Numerical Solution of Nonlinear Fredholm and Volterra Integrals by Newton–Kantorovich and Haar Wavelets Methods

The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.


Introduction
The application of integral equations can be found in various fields which include mathematics, physics and engineering. The process of solving the integral equations analytically is very complicated and for application purposes, it will be sufficient to solve the latter numerically. Previously, many methods have been established to find numerical solutions for integral equations. These methods include the polynomial approximation [1,2], linear multistep methods [3], modified homotopy perturbation [4], wavelets [5][6][7][8][9], triangular functions [10] and Newton-Kantorovich method [11][12][13][14]. A few of the mentioned methods are applicable to merely linear integrals whereas the latter are capable of approximating nonlinear integral equations. Finding the numerical solutions for integral equations are often a complicated process and requires a large number of arithmetic computations. Therefore, a simple and efficient technique that can solve both linear and nonlinear integral equations is needed.
This research aims to present a numerical method which approximates nonlinear Fredholm integrals and nonlinear Volterra integral both of the second kind. The functions f , f l ∈ L 2 [0, 1], l = 0, 1, . . . , m, K(t, s) ∈ L 2 [0, 1] × [0, 1] are given functions whereas, x(t) is the unknown function with m as a positive integer. This article is organized as follows: following the introduction section, Sections 2 and 3 provides the preliminaries and approximation of the functions by Haar wavelets; numerical solutions for both nonlinear Fredholm and Volterra integrals are then constructed in Sections 4 and 5 respectively; error analysis for Haar wavelets which is derived in Section 6 satisfies the conditions in Holder space; the numerical results are reported in Section 7; while the conclusion is elaborated in Section 8.

Preliminaries
Theorem 1. The solution of system is established as below: where See [6,15].

Corollary 1.
The solution of the system is defined below: where τ i , σ i , β i , α i and ρ i are illustrated in Theorem 1. See [8].
is called a Holder space of order s and denoted by H s [0, 1] whereas, the norm is given by for all t, y ∈ [0, 1].

Establishing Haar Wavelets
Haar wavelets are selected to solve the Equations (1) and (2) due to fact that they are the most simple orthonormal wavelets with compact support. The Haar wavelets are defined on the interval [0, 1) by which forms an orthonormal system: Any square integrable function, x(t) over the interval (0, 1) can be approximated using Haar wavelets as follows: In practice, the series (6) is truncated as where N = 2 β , β = 0, 1, 2, . . . , and c n are the unknown coefficients. To evaluate the coefficients c n , we need to consider these collocation points Next, by substituting the collocation points (8) into (6), we obtain the following 2N × 2N linear system of equations which could be written in the matrix form where H is an 2N × 2N with h ij = h i (t j ) and unknown coefficients, C = [c 1 , c 2 , . . . , c n ] T . The H is an asymmetric Haar with only element 1, −1 or 0. Here we do not need to solve the above system which is computationally expensive for large values of N, thus Theorem 1 gives us a simple formula to calculate the coefficients c n .
Let K(t, s) be a square integrable function with variables t and s where the function K(t, s) is approximated using the Haar wavelet basis as Substituting (8) into (10) yields the following system of 2N × 2N linear equations where the corollary 1 above denotes an algorithm for finding the unknown coefficients c i (t), i = 1, 2, . . . , 2N.

Nonlinear Fredholm Integral of the Second Kind
This section provides the derivation for solving nonlinear Fredholm integral of the second kind. Firstly consider the following nonlinear Fredholm integral of the second kind which can be rewritten in the operator form The initial iteration of Newton-Kantorovich method is described as where x 0 = x 0 (t) is the initial guess and is a continuously differentiable function in the interval [0, 1]. P (x 0 ) is the derivative of P(x) at the point x 0 . By the method of iteration, we obtain a sequence of linear integral equations as follows: where Therefore, the nonlinear Fredholm integral Equation (12) is reduced to a sequence of linear integral Equations (14). The Newton-Kantorovich method constructs a sequence of functions that will converge to the solution (13). The Newton-Kantorovich theorem was first introduced by Leonid Kantorovich in 1948 since then, many studies and applications have been done in this area (see [13,14,[16][17][18][19]).
In general, the sequence of x i (t) can be evaluated by solving the following equation . . , in order to achieve better approximation of x(t).
After a few iterations, the approximated solution x i (t) stacks at a certain level of iteration and is not gaining better results. Therefore, we need to increase the number of bases N for the Haar wavelets as to improve the approximated solution of x i (t). The consequence of increasing the number of bases N increases the computational effort. Fortunately, the problem can be avoided. Due to the Haar wavelets figure (step functions), one's could take the average of the approximated solution x i (t) below: and allign back the approximated solution as The Equation (21) will perform a better approximation solution than x i (t) without any increase in the number of bases N for the Haar wavelets. Next, we proceed with the nonlinear Volterra integral of the second kind.

Nonlinear Volterra Integral of the Second Kind
The general nonlinear Volterra integral of the second kind is formulated as.
Using similar techniques, the Fredholm integral is reduced via the Newton-Kantorovich method produces a sequence of linear Volterra integral equations where where K 1 (t, s) = K(t, s) m ∑ l=0 l f l (s)x l−1 0 (s). By approximating the functions ∆x 1 (t) and K 1 (t, s)∆x 1 (s) using the Equations (16) and (17) and substituting it into Equation (23) yields Similarly to the Fredholm integral, we only expand the left side of the Equation (24) by the Haar wavelets. Then, we substitute Equation (24) using the collocation points from (8) and get the following 2N × 2N linear system of equation: By solving this linear system, we obtain x 1 (t) where, x 1 (t) = ∆x 1 (t) + x 0 (t). For a better approximation of x(t), this process is repeated to find x 2 (t), x 3 (t), . . . until a sufficient solution for x i (t) is obtained. Finally, to achieve a better approximation than x i (t) without increasing the number of Haar wavelet bases N as mentioned in the previous section, the sufficient solution x i (t) is substituted into Equation (21).

Error Analysis
Let x i (t) ∈ L 2 [0, 1] be sequence of functions which satisfy

From definition of the Haar wavelets:
h n (t) = 2 j 2 H(2 j t − k), k = 0, 1, . . . 2 j − 1, j = 0, 1, . . . , By changing the limit of integration in the above equation from Taking into account that the sequence of function x i (t) The latter allow us to estimate the coefficients c n as follows Hence, we have arrived to the final estimation for the coefficients Therefore, This completes the proof of the Theorem 2.
We note that Ahmedov et al. [20] have shown that the error in L 2 [0, 1] norm is given by This ensures that the Haar wavelet approximation converges if N increases.

Numerical Examples
In this section, we demonstrate the efficiency of the Newton-Kantorovich-Haar wavelets method to solve nonlinear Fredholm and Volterra integral equations of the second kind. The calculation for each of these examples was performed in Maple 15. Example 1. Consider the following nonlinear Fredholm integral equation [11] x(t) = sin (πt) + 1 5 1 0 cos (πt) sin (πs)(x(s)) 3 ds.
In order to approximate this integral equation, we repeat the Haar wavelets method until x 3 (t) and used the approach in Equation (4) to obtain S 10 (t).
We apply the same initial condition as x 0 (t) = 0 stated in [11] for N = 8 and N = 16. In Table 1, we compare the absolute error of the proposed method with those from Newton-Kantorovich-Simpson quadrature method [11]. Figure 1 shows the comparison of the approximated solution x 3 (x), S 10 (t) and the exact solution of this integral equation x(t) = sin(πt) + 1 3 (20 − √ 391) cos(πt). The advantage of the proposed method is that we can obtain an estimate of all t values in the interval [0, 1] for any value of N, unlike the [11] method which can only obtain a certain t value depending on the subinterval number they divide.  Example 2. Consider the following nonlinear Volterra integral equation [11] x(t) − t 0 x 2 (s)ds = sin(x) − t 2 + 1 4 sin(2x).
For this example, we intend to solve the nonlinear Volterra integral equation by choosing the initial x 0 (t) = 1. In this case, we use four iterations to get the estimated settlement of x 4 (t) in Figure 2. In Table 2, we compare the absolute error of the proposed method with Newton-Kantorovich block-by-block methods [11]. The exact solution for the integral equation is sin(t).   [9] x(t) + 1 0 e t−2s [x(s)] 3 ds = e t+1 .
We solve the Fredholm integral equation by choosing the initial condition x 0 (t) = 1. The approximated solution x 3 (t) is then converted to S 10 to obtain better accuracy for N = 4. For the case of N = 8, an adequate solution is obtained using a single iteration x 1 (t) but with a slight alteration where the approximated solution, x 3 (t) for N = 4 is taken as the new initial condition. Similarly, we attain x 1 (t), for N = 16 using the initial condition x 1 (t) of N = 8. Table 3 shows the absolute error using the Haar wavelets method [9] and the presented method. It is evident in Table 3 that our method is better than the Haar wavelets method [9] for this problem. Moreover, the presented method yields good estimation results with only N = 4 compared to the Haar [9] method using N = 16.
in which the exact solution is x(t) = e t . In this example, we acquire an approximated solution of x 3 (t) using N = 4 and by letting the initial condition x 0 (t) = 1 + x. Next, the first iterations for N = 8 and N = 16 are attained in similar manner as Example 3. Table 4 describes both the absolute errors for the presented method S 10 (t) and Haar wavelets method [9]. Example 5. Consider the following nonlinear Volterra integral equation [21] x(t) + t 0 (x 2 (s) + x(s))ds = with exact solution e −t . The initial guess is x 0 (t) = 1 in finding the approximate solution x 4 (t) and we use the same procedure as the example above in finding the approximate solution for N = 8 and N = 16.
In Table 5, we compare the absolute error of the proposed method with hybrid Taylor polynomials and Block-Pulse functions [21].

Conclusions
This study describes new techniques using a combination of Newton-Kantrovich and Haar wavelets to solve the second kind of nonlinear Fredholm and Volterra integral equations. It has been proven that an appropriate initial guess is required in the Newton-Kantrovich method. Therefore, when applying large values of N such as N = 16 to the approximation of the Haar wavelets x 16 (t), we recommend using the previous approximate solution x 8 (t) or x 4 (t) which considers the smaller value of N as an initial condition. This will ultimately reduce the number of iterations and improve the accuracy of the estimated solution. We also provide a simple approach at the end of Sections 4 and 5 that can improve the approximation solution without increasing the number of bases N for the functions of Haar wavelets.