triangular functions to approximate known and unknown functions, whose properties of these functions were shown in Section 3. Consider the following two-dimensional nonlinear fractional integral equation where r 1 > 0, r 2 > 0, the functions k(x, y, s, t) and g(x, y) are known and f(x, y) is the unknown function to be determined. Also, p ≥ 1 is a positive integers. Using the methods mentioned in Section 4, the functions f(x, y), g(x, y), [f (x, y)] p , and k(x, y, s, t) can be approximated by: where T(x, y) is defined in Equation (3.3), the vectors F, G, F p and matrix K are 2D-TFs coefficients of f(x, y), g(x, y), [f (x, y)] p , and k(x, y, s, t), respectively. Now, by substituting Equation (5.2) in Equation (5.1), we have Using Equation (3.7), we conclude that k(x, y, s, t) = T(x, y) T KT(s, t), Substituting Equation (4.1) in Equation (5.3), we have Clearly, by assuming H = KF p P r 1 ,r 2 , we will get Now, using (3.7), we have: Hence, we will get the following nonlinear algebraic system: Clearly, this system can be solved by known methods such as Newton's method. After solving (5.5), we can obtain the approximate solution of (5.1) using (3.8).

Illustrative examples
To illustrate the effectiveness of the proposed method in the Section 5, we present three test examples. In these examples, we assume that T 1 = T 2 = 1, m 1 = m 2 . Also, in this section, we apply the following error function where f(x, y) and f m 1 ,m 2 (x, y) are the exact and the approximate solutions of the two-dimensional fractional integral Equation (5.1), respectively. Example 6.1 (Najafalizadeh & Ezzati, 2016) Consider the following two-dimensional fractional integral equation: whose exact solution is given by f (x, y) = √ 3xy 3 .
The approximate solution of f(x, y) is obtained using 2D-TFs method described in Section 5. Table 1 shows a comparison of the proposed method and the method of Najafalizadeh and Ezzati (2016). The displayed results show that the proposed method is more accurate than the proposed method in Najafalizadeh et al. (2016).
Example 6.2 Consider the following two-dimensional fractional integral equation: The exact solution of this example is f (x, y) = 1 2 xy. Table 2 illustrates the numerical results for this example.
Example 6.3 As a last example, we present the following two-dimensional fractional integral equation: