A Novel Method for Solving Nonlinear Volterra Integro-Differential Equation Systems

and Applied Analysis 3 Table 1: The L2 and L∞ errors for Example 1. N 6 8 10 12 14 16 L2 − error y1(t) 3.1e − 005 1.4e − 007 4.3e − 10 9.0e − 13 8.5e − 16 2.0e − 15 L∞ − error y1(t) 2.0e − 005 8.1e − 008 2.2e − 10 4.1e − 13 4.4e − 16 8.8e − 16 L2 − error y2(t) 8.4e − 006 2.3e − 008 6.7e − 11 1.3e − 13 9.6e − 16 1.1e − 15 L∞ − error y2(t) 6.7e − 006 1.5e − 008 4.4e − 11 8.4e − 14 4.4e − 16 6.6e − 16 No. of Itre. 23 21 31 17 21 25 (yi)N = IN,:yi, where this notation IN,: denotes the last row of the (N × N) identity matrix. For the interpolating polynomial to satisfy the ith differential equation of the system of Volterra integro-differential equations (1) at each interior node, the collocation equation ?̇?i (tm) = fi (tm, p1 (tm) , . . . , pn (tm)) + ∫t o ki (t, s) Fi (p1 (s) , . . . , pn (s)) ds, pi (tN) = y0 i , i = 1, . . . , n (10) should be satisfied. Substituting the differentiation and the integration matrix relations into (10), we get [D m,: IN,: ] yi = [ Im,:fi (tm, y1, . . . , yn) y0 i ] + [Im,: (V ⋅ Fi (y1, . . . , yn)) 0 ] , i = 1, . . . , n. (11) Now, in view of (3) and the definitions of L and A, by substituting the differentiation and integration matrix relations, we will have the following explicit PIM for solving (1) which is called the spectral PIM (SPIM): y i = y i + h[D (1) m,: IN,: ] −1 ([D m,: IN,: ] y k i − [Im,:fi (tm, y1, . . . , yn) y0 i ] − [Im,: (V ⋅ Fi (y1, . . . , yn)) 0 ]) . (12) If we define L = [D(1) m,:, IN,:]T, f = [fi(tm, y1, . . . , yn), y0 i ]T, and Nyk = [Im,: (V ⋅ Fi(y1, . . . , yn)), 0]T, then we will have the following explicit iterative relation for finding the solution vector y i y i = y i + hL−1 (Lyk i − f − Ny i ) . (13) Here the vector y i is defined as y i = {yk+1 i (t1) , . . . , yk+1 i (tN)} . (14) Now one can start with the initial guess y i for obtaining the approximations yi. 4. Illustrative Examples In this section, we give several test examples to confirm our analysis. To examine the accuracy of the results, L2 and L∞ are employed to assess the efficiency of the method SPIM. All the computations were performed using software Matlab and terminated when the current iterate satisfies ‖yk− yk−1‖ ≤ 10−16, where yk is the solution vector of the kth SPIM iteration. Example 1. Consider the following system of Volterra integro-differential equations [20]: ̇ y1 (t) = 1 − 1 2 ̇ y2 2 (t) + ∫t 0 ((t − s) y2 (s) + y2 (s) y1 (s)) ds, ̇ y2 (t) = 2t + ∫t 0 ((t − s) y1 (s) − y2 2 (s) + y2 1 (s)) ds, (15) with the initial conditions y1(0) = 0 and y2(0) = 1. The exact solution of this system is (y1(t), y2(t)) = (sinh t, cosh t). Table 1 illustrates theL2 andL∞ errors for different values of N as well as the number of iterations to reach the abovementioned stopping criteria. Also Figure 1 shows the absolute error of the proposed method for N = 16. As expected, the exponential rate of convergence is observed for the system of nonlinear Volterra integro-differential equations, which confirmed our theoretical predictions. Example 2. Consider the following system of Volterra integro-differential equations with the exact solution (y1(t), y2(t)) = (t + et, t − et) [8]: ̇ y1 (t) = 1 + t − t2 − y2 (t) + ∫t 0 (y1 (s) + y2 (s)) ds, ̇ y2 (t) = −1 − t + y1 (t) − ∫t 0 (y1 (s) − y2 (s)) ds, (16) subjected to initial conditions y1(0) = 1 and y2(0) = −1. The results of L2 and L∞ errors of y1(t) and y2(t) for the different values of N as well as the number of iterations to reach the stopping criteria are given in Table 2. Figure 2 depicts the absolute error of the presented method for N = 16. Again, the exponential rate of convergence is observed for the system of nonlinear Volterra integrodifferential equations. 4 Abstract and Applied Analysis


Introduction
Systems of integro-differential equations and their solutions play a pivotal role in the fields of science, industrial mathematics, control theory of financial mathematics, and engineering [1][2][3].Physical systems, such as biological applications in population dynamics, and genetics where impulses arise naturally or are caused by control are modeled by a system of integro-differential equations [4,5].The initial value problem for a nonlinear system of integro-differential equations were used to model the competition between tumor cells and the immune system [6].In [7], two systems of specific inhomogeneous integro-differential equations are studied in order to examine the noise term phenomenon.Thus applications of numerical methods for solving these equations are attractive.This has led to a great deal of research in recent years with the use of numerical methods such as the variational iteration method [8], differential transform method [9], Bezier curves method [10], radial basis function networks [11], biorthogonal systems [12], the block pulse functions method [13], and a collocation method in combination with operational matrices of Bernstein polynomials [14].
The parametric iteration method (PIM) is an analytic approximate method that provides the solution of linear and nonlinear problems as a sequence of iterations.In fact, the PIM as a fixed-point iteration method is a reconstruction of the variational iteration method.Since the implementation of the PIM generally leads to the calculation of unneeded terms, where more time is consumed in repeated calculations for series solutions, so to overcome these shortcomings, a useful improvement of the PIM was proposed in [15].
To demonstrate the utility of the proposed method, some examples of system of Volterra integro-differential equations are given, which are solved using the established method.The

Parametric Iteration Method
The PIM gives rapidly convergence by using successive approximations of the exact solution if such a solution exists; otherwise the approximations can be used for numerical aims.To convey the basic idea of this method, we first consider (1) as below: where  denotes the auxiliary linear operator with respect to   .In (2)  is a nonlinear continuous operator with respect to   and   () is the source term.
Next, we construct a family of explicit iterative processes for (2) [15,16] where with the initial conditions  +1  (0) =  0  .Also, we can construct a family of the implicit PIM for (2) as follows: with the above initial condition. 0  is the initial guess which can be freely found from solving its corresponding linear equation ([ 0  ] = 0 or [ 0  ] =   ()) and the subscript  denotes the th iteration.The parameter ℎ ̸ = 0 denotes the so-called auxiliary parameter, which can be identified easily and efficiently by the technique proposed in [15].Also we are free to choose the auxiliary linear operator , the auxiliary parameter ℎ, and the initial approximation  0  , which is fundamental to the validity and flexibility of the PIM.Accordingly, the successive approximations of    (),  ≥ 0, for the PIM iterative relation will be obtained readily in the auxiliary parameter ℎ.Consequently, the exact solution can be obtained by using the following: When the original PIM fails, then the presence of the parameter ℎ in (3) or ( 5) can play an important role in the frame of the PIM.However, we can always discover a valid region of ℎ for every physical problem by plotting the solution or its derivatives versus the parameter ℎ in some points.An approximate optimal value of the convergence accelerating parameter ℎ can be determined at the order of approximation by the residual error [17]: Now, one can minimize (7) by imposing the requirement   (ℎ)/ℎ = 0.

Description of the Method
The PIM procedure provides the solution of the system of Volterra integro-differential equations as a sequence of iterates; its successive iterations may be very complex so that the resulting integrals in its iterative relation may not be performed analytically.Here, we will overcome this shortcoming of the PIM for solving (1) by proposing a spectral collocation PIM.As will be shown in this paper later, the new method will be very simple to implement and save time and calculations.
Generally, in order to solve system (1) using a spectral collocation scheme, the interpolating polynomials   () ( = 1, . . ., ) are required to satisfy the equations of the system at the interior nodes.The values of the interpolating polynomials at the interior nodes  2 , . . .,   are   (t  ) = (y  )  =  ,: y  ( = 1 :  − 1), where  ,: denotes the  row of the  ×  identity matrix and the derivative values are ṗ  (t  ) =  (1)  ,: y  .The initial condition that involves the interpolating polynomials can be handled by using the formulas   (t  ) = should be satisfied.Substituting the differentiation and the integration matrix relations into (10), we get ,: Now, in view of (3) and the definitions of  and , by substituting the differentiation and integration matrix relations, we will have the following explicit PIM for solving (1) which is called the spectral PIM (SPIM): )   ,: )   ,: If we define L = [ (1)  ,: ,  ,: ]  , f = [  (t  , y 1 , . . ., y  ),  0  ]  , and Ny  = [ ,: ( ⋅   (y 1 , . . ., y  )), 0]  , then we will have the following explicit iterative relation for finding the solution vector y +1 Here the vector y +1  is defined as Now one can start with the initial guess y 0  for obtaining the approximations y  .

Illustrative Examples
In this section, we give several test examples to confirm our analysis.To examine the accuracy of the results,  2 and  ∞ are employed to assess the efficiency of the method SPIM.All the computations were performed using software Matlab and terminated when the current iterate satisfies ‖y  − y −1 ‖ ≤ 10 −16 , where y  is the solution vector of the th SPIM iteration.
Table 1 illustrates the  2 and  ∞ errors for different values of  as well as the number of iterations to reach the abovementioned stopping criteria.Also Figure 1 shows the absolute error of the proposed method for  = 16.As expected, the exponential rate of convergence is observed for the system of nonlinear Volterra integro-differential equations, which confirmed our theoretical predictions.
The results of  2 and  ∞ errors of  1 () and  2 () for the different values of  as well as the number of iterations to reach the stopping criteria are given in Table 2. Figure 2 depicts the absolute error of the presented method for  = 16.Again, the exponential rate of convergence is observed for the system of nonlinear Volterra integrodifferential equations.
The errors and number of iterations for the different values of  are presented in Table 3.The other results are presented in Figure 3 with  = 16 similar to Examples 1 and 2.

Conclusion
In this paper, we presented a powerful numerical approach based on a combination of the Chebyshev spectral collocation technique and the parametric iteration method for solving the linear and nonlinear system of Volterra integrodifferential equations.This method inherits the strengths of the PIM and is easy to implement and is accurate when applied to the linear and nonlinear system of Volterra integrodifferential equations.The comparison of the approximate solution and the exact solution reveals that the present method is very accurate and convenient for solving the linear and nonlinear system of Volterra integro-differential equations.

Table 2 :
The  2 and  ∞ errors for Example 2.