A Computational Method for Nonlinear Fredholm Integro-Differential Equations Using Haar Wavelet Collocation Points

Haar wavelet collocation points method is developed to the computational solution for nonlinear Fredholm integral and integro-differential equations on interval [0, tf ] using Leibnitz-Haar wavelet collocation points method. Essential principle is transmutation of the integral equation to equivalent higher order differential equation together with initial conditions. The transmutation is carried out using the Leibniz law. Haar wavelet collocation points and its operational matrix is employed to transform the higher order differential equation to a set of algebraic equations, then resolving these equations usage MATLAB program to calculate the demanded Haar coefficients. The computational results of the proposed approach is presented in four problems and make a simulation against the accurate solution. In addition, Error analysis is exhibited the proficiency of the proposed technique and when Haar wavelet resolutions increases the results are close to the accurate solutions.


Introduction
Recently, Integral and integro equations find out its applications in many subjects of mathematics, science and engineering. in much of the situations, it is hard to resolve these problems, in particular analytically. Especially, nonlinear integral and integro equations of Fredholm type. For this cause, numerous computational techniques have been devoted for obtaining solutions of the most interesting topics is the nonlinearity [1]. Babolian et al. [2] solved nonlinear Volterra-Fredholm integro-differential equations thru direct technique utilizing triangular functions. Rashed in [3] introduced numerical solution for differential, integral and integro-differential equations using Lagrange interpolation. In addition, Ordokhani et al. [4] employed Bernstein polynomials to find the numerical solution for the nonlinear Fredholm integro-differential equations. Also, Haleema et al. [5] used new technique to resolve nonlinear high-order Volterra and Fredholm differential equations utilizing Boubaker ICMAICT 2020 Journal of Physics: Conference Series 1804 (2021) 012032 IOP Publishing doi: 10.1088/1742-6596/1804/1/012032 2 polynomial technique. Adomian decomposition approach have been used to solve Fredholm integrodifferential equation as in [6].
In of late years, Haar wavelets is a comparatively modern and an emerging instrument into applied mathematical research domain. It has been employed into a broad scope of engineering areas; specially, signal test for wave shape illustration and segmentations, time-frequency analysis and fast algorithms for convenient application. Wavelets allow the correct representation of a set of functions and factors. Furthermore, wavelets set a connection with rapid numerical algorithms [7] [8]. For these advantages, many researchers have been successfully applying Haar wavelet collocation points technique (HWCP) to various linear and nonlinear integral and integro-differential equations. For instance, Lepik and Enn [9] used Haar wavelet method for solution of nonlinear Fredholm integral equations. Babolian and Shahsavaran [10] proposed numerical solution of nonlinear Fredholm integral equations of the second kind utilizing Haar wavelets. Aziz and Imran [11] presented unprecedented algorithm for the numerical solution of nonlinear Fredholm and Volterra integral equations utilizing Haar wavelets. Mishra et al. [12] employed HWCP technique to obtain solutions of problems involving integral, differential and integro-differential equations. Shiralashetti  and a new operational matrix on new collection points instead of the one   1 , 0 and employing the Leibnitz and Haar wavelet collocation technique in which the nonlinear Fredholm integral and integro-differential equation that is converted into equivalent higher order differential equation with initial conditions, then by converting this equation into a set of linear algebraic equations using the Haar wavelet collocation points and its operational matrix, it is easy to resolve using the MATLAB program.
The manuscript is ordered as follow up: In Section two, the problem statement and its variables are given. Section three is devoted to the Haar wavelets functions and its operational matrix onto the collocation points. In section four, the computational technique is debated. In section five, we put in the picture our computational results and displayed the precision of the suggested diagram. Conclusion is argued in section six.

PROBLEM STATEMENT
Consider the nonlinear Fredholm integral and integro-differential equations is [1]: tf ]) can be expressed as in finite sum of Haar wavelets as: Which has the form in matrix as: is the Haar function vector and  is the coefficients vector which can be obtained from So, the Haar functions vector can be characterized into the matrix form as m HI next, the components are arranged as (10) ).
As instance, the fourth-order Haar wavelet matrix 4 HI expressed into matrix form on the collocation points on interval of [0 , 1) as: indeed , the coefficients vector T b m can be easily got as: . In this research, our goale is applying Haar wavelets collocation points method on interval [0, tf ] . So we expanded the Haar wavelet functions integration of ) (t h i  the same as in equation (15) : Q is an m m  the operational matrix of integration, which is recursively acquired as [14][15]: Where the form into the interval of [0,1] was given firstly by [16]

Leibnitz Rule and Haar Wavelet Collocation Points Technique
In order to obtain solution for nonlinear Fredholm integral and integro-differential equation on interval . We introduce a Leibnitz law and Haar wavelets collocation points technique (LHWCP) as: The transformation of the integral equations (17) into an equivalent differential equation with initial conditions. The transmutation is carried out using the Leibniz law [1] for differentiation of integrals.
Let us assume Then differentiation of the integral in (18) , where 1 and 0 are corrected constants, then the Leibnitz law (18) reduces to A numerical computation procedure is as follows: Step 1: Differentiating equation (17) using Leibnitz rule as equation (19) we get, Step 2: Employing Haar wavelet collocation points method . Let us suppose that, Step 3: By integrating (23) two times and utilizing initial conditions (24) and (25), is an m row vector.
Step 4: Substituting (23) into equation (22), which reduces to the linear algebraic set of m equations with m unknowns Haar wavelet coefficients T  row vector. Then by using MATLAB program [17] to find the Haar coefficients. Finally, substituting Haar coefficients row vector in equation (25) to find the demand approximate solutions of equation (22).

Illustrative Problems
In order to show the efficiency and accuracy of the numerical results of the proposed Leibnitz-Haar wavelet collocation points method on new interval   tf , 0 , we illustrative four numerical problems from the sources to prove the ability of the HWCP technique and error function.
where the error function can be expressed as: and Exact G are the approximate and exact solution respectively.
and the correct solution is as follows: According to the LHWCP technique procedure that we set on section 4. Successively differentiating equation (27) two times and employing Leibnitz law reduces to the second order differential equation as following: By Integrating equation (30) two times we get: , Next, substituting equation (30) in equation (29), we get the algebraic system of m equations with m unknown Haar wavelet coefficients.
 is the coefficient vector of Haar wavelet function which can be calculated from equation (13).
Solving equation (33) Figure 1 and Table 1 respectively. In addition, the Figure  1 shows that the m = 32 it is enough to matches with the exact solution.   Next, consider nonlinear Fredholm Integro-differential equation [18], the correct solution is as follows: Similarly to Problem 1, the problem 2 reduces to third order differential equation as following: By Integrating equation (37) three times we get:    1.2500 1.2500 0.0000 3.7500 3.7500 3.7500 0.0000 6.2500 6.2500 6.2500 0.0000 8.7500 8.7500 8.7500 0.0000 Problem 3: Consider the nonlinear Fredholm-Hammerstein Integro-differential equation [19], . The correct solution is as follows: Similarly to Problem 1, the equation (42) and employing Leibnitz law reduces to the second order differential equation as following : , Substituting equation (44) in equation (43), we get the algebraic set of m equations with m unknown Haar wavelet coefficients.
 is the coefficient vector of Haar wavelet function which can be calculated from equation (13).
Solving equation (47) by employing MATLAB program to find Haar wavelet coefficients T  .
Substituting these coefficients in equation (46) to obtain the solution according to the proposed LHWCP technique. The results have been illustrated in Figure 3 and Table 3. Numerical results for m = 4 , 8 , 16 and m = 16 are displayed in Figure 3 and Table 3 respectively. In addition, the Figure 3 shows that the m = 16 it is enough to matches with the exact solution.  Consider the nonlinear Fredholm Integro-differential equation [10], . The correct solution is as follows . Similarly to Problem 1, the equation (48) and using Leibnitz rule reduces to the first order differential equation as following : , Substituting equations (51) and (52) in equation (50), we get the algebraic system of n equations with m unknown Haar wavelet coefficients.
Solving equation (53) by employing MATLAB program to find Haar wavelet coefficients T  .
Substituting these coefficients in equation (52) to obtain the solution according to the proposed LHWCP technique. The results have been illustrated in Figure 4 and Table 4. Numerical results for m = 4 , 8 , 16 32 and m = 32 are displayed in Figure 4 and Table 4 respectively. In addition, the Figure 4 shows that the m = 32 it is enough to matches with the exact solution.

Conclusion
In the present work, a new interval used for resolving nonlinear Fredholm integral and integrodifferential equations on interval   tf , 0 using LHWCP technique. First of all, the nonlinear integral and integro-differential equations are transformed to higher order differential equations with initial conditions by using Leibnitz rule. Then we transforms the higher order differential equations to a set of algebraic equation using the Haar wavelet collocation points and its operational matrix. Then resolving these equations utilizing MATLAB program to find the demanded Haar coefficients. LHWCP technique is mathematically simple and easy to utilize, then the required less computational complexity and provide more quantitatively reliable results. From the numerical problems, we could conclude that the proposed method almost coincides with the exact solutions of the problems. Illustrative problems clearly depict the validity and applicability of the technique and error analysis shows that, when the level of Haar wavelet resolution increases, gives the better results.