Approximation of functions by Bernoulli wavelet and its applications in solution of Volterra integral equation of second kind

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Introduction
Wavelet analysis is recent and advanced analysis in pure as well as applied mathematical research area. It has been applied in different branches of science and technology such as engineering, image processing, signal analysis, time frequency analysis, and fast numerical algorithms. There are several problems of science and technology in the form of integral equations. There are various types of methods to find the numerical solution of integral equations. Sometimes it is difficult to find the solution of the integral equations analytically. Therefore, there is need of approximate solution.
Bernoulli wavelets have been applied to solve fractional order differential equations and calculus of variation problems by Keshavarz [3,4]. We have presented Bernoulli wavelet method to solve linear Volterra integral equations of the form: (1.1) and nonlinear Volterra integral equation of the form: K (x, t)F(y(t))dt (1.2) where f ∈ L 2 [0, 1) and K ∈ L 2 [0, 1) × L 2 [0, 1) are known functions, F is a function of y and y is unknown function to be determined [1,[9][10][11].
The main objectives of this research paper are following: It is remarkable that the solution of (1.1) and (1.2) obtained by Bernoulli wavelet technique and its exact solution are almost same.

Bernoulli wavelets
Bernoulli wavelets B n,m have four arguments:n = n − 1, n = 1, 2, . . . , 2 k−1 , where k is some positive integer, m is the order of Bernoulli polynomials and t is the normalized time. Then, B n,m are defined on the interval [0,1) as follows: for m = 0, 1, 2, . . . , M − 1, and n = 1, 2, . . . , 2 k−1 . The coefficient is Bernoulli polynomial of order m, which is given by α j are Bernoulli numbers and can be defined by the identity for some positive constant A (Zheng and Wei [21]).

Approximation of function
A function f ∈ L 2 [0, 1) can be expanded as where the coefficients c n,m are given by where < , > represents inner product in L 2 [0, 1).
Bernoulli wavelet basis functions are given by

Bernoulli wavelets operational matrix of integration
For k = 3 and M = 3, the 2 k−1 M × 2 k−1 M Bernoulli wavelet operational matrix of integration given by Razzaghi [4] as follows:

Bernoulli wavelets operational matrix of product
The product of two Bernoulli wavelet functions is given by

Results
Next, we prove the following Theorems:

such that its Bernoulli wavelet expansion is
given by series Then approximation error E To prove the above theorems, first we prove the following Lemma: is a function such that its Bernoulli wavelet expansion is given by the series (4.1) .
Hence, Theorem 4.1 has been established.

Proof of Theorem 4.2 c n,m =
(by Lemma 4.1) Thus, Theorem 4.2 has been established.

Solution of linear Volterra integral equation of second kind
Consider the linear Volterra integral equation of second kind given by eq n (1.1). Using eq n (3.4), wehave

B(t) dt = B T (x)KỸ P B(x)
Then By evaluating the eq n (5.1) in the intervals, we have the following system of linear equations: Solving the system given by eq n (5.2), we can find Y and hence the solution y(x).

Solution of nonlinear Volterra integral equation of second kind
Consider the Non-linear Volterra integral equation of second kind given by eq n (1.2). Using the Bernoulli wavelet approximation, we have Hence, we have following system of 2 k−1 M equations: Using these, Eq n (5.6) becomes This is the nonlinear system of algebraic equations which can be solved by Newton's method. After obtaining the coefficients, we can find the solution of the integral equation (1.2).

Numerical examples
Some linear and nonlinear Volterra integral equations have been solved by Bernoulli wavelet technique discussed in this paper and are compared with their exact solutions. The graphs of these solutions are plotted. It is observed that exact solution and approximate solutions of Volterra integral equations are almost equal.

Example 1
Consider the linear Volterra integral equation y(x) = 2x 3 is the exact solution of the integral equation (6.1). Table 1 shows a numerical comparison between exact solution and approximate solutions obtained by Bernoulli wavelet technique. In addition, Fig. 2 shows a graphical representation of exact and approximate solution of linear Volterra integral equation (6.1).     Table 2 shows a numerical comparison between exact solution and approximate solution obtained by Bernoulli wavelet technique. In addition, Fig. 3 shows a graphical representation of exact and approximate solution of linear Volterra integral equation (6.2).

Example 3
Consider the following nonlinear Volterra integral equation: y(x) = x 2 − x 5 20 + x 0 (x − t) (y(t)) 3 dt (6.3) y(x) = x 2 is its exact solution. Table 3 shows a numerical comparison between exact solution and approximate solutions obtained by Bernoulli wavelet technique. In addition, Fig. 4 shows a graphical representation of exact and approximate solutions of