System of Nonlinear Volterra Integro-Differential Equations of Arbitrary Order

In this paper, a new approximate method for solving the system of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional) order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs). Also, we construct the fractional derivative operational matrix of order α in the Caputo’s definition for GFCFs. This method reduces a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.

In this study, by substituting the GFCFs with the unknown coefficient that satisfies in the boundary conditions, we convert the nonlinear Volterra IDEs system of arbitrary order into nonlinear algebraic system.Because of the fractional derivatives can be in the IDE, we use the fractional derivative operational matrix of GFCFs to convert nonlinear Volterra IDEs system into nonlinear algebraic system.By solving such system, the approximate solution of exact solution is obtained.
The organization of the paper is expressed as follows: in section 2, some basic definitions and theorems are expressed.In section 3, the GFCFs and their properties are obtained.In section 4, the operational matrix of fractional derivative to GFCFs is obtained.In Section 5, application of the method is explained.Illustrative examples of the proposed method are shown in section 6.Finally, a conclusion is provided.

Basical definitions
In this section, some basic definitions and theorems which are useful for our method have been introduced [12,13].
Theorem 2. Suppose that {P i (t)} be a sequence of orthogonal polynomials, w(t) is weight function for {P i (t)}, and q(t) is a polynomial of degree at most n − 1, then for p n (t) ∈ {P i (t)} we have: p n (t), q(t) w = 0. Proof: See the section 2.3 in Ref. [16].

Generalized Fractional order of the Chebyshev Functions
In this section, the generalized fractional order of the Chebyshev functions of the first kind have been defined, and then some properties and convergence of them for our method have been introduced.

The Chebyshev functions
The Chebyshev polynomials have been used in numerical analysis, frequently, including polynomial approximation, Gauss-quadrature integration, integral and differential equations and spectral methods.Some of Chebyshev polynomials properties are: orthogonality, recursive, real zeros, complete for the space of polynomials, etc.For these reasons, many researchers have employed these polynomials in their research [17,18,19,20,21,22,23].
The number of researchers by using some transformations extended Chebyshev polynomials to semi-infinite or infinite domain for example by using x = t−L t+L , L > 0 the rational Chebyshev functions are introduced [24,25,26,27,28,29,30].
In this study, by transformation x = 1 − 2( t η ) α ; α, η > 0 on the Chebyshev polynomials of the first kind, the generalized fractional order of the Chebyshev orthogonal functions (GFCF) in interval [0, η] have been introduced, that we can use them to solve nonlinear Volterra IDEs.

The GFCFs definition
The efficient methods have been used by many researchers to solve the differential equations (DE) is based on series expansion of the form n i=0 c i t i , such as Adomian's decomposition method [31] and Homotopy perturbation method [32].But the exact solution of many DEs can't be estimated by polynomials basis, for a simple example: the ODE of 4yy ′′ = 3t, y(0) = y ′ (0) = 0, that the exact solution is y(t) = t 3 2 , therefore we have defined a new basis for Spectral methods to solve them as follows: The analytical form of η F T α n (t) of degree nα is given by where Note that η F T α n (0) = 1 and η F T α n (η) = (−1) n .The GFCFs are orthogonal with respect to the weight function where δ mn is Kronecker delta, c 0 = 2, and c n = 1 for n ≥ 1.The Eq. (3.2) is provable using the properties of orthogonality in the Chebyshev polynomials.

Approximation of functions
Any function y(t), t ∈ [0, η], can be expanded as the follows [16]: where the coefficients a n are obtained by inner product: and using the property of orthogonality in the GFCFs: In practice, we have to use first m-terms GFCFs and approximate y(t): with (3.5)

Convergence of method
The following theorem shows that by increasing m, the approximation solution where .
Since A T Φ(t) is the best approximation to f (t) from η F α m , and y ∈ η F α m , one has Now by taking the square roots, the theorem can be proved.* Theorem 4. The generalized fractional order of the Chebyshev function η F T α n (t), has precisely n real zeros on interval (0, η) in the form

Moreover, d
dt η F T α n (t) has precisely n−1 real zeros on interval (0, η) in the following points: Proof.The Chebyshev polynomial T n (x) has n real zeros [33,34]: System of Nonlinear Volterra Integro-Differential Equations 39 Therefore T n (x) can be written as . Also, we know that, the real zeros of d dt T n (t) occurs in the following points [34]: Same as in the previous, the absolute extremes of

The fractional derivative operational matrix of GFCFs
In the next theorem, the operational matrix of the Caputo fractional derivative of order α > 0 for GFCFs is generalized, which can be expressed by: (4.1) Theorem 5. Let Φ(t) be GFCFs vector in Eq. (3.5), and D (α) is an m × m operational matrix of the Caputo fractional derivatives of order α > 0, then: , and orthogonally property of GFCFs, the Eqs.(2.1) and (3.1), for i, j = 0, 1, ..., m − 1: ), therefore by theorem 2, D (α) i,j = 0 for any i ≤ j.Now for i > j we have: Now, by integration of above equation, the theorem can be proved.* Remark: The fractional derivative operational matrix of GFCFs is an lowertriangular matrix and for α = 1 is same as shifted Chebyshev polynomials [35].

Application of the GFCF collocation method
In this section, we apply the GFCFs collocation method to solve the system of nonlinear Volterra IDEs of arbitrary order.
To apply the collocation method, we construct the residual functions by substituting y jm (t) in Eq. (5.1) for y j (t) in the system of nonlinear Volterra IDEs of arbitrary order the Eq.(1.1), for i = 1, 2, ..., M : The equations for obtaining the coefficient {a jn } M,m−1 j=1,n=0 arise from equalizing Res i (t) to zero on m collocation points: (5.3) In this study, the roots of the GFCFs in the interval [0, η] (Theorem 4) are used as collocation points.By solving the obtained set of 2m × 2m equations, we have the approximating functions y jm (t), j = 1, 2, ..., M .
Note that if we have a fractional derivative in system, we can not use the Eq.(5.1) for the unknown function, instead, we use Eqs.(3.3) and (4.1), and then the boundary conditions will applied.
And also consider that all of the computations have been done by Maple 2015 on a laptop with CPU Core i7, Windows 8.1 64bit, and 8GB of RAM.

Illustrative examples
In this section, by using the present method we solve some well-known examples to show efficiently and applicability GFCFs method based on Spectral method.These examples have been studied by other researchers.We apply the present method to solve the system of nonlinear Volterra IDEs of arbitrary order, and their outputs are compared with the corresponding analytical solution.

Nonlinear system of Volterra IDEs of integer order
We applied the present method to solve some system of Volterra IDEs of integer order.
Example 1.Consider nonlinear system of Volterra IDEs with boundary conditions The exact solutions of this problem are u(t) = sin(t) and v(t) = cos(t).By applying the technique described in last section, we satisfy the boundary conditions as follows: We construct the residual functions as follows: Therefore, to obtain the coeffcient {a jn } 2,m−1 j=1,n=0 ; Res i (t), i = 1, 2 is equalized to zero at m collocation point.By solving this set of nonlinear algebraic equations, we can find the approximating function y jm (t).
Figure 2 shows the logarithmic graph of the absolute and the residual errors of the approximate solutions and the analytic solutions for m = 20, and α = 0.50.Example 2. Consider the following linear system of Fredholm IDEs with boundary conditions u The exact solutions of this problem are u(t) = 3t 2 + 1 and v(t We satisfy the boundary conditions: and construct the residual functions as follows: Figure 3 shows the logarithmic graph of the absolute and the residual errors of the approximate solutions and the analytic solutions for m = 15, and α = 0.50. The exact solutions of this problem are u(t) = sinh(t) and v(t) = cosh(t).We satisfy the boundary conditions: and construct the residual functions as follows: Figure 4 shows the logarithmic graph of the absolute and the residual errors of the approximate solutions and the analytic solutions for m = 15, and α = 0.25.
The exact solutions of this problem are u(t) = t + e t and v(t) = t − e t .We satisfy the boundary conditions: and construct the residual functions as follows: The exact solutions of this problem are u(t) = e t and v(t) = e 2t .We satisfy the boundary conditions: and construct the residual functions as follows: Figure 6 shows the logarithmic graph of the absolute and the residual errors of the approximate solutions and the analytic solutions for m = 17, and α = 0.50.
The exact solutions of this problem are u(t) = sin(t) and v(t) = cos(t).We satisfy the boundary conditions: and construct the residual functions as follows: Figure 7 shows the logarithmic graph of the absolute and the residual errors of the approximate solutions and the analytic solutions for m = 20, and α = 0.50.

Nonlinear system of Volterra IDEs of fractional order
We applied the present method to solve some system of Volterra IDEs of fractional order.
The exact solutions of this problem are u(t) = t 3 + t + 1 and v(t) = e t .By applying the technique described in last section, we satisfy some boundary conditions as follows: y 1m (t) = y 1m (t) = A T 1 Φ(t), y 2m (t) = 1 + t + t 2 y 2m (t),

Conclusion
In this paper, the generalized fractional order of the Chebyshev functions (GFCFs) of the first kind have been introduced as a new basis for Spectral methods and this basis can be used to develop a framework or theory in Spectral methods.Next an operational matrix of fractional derivative for these orthogonal functions is obtained.These functions and matrix can be used to solve the system of nonlinear Volterra integro-differential equations of arbitrary order.The comparison of the approximate solutions and the exact solutions shows that the proposed method is more efficient tool and more practical for solving linear and non-linear systems of integro-differential equations and plots confirm.

Figs. 1 Figure 1 :
Figure 1: Graphs of the GFCFs for various values of n and α.
(a) Graphs of the absolute errors (b) Graphs of the residual errors

Figure 2 :
Figure 2: Log graphs of the absolute and the residual errors for example 1.
(a) Graphs of the absolute errors (b) Graphs of the residual errors

Figure 3 :Example 3 .
Figure 3: Log graphs of the absolute and the residual errors for example 2.
(a) Graphs of the absolute errors (b) Graphs of the residual errors

Figure 4 :
Figure 4: Log graphs of the absolute and the residual errors for example 3.

Example 4 .
Consider the nonlinear Volterra IDEs problem with boundary conditions u

Figure 5
Figure5shows the logarithmic graph of the absolute and the residual errors of the approximate solutions and the analytic solutions for m = 20, and α = 0.50.
(a) Graphs of the absolute errors (b) Graphs of the residual errors

Figure 5 :Example 5 .
Figure 5: Log graphs of the absolute and the residual errors for example 4.
(a) Graphs of the absolute errors (b) Graphs of the residual errors

Figure 6 :Example 6 .
Figure 6: Log graphs of the absolute and the residual errors for example 5.

Figure 8
Figure8shows the logarithmic graph of the absolute and the residual errors of the approximate solutions and the analytic solutions for m = 10, and α = 0.50.
(a) Graphs of the absolute errors (b) Graphs of the residual errors

Figure 8 :
Figure 8: Log graphs of the absolute and the residual errors for example 7.