Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem

Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semi-analytical method that is based on weighted mean-value theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.


Description of the method: weighted mean-value method for integrals (WMVM)
Mean value theorems for both derivatives and integrals are very powerful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. Recently, some applications of the mean-value theorem for solving different classes of Fredholm integral equations from one dimensional to higher dimensional have been introduced (Avazzadeh et al. 2011 ;Heydari et al. 2013;Li and Huang 2016). The results are promising and the method is very simple.
In this article, the weighted mean-value theorem will be used to obtain solutions for a wide class of Fredholm integral equations. As it will be seen in the subsequent sections that under some mild conditions the weighted mean-value theorem can be applied to Fredholm integral equations and significant results are obtained.
Theorem 1 [Weighted mean value theorem for integrals (Apostol 1967)] Let φ, ψ: [a, b] → R be continuous on [a, b]. If ψ never changes sign in [a, b], then there exists a number c ∈ [a, b] such that Results in this paper include application of the weighted mean-value theorem for integrals to the following classes of Fredholm integral equations: • Linear and nonlinear Fredholm integral equations of the second kind ("Solving linear and nonlinear Fredholm integral equations via WMVM" section) • Linear and nonlinear Fredholm integro-differential equations of the second kind ("Solving Fredholm integro-differential equations via WMVM" section) • Linear and nonlinear systems of Fredholm integral equations of the second kind ("Solving linear and nonlinear systems of Fredholm integral equations via WMVM" section) • Linear and nonlinear systems of Fredholm integro-differential equations of the second kind ("Solving systems of Fredholm integro-differential equations via WMVM" In addition, illustrative examples (see "Numerical results" section) are provided to show the ability of the method and to compare with the existing approaches in the literature (see "Comparison and discussions" section). I would like to point out that I do not aim for complete generality, but making simplifying assumptions that produce significant results. In Avazzadeh et al. (2011), the authors obtained significant results under the assumption that an application of the mean-value theorem to Fredholm integral equations produces a number c rather than a function c(x). For some cases, this assumption produces an error in numerical solution (Zhong 2013). Throughout the paper I also assume c(x) = c.

Solving linear and nonlinear Fredholm integral equations via WMVM
In this section, consider the following Fredholm integral equation of the second kind: where is a real number, F, f, and K are continuous functions, and u is the unknown function to be determined. Since the Eq. (1) will stand for both linear and non-linear Fredholm integral equations, the case that F (u(·)) = u(·) is allowed.
In this and all subsequent sections, the assumption on the kernel function is as follows: After applying WMVM to (1), one gets where γ (x) = b a K (x, t) dt and c ∈ [a, b]. Notice that to obtain a solution for (1), one just needs to find the value of u(c) for c whose existence guaranteed by weighted meanvalue theorem. To reach u(c) and c, the following steps are proposed: First substitute c for x in (2) to get Then, substitute (2) into (1) to get Next, plug c into (4) which lead to After that, solve (3) and (5) simultaneously to obtain c and u(c). Finally, substitute c and u(c) into (2) to get a solution.

Solving Fredholm integro-differential equations via WMVM
In this section, consider Fredholm integro-differential equation given by where , F, f and K are defined as before, u (n) (x) stands for the nth derivative, and a k are constants that represent the initial conditions.
In operator notation, Eq. (6) can be written as where the differential operator is given by L = d n dx n · The inverse operator L −1 is an n-fold integral operator given by Applying WMVM to (6), one can obtain An application of the integral operator L −1 to both sides of Eq. (9) along with initial conditions yields Now, replace x with c in (10) to get In addition, substitute Eqs. (9) and (10) into (6) to get Then, replace x with c in (12) to get Finally, considering Eqs. (13) and (11) together, a system of equations with c and u(c) appearing as unknowns are obtained. Solution of this system will give a numerical approximation of desired function u(x).

Solving linear and nonlinear systems of Fredholm integral equations via WMVM
In this section, consider systems of Fredholm integral equations given by It is assumed that there is n × n system of equations. One particular equation can be represented by If applying WMVM to (15), one gets . For simplicity and notational convenience , without loss of generality, it is assumed that there are two unknowns and two functions, i.e., n = 2.
Thus, one has After applying WMVM to (17) (20), and c 2 and c 4 in (21) one can get 4 more equations. Combining these equations with (19), a system of algebraic equations will be obtained. By solving this algebraic system of equations, the desired solution for the system of integral equations will be reached.

Solving systems of Fredholm integro-differential equations via WMVM
In this sections, systems of Fredholm integro-differential equations of the second kind will be studied. Consider After applying WMVM to (22), one gets where c m ∈ (a, b) and An application of the integral operator L −1 introduced in (8) to both sides of Eq. (23) along with initial conditions yields Substituting c 1 and c 3 into first equation in (24) and c 2 and c 4 into second equation in (24), 4 equations will be obtained. Then, by substituting (23) and (24) into (22), 2 new equations will be obtained. Replacing x with c 1 and c 3 in the first equation and c 2 and c 4 in the second equation, there will be 4 more equations. Solving this nonlinear system of equations will give the desired solution.

Numerical results
In this section, numerical results are presented for various types of Fredholm integral equations mentioned in the previous sections. The results show the validity and efficiency of the method. It is important to note that all numerical computations are performed using Matlab software. For solving a non-linear system of equations, the Matlab built-in functions use the Newton's method with an initial guess or some modified versions of it. Since these methods are, in general, local, the initial guess plays a decisive role in obtaining solutions.

Example 1 (Linear Fredholm integral equation) Consider the following linear Fredholm integral equation of the first kind (Wazwaz 2011):
The exact solution for the equation is that u(x) = e x .
Applying the presented method, the following system of equations are obtained:

Solving this system of nonlinear equations results in
The approximate solution can be evaluated from which leads to the exact solution. The graph of the equations in (25) is given in Fig. 1.

Then plug (33) into (32) to get
Now replace x with c 1 and c 3 in the first equation in (35) and c 2 and c 4 in the second equation in (35) so that there are 4 more equations. Combining (34) with these equations, one finally gets a nonlinear system of 8 equations with 8 unknowns. Solving this system and the result is as follows: Substitute these values into (33), the exact solutions are obtained, namely, Example 6 (System of Fredholm integro-differential equation) Consider the following system of Fredholm integro-differential equation: The exact solution for the equation is that u 1 (x) = x and u 2 (x) = x 2 . Applying the presented method, the following system of equations are obtained: where c 1 , c 2 , c 3 , and c 4 ∈ [0, 1]. An application of the integral operator L −1 introduced in (8) For a detailed treatment of application of the ADM to integral equations the reader is referred to Wazwaz (2011).

Example 7 (Nonlinear Fredholm integral equation) Consider the following nonlinear
Fredholm integral equation of the second kind: This was the second example in the previous section. Applying the ADM, one gets where A n are the Adomian polynomials given in (43). The ADM admits the following recursion relation:

This yields
Combining these components of the solutions to get It is important to note here that an application of the ADM produced one approximate solution. On the other hand, applying the WMWM (see Example 2) 3 exact solutions were obtained. As the final example, consider an equation for which the method introduced in Avazzadeh et al. (2011) does not provide a number c ∈ [0, 1] when solving the nonlinear system of equations obtained after applying the method. This is shown by a geometric reasoning. it is also shown that applying WMVM will produce the exact solution.
The exact solution for the equation is that u(x) = 1.
Applying the method introduced in Avazzadeh et al. (2011), the following system of equations are obtained: The graph of the equations in (44) is given in Fig. 3.
From the Fig. 3, it is clear that one cannot find a number c between 0 and 1 satisfying both equations given in (44). On the other hand, applying the presented method, one gets Substitute c for x to get Using (46), the second equation [see (4)] directly gives the exact solution. That is, u(x) = 1.

Conclusion
In this article, an effective method based on weighted mean-value theorem for solving different types of Fredholm integral equations of the second kind, from linear to nonlinear equations and integro-differential to the systems of equations involving them, is (45) (46) u(c) = 1.