Solving nonlinear Fredholm integro-di erential equations via modi cations of some numerical methods

This paper presents the modi cations of the variational iteration method (MVIM), the Laplace Adomian decomposition method (MLADM), and the homotopy perturbation method (MHPM) for solving the nonlinear Fredholm integro-di erential equation of the second kind. In these techniques, a series is established, the summation of which gives the solution of the discussed equation. The conditions ensuring convergence of this series are presented. Some examples to illustrate the investigated methods are presented as well, and the results reveal that the proposed methods are very e ective. Moreover, we present the comparison between our proposed methods with the exact solution and some traditional methods during numerical examples. The results show that (MHPM) and (MLADM) lead to an exact solution, whereas (MVIM) leads to limited solutions. Finally, the uniqueness of solutions and the convergence of the proposed methods are also proved.


Introduction
Mathematical modeling of many physical systems leads to integrodierential equations in various elds of engineering and physics. There are some methods to obtain approximate solutions to this kind of equations.
From these methods are the homotopy perturbation method, Laplace Adomian decomposition method, and variational iteration method, which have undergone many modications in the recent period. Fredholm integro-dierential equation has been solved by some other methods, such as weighted mean value theorem [9]. The approximate solution for solving the nonlinear Fredholm integro-dierential equation of the second kind in the complex plane by using the properties of rationalized Haar wavelet has been obtained in [14]. A two-dimensional nonlinear Volterra-Fredholm integro-dierential equation by using some iterative methods is presented [13]. In [16], the author modied the existing homotopy perturbation method to solve the high-order integro-dierential equations through canonical polynomials basis function. MADM is applied to nd the approximate solution for Fredholm integral equation and its system in [4,5]. Solving linear and nonlinear Volterra integral equations and their system by using some numerical methods are discussed by [6,8]. Analytical methods with Laplace transform are implemented in [7] to nd an approximate solution for Volterra integral equation with a convolution kernel. Non-standard nite dierence methods to nd the numerical solution of linear Fredholm integro-dierential equation have been introduced by Pandey [17]. Al-Mdallal [2] presented the monotone iterative sequences for solving nonlinear integro-dierential equations of the second-order. Whereas Atabakan et al. [12] used the spectral homotopy analysis method to solve nonlinear Fredholm integro-dierential equations. Also, the same authors in [11] have applied the composite Chebyshev nite dierence method to nd the solution of Fredholm integro-dierential equations. In this regard, Aloko et al. [3] discussed the new variational iteration method to nd the numerical solutions of the second kind of nonlinear Fredholm integro-dierential equations. Recently, Safavi and Khajehnasiri [18] have proposed two-dimensional block-pulse functions for solving nonlinear mixed Volterra-Fredholm integrodierential.
On the other hand, the Fredholm integral equation is solved in [1] using the homotopy analysis method. Legendre multi-wavelets collocation method for the numerical solution of linear and nonlinear integral equations was discussed lately by Asif et al. [10]. Variational iteration method and homotopy perturbation method to nd the approximate solution of Volterra integral equations were achieved by Mirzaei [15]. Syam et al. [19] employed an ecient numerical algorithm for solving fractional higher-order nonlinear integrodierential equations.
Until recently, the applications of the HPM, LADM, and VIM have been developed by scientists and engineers in nonlinear problems, because these methods are the most convenient and powerful. Motivated by the above works, in this paper, we consider a nonlinear Fredholm integro-dierential of the form where f (x) and ξ n (x) are given real-valued functions and analytic functions, k(x, t) is the kernel of the equation, y (n) (x) present the n-th derivative of y(x) and G(y(x)) is a nonlinear function of y(x).
The main motive for this research is to develop the applications of the modications of HPM, LADM, and VIM in nonlinear problems because these methods are the most convenient for solving such types of equations, especially the nonlinear Fredholm integro-dierential equations. Consequently, we apply the modications of HPM, LADM, and VIM for solving some equations of type (1). Numerical examples are given to demonstrate the exact and approximate solutions. Also, we use the absolute error table and comparisons with current approaches to show the precision and eectiveness of these methods. Besides, we prove the uniqueness of the solution and the convergence of the proposed methods.
The remainder of the paper is displayed as follows. In Section 2, we give the formulation of MHPM, MLADM, and MVIM. Section 3 proves the uniqueness of the solution of Eq. (1) and the convergence of the proposed methods. Numerical examples to solve the nonlinear Fredholm integro-dierential equations of the second kind are provided in Section 4. The comparison between the analytical and approximate solution obtained by the proposed methods and the other methods is discussed in Section 5. In the last section, we close this work with a conclusion.

Formulation of methods
Some eective methods have centered on the development of more advanced and ecient methods for solving nonlinear Fredholm integro-dierential equations, such as the modied variational iteration method (MVIM), the modied homotopy perturbation method (MHPM), and the modied Laplace Adomian decomposition method (MLADM).

Modied variational iteration method
To illustrate the fundamental principles of MVIM, we consider the dierential equation as follows: where L, N are linear, nonlinear terms respectively and g(x) is an inhomogeneous term. The correction function for Eq. (2) using variational iteration method are presented as the form: where λ is a general Lagrange multiplier that can be optimally dened by variational theory, that is by part integration and by the use of restricted variation. Putting Ly n (ξ) = y (ξ) we get Generalized integration of the parts is 1 0 λ(ξ) y n n (ξ) dξ = λ(ξ)y n−1 n (ξ) − λ (ξ)(y n−2 n (ξ) + λ y n−3 n (ξ)) − · · · − (−1) we can note that there may be a constant or a function in this method, and δ is the restricted value behaves as a constant,ỹ n (ξ) is considered as δỹ n (ξ) = 0 restricted variation and so, the extreme condition demands that the stationary conditions should be met: Thus, the general multiplier of Lagrange easily can be identied as: Successive approximations of y n (x), n ≥ 0, of solution y(x) can be easily obtained by using selective function y 0 (x). We will consider a particular case of Eq. (1) of the form: with the initial condition y b = c b , b = 0, 1, . . . , (n − 1), where k, m are integers with k ≥ m ≥ n and y b are real constant. Now we consider Eq. (2), where g(x) is a known analytical function and the nonlinear operator N (y n ) can be decomposed as where y j are the polynomials of x and the relationship of recurrence is determined as The nonlinear term in Eq. (3) can be written as Nỹ(ξ) = N y n (ξ) and the n th term approximate solution in Eq. (8) is Appling L −1 to the recurrence relation for the nding of the components, the (n + 1)th approximation of the exact solution for the unknown function y(x) is determined as y n+1 (x) = L −1 (N (y 0 + y 1 + · · · + y n )) − L −1 (N (y 0 + y 1 + · · · + y n−1 )), we construct the solution as The modication for Eq. (3) has formulated as The zeroth approximation y can be any eclectic function. However, the initial values have been preferably using for the selective zeroth approximation y 0 . Consequently, the solution is given by

Modied homotopy perturbation method
The method of homotopy perturbation rst proposed by Ji-Huan He in (1997) [15,16]. Consider the general form of nonlinear Fredholm integro-dierential equations with initial conditions where M (y) and N (y) are linear and nonlinear functions of y, respectively. To explain the basic concept of this approach, we consider the following nonlinear dierential equation: with boundary conditions where A, B are general dierential operator and boundary operator respectively, Γ is the boundary of the domain Ω, and f (r) is a known analytic function. Dividing the operator A into two parts: M and N . Therefore, Eq. (15) can be rewritten as follows: Using the homotopy technique, we construct a homotopy v(z, p) : or where p is an embedding parameter, and y 0 is an initial approximation of Eq. (15) which satises the boundary conditions. From Eqs. (18), (19), we have The changing in the process of p from zero to unity is just that of v(z, p) from y 0 (z) to y(z). In topology this is called deformation and M (v) − M (y 0 ), and A(v) − f (z) are called homotopic. Now, we assume that the solution of Eqs. (18), (19) can be expressed as The approximate solution of Eq. (15) can be obtained by setting p = 1.
To apply MHPM for solving the Fredholm integro-dierential equations, we dene a convex homotopy by where M (y) is a functional operator with known solution v 0 , which has obtained as which is dependent on the order of dierentiation. In most cases, we may choose a convex homotopy by HPM reduces the disadvantages of the conventional perturbation method while retaining all its benets. The series in Eq. (22) is convergent in most cases, and the convergence rate depends on A(y)−f (z) = 0. Note that the components v n for n = 1, 2, . . . must be determined in the HPM in order to achieve an approximate solution. Particularly for n ≥ 3, large and sometimes complicated computations have needed. To avoid this problem, the MHPM is implemented in which v 0 is calculated in such a way that v 1 = 0 for n ≥ 1. As a result, the number of calculations decreases relative to those in the HPM.

Modied Laplace Adomian decomposition method
Consider the general form of nonlinear Fredholm integro-dierential equations in Eq. (14).
The general form of second-order nonlinear partial dierential equations with initial conditions in the form where L = ∂ n ∂x n is the second order dierential operator, M, N represent the remaining linear operator and the general non-linear dierential operator respectively and z(x, t) is the source term. Using Laplace transform on both sides of Eq. (23), we have applying Laplace Transform's dierentiation property, we get: In the Laplace decomposition method, the next step is to represent the solution as an innite series given below: decomposing the nonlinear operator as where A n is the Adomian polynomial given below: , ∀n ∈ N, using (24), (25) and (26), we get we have to compare both sides of (27) as , n ≥ 1, where k 1 (x, s) and k 2 (x, s) are Laplace transform of k 1 (x, t) and k 2 (x, t) respectively.
The application of the inverse Laplace transformation to Eq. (28) provides our requisite recursive relation as follows: The solution by using the modied Adomian decomposition method depends highly on the choice of k 0 (x, t) and k 1 (x, t), where k 0 (x, t) and k 1 (x, t) represent the terms resulting from the source term and the initial conditions prescribed.

MAIN RESULTS
In this section, we will prove the uniqueness of solution for Eq. (1) and the convergence of the proposed methods.
Theorem 3.2. If problem (1) has a unique solution, then the solution y n (x) obtained from the recursive relation (12) using VIM converges when Proof. We have the iteration formula: where L −1 is the multiple integration operator given as From the above equations, we get If we set, ξ k (x) = 1, and z n+1 (x) = y n+1 (x) − y n , z n (x) = y n (x) − y n , since z n (a) = 0, then Therefore, Proof. We have to prove a special case from theorem 3.3 when G(y(t)) = y 2 (t), rts we will start to verify the convergence of L(y) − L(z), y − z ≥ k y − z 2 , k > 0, ∀y, z ∈ H, for the operator L(y) : i.e., ∃k ≥ 0, ∀y, z ∈ H, we have Then we get According the Schwartz inequality, we get Now we use the mean value theorem, then we have where y ≤ η ≤ z and y ≤ M, z ≤ M. Therefore: where C(M ) = M 3 and therefore L(y) − L(z), y − z ≥ C(M ) y − z z is hold. The proof is complete.

Numerical Examples
This section contains numerical examples to illustrate the accuracy and eectiveness properties of the methods to solve the nonlinear Fredholm integro-dierential equation of the second kind. The absolute errors used is dened as |y(x) − y n (x)|, where y(x), y n (x) are the exact and approximate solutions respectively. The numerical solutions of our proposed methods will compare with the numerical solutions of other known methods.
Example 4.1. Consider the following nonlinear Fredholm integro-dierential equation of the second kind with the initial condition y(0) = 0 and the exact solution y(x) = xe x .
• Using MVIM The correction functional for Eq. (30) is constructed as Making the functional stationary and noting that,ỹ n is a restriction variation, δỹ n = 0. To nd the optimal λ(ξ) and calculate variation with respect to y n , we have the stationary conditions by applying Eq. (4): The Lagrange multiplier can be identied as λ = −1 • Using MHPM By letting g 1 (x) = xe x + e x and g 2 (x) = −x in order to obtain Hence, which is the exact solution.
• Using MLADM Applying the Laplace transform and by using the initial condition, we have Using the inverse Laplace transform we get Decomposing the solution as an innite sum given below Substituting (31) on (32) we get in which A n = n j=0 y j . The recursive relation is given below xy n (t))dt = 0, n ≥ 0, in which of A n = n j=0 y j , where for every n ≥ 1, A n = 0. Hence, the exact solution is with the initial condition y(0) = 0 and exact solution y(x) = x.
The Lagrange multiplier can be identied as λ = −1 • Using MHPM By letting g 1 (x) = 1 and g 2 (x) = 1 4 − 1 3 x 2 in order to obtain Hence, which is the exact solution.
• Using MLADM Applying the Laplace transform and by using the initial condition, we have Using the inverse Laplace transform we get Decomposing the solution as an innite sum given below Substituting (34) on (35) we get in which A n = n j=0 y 2 j . The recursive relation is given below in which of A n = n j=0 y 2 j , where for every n ≥ 1, A n = 0. Hence, the exact solution is y(x) = y 0 (x) = x.
• Using MVIM The correction functional for Eq. (36) is constructed as Making the functional stationary and noting that,ỹ n is a restriction variation, δỹ n = 0. To nd the optimal λ(ξ) and calculate variation with respect to y n , we have the stationary conditions: The Lagrange multiplier can be identied as λ = − Consequently, we have the following approximations: y 1 (x) = cos x − 0.04166666667x 4 + 0.008971723581x 7 y 2 (x) = cos x − 0.04166666667x 4 + 0.007137527979x 7 y 3 (x) = cos x − 0.04166666667x 4 + 0.007550623256x 7 • Using MHPM By letting g 1 (x) = sin x and g 2 (x) = −x in order to obtain which is the exact solution.
• Using MLADM Applying the Laplace transform and by using the initial condition, we have Using the inverse Laplace transform we get Decomposing the solution as an innite sum given below Substituting (37) on (38) we get in which A n = n j=0 y j . The recursive relation is given below: xty 0 (t))dt = 0, y n+1 (x) = −L −1 1 s 3 L π 2 0 xty n (t))dt = 0, n ≥ 0, in which of A n = n j=0 y j , where for every n ≥ 1, A n = 0. Hence, the exact solution is y(x) = y 0 (x) = cos x.   Tables 1, 2, 3, and 4 show the comparison between the analytical and approximate solution obtained by the proposed methods and the other methods viz VIM, ADM, and Bernstein Polynomials Method (BPM). The simplicity and accuracy of the proposed methods illustrate by computing the absolute error. The accuracy of the result can improve by introducing more terms of the approximate solutions. There is good agreement between the exact and approximate solution obtained by MVIM. MHPM and MLADM converge more easily than MVIM, and both techniques give us the same exact solutions as the examples. Our results show that the proposed methods are ecient and powerful techniques that provide higher accuracy and closed-form solution approximations. And the approximate solution error is obtained by considering only the partial sum of the series. 6. Conclusion In this paper, the modications of the MVIM, MHPM, and MLADM have successfully been applied to nd the solution of nonlinear Fredholm integro-dierential equations of the second kind. The methods can be concluded that is very powerful and ecient techniques in nding exact solutions or approximate solutions for wide classes of problems. The numerical results show that our proposed methods provide a sequence of functions that converges to the exact solution of the problem and reduce the computational diculty for solving nonlinear Fredholm integrodierential equations of the second kind when compared to other traditional methods. The eectiveness of our methods examine in some examples and the results show that the techniques are easier than many other numerical techniques. These modications are promising and readily implemented, which makes it a more ecient tool and more practical for solving linear and nonlinear integro-dierential equations as well as give us an analytical solution for these type of equations.