Corrections on A numerical method for solving nonlinear Volterra--Fredholm integral equations
Ngo Thanh Binh, Khuat Van Ninh
TL;DR
This paper provides corrections to a previous numerical method for solving nonlinear Volterra-Fredholm integral equations, improving discretization accuracy without altering the original main results.
Contribution
The authors refine the discretization approach for nonlinear Volterra-Fredholm integral equations, enhancing accuracy while maintaining the original method's validity.
Findings
Improved discretization leads to more accurate solutions.
Corrections do not affect the original main results.
Enhanced numerical stability in the method.
Abstract
Some corrections are made in our article, which was published in Appl. Anal. Optim. Vol. 3 (2019), No. 1, 103--127. These corrections are intended to transform the equation \eqref{eq:1.1} \begin{equation}\label{eq:1.1} x(t) + \int\limits_a^t {K_1(t,s,x(s)) ds} + \int\limits_a^b {K_2(t,s,x(s)) ds} = g(t),\;\,a \le t \le b \tag{1.1} \end{equation} into a discretized form in a tighter and more accurate way without affecting the main results of the article.
| Nodes | Exact | Approximate | Absolute | Nodes | Exact | Approximate | Absolute |
|---|---|---|---|---|---|---|---|
| solutions | solutions | error | solutions | solutions | error | ||
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TopicsFractional Differential Equations Solutions · Iterative Methods for Nonlinear Equations · Numerical methods in engineering
A numerical method for solving nonlinear Volterra–Fredholm integral equations
Ngo Thanh Binh
Faculty of Fundamental Science, Namdinh University of Technology Education, Nam Dinh, Vietnam
and
Khuat Van Ninh
Department of Mathematics, Hanoi Pedagogical University 2, Phuc Yen, Vietnam
Abstract.
In this paper, we introduce a numerical method for solving nonlinear Volterra–Fredholm integral equations. Our method consists of two steps. First, we define a discretized form of the integral equation by quadrature methods. We then propose an iterative method, which is based on a hybrid of the method of contractive mapping and parameter continuation method, to solve the perturbed system of nonlinear equations obtained from discretization of the considered problem. Finally, an example is given to demonstrate the validity and applicability of our method.
Key words and phrases:
Volterra–Fredholm integral equations, quadrature method, parameter continuation method, discretization, approximation.
2010 Mathematics Subject Classification:
45L05, 45G10, 47J25.
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.92.2014.51.
1. Introduction
Many problems which arise in mathematics, physics, biology, etc., lead to integral equations. Volterra–Fredholm integral equations play an important role in the theory of integral equations. Since Volterra–Fredholm integral equations are usually difficult to get their exact solution, therefore, many authors have worked on analytical methods and numerical methods for approximate solution of this kind of equations. For example, Adomian decomposition method was used to solve Volterra–Fredholm integral equations in [23]. H. Brunner [4] and P. J. Kauthen [8] have employed the Collocation method to solve mixed Volterra–Fredholm integral equations. The numerical solutions of the nonlinear Volterra–Fredholm integral equations by using Homotopy perturbation method was introduced in [7]. Y. Mahmoudi [11] and S. Yalçinbaş [25] developed the Taylor polynomial solutions for the nonlinear Volterra–Fredholm integral equations. In [16], Y. Ordokhani applied the rationalized Haar functions to solve nonlinear Volterra–Fredholm–Hammerstein integral equations. Recently, M. Zarebnia [26] used the sinc functions to solve the nonlinear Volterra–Fredholm integral equations. F. Mirzaee and A. A. Hoseini [13] obtained an approximate solution for the nonlinear Volterra–Fredholm integral equations by the hybrid of block–pulse function and Taylor series (HBT). An iterative scheme for extracting approximate solutions of the Volterra–Fredholm integral equations has been presented by A. H. Borzabadi and M. Heidari [3]. J. Xie et al. [24] developed the numerical technique based on block–pulse functions (BPFs) to approximate the solutions of nonlinear Volterra–Fredholm–Hammerstein integral equations in two–dimensional spaces.
In this paper, we intend to present a numerical method for approximating the solution of nonlinear Volterra–Fredholm integral equation as follows
[TABLE]
where is an unknown function that will be determined, are known functions and are known constants. At first, we use one of frequently used quadrature methods to reduce the equation (1.1) into a perturbed system of nonlinear equations. For further information on quadrature methods in this respect, see [5, 9, 17]. Next, we propose an iterative method to solve the obtained perturbed system of nonlinear equations. The method is based on a hybrid of the method of contractive mapping and parameter continuation method. Parameter continuation method was suggested and developed by S. N. Bernstein [2] and J. Schauder [10]. Later on, V. A. Trenoghin [18, 19, 20, 21] has developed a generalized variants of the parameter continuation method and used to prove the invertibility of nonlinear operators, which map a metric space or a weak metric space into a Banach space. Y. L. Gaponenko [6] proposed and justifed the parameter continuation method for solving operator equations of the second kind with a Lipschitz - continuous and monotone operator, which operates in an arbitrary Banach space. K. V. Ninh [14, 15] has studied parameter continuation method for solving the operator equations of the second kind with a sum of two operators. In [22], V. G. Vetekha presented the application of parameter continuation method to solving the boundary value problem for the ordinary differential equations of second order. Parameter continuation method has some advantages that encourage us to use it. Firstly, the properties of contractions such as iteration, error estimates are used to find approximate solutions and estimate the errors of approximate solutions. Furthermore, this method is very simple to apply and to make an algorithm.
The paper is organized as follows. In Section 2, the parameter continuation method for solving operator equations of the second kind is briefly presented. In this section, we recall some definitions and results that will be useful in the sequel. In Section 3, we transform the equation (1.1) into a perturbed system of nonlinear equations. Then we discuss the existence and uniqueness of the solution of the obtained perturbed system of nonlinear equations and prove that its solution converges to the exact solution of the problem. We also study approximate solutions of the perturbed system of nonlinear equations and their error estimates. An illustrative example is given in Section 4 to illustrate the efficiency of the introduced method. Finally, Section 5 draws some conclusions from the paper.
2. Parameter continuation method for solving operator equations of the second kind
In this section, we recall some definitions and results which we will use in the sequel. For details, we refer to [6].
Let be a Banach space and be a mapping, which operates in the space . Consider the operator equation of the second kind
[TABLE]
Definition 2.1** ([6]).**
The mapping , which operates in the Banach space is called monotone if for any elements and any the following inequality holds
[TABLE]
Remark 2.2** ([6]).**
If is Hilbert space then the condition of monotonicity (2.2) is equivalent to the classical condition
[TABLE]
where is an inner product in the Hilbert space .
Lemma 2.3** ([6]).**
Assume that is a monotone mapping which operates in the Banach space . Then for any elements and any positive numbers , the following inequality holds
[TABLE]
The basic idea of the parameter continuation method for solving operator equations of the second kind (2.1) is as follows. Consider a one-parametric family of equations
[TABLE]
which when gives the trivial equation and when gives the initial equation (2.1). Dividing into equal parts with is a natural number such that , where is Lipschitz coefficient of the operator . After changes of variables
[TABLE]
we construct intermediate equations with contractive operators in new variables. By virtue of the monotonicity and Lipschitz continuity of the operator , contraction coefficients of these contractive operators equal . By shifting the parameter step by step from [math] to we can verify that the equation (2.1) has a unique solution.
Theorem 2.4** ([6]).**
Suppose that the mapping , which operates in the Banach space is Lipschitz - continuous and monotone. Then the equation (2.1) has a unique solution for any element .
To find approximate solutions of the equation (2.1), Y. L. Gaponenko has constructed the following iteration process
[TABLE]
The symbolic notation (2.4) should be understood as the following iteration processes, which consist of iteration processes
[TABLE]
For simplicity, assume that and the number of steps in each iteration scheme of the iteration process (2.4) is the same and equals . Denote as the approximate solutions of the equation (2.1), which is constructed by the iteration process (2.4). In this case, Y. L. Gaponenko received the error estimations of approximate solutions of the equation (2.1), which are presented in the following theorem.
Theorem 2.5** ([6]).**
Assume that the conditions of Theorem 2.4 are satisfied. Then the sequence of approximate solutions constructed by iteration process (2.4) converges to the exact solution of the equation (2.1). Moreover, the following estimates hold
[TABLE]
where is Lipschitz coefficient of the operator is the smallest natural number such that .
3. Main results
Supposing that the discussed integral equation (1.1) has solution. We will consider the equation (1.1) under the following assumptions:
- (i)
; 2. (ii)
are differentiable continuous functions up to order on , where ; 3. (iii)
satisfies a Lipschitz condition of the type
[TABLE]
for all and for all reals , where and ; 4. (iv)
satisfies a Lipschitz condition of the type
[TABLE]
for all and for all reals , where ; 5. (v)
satisfies the condition
[TABLE]
for all with .
At the beginning, we transform the equation (1.1) into a discretized form. Let be an equidistant partition of where is the discretization parameter of the partition. Now, if is an analytical solution of (1.1), then for the partition on , we have
[TABLE]
In (3.1), the integral term can be estimated by a numerical method of integration, e.g. Newton–Cotes methods. Therefore, by taking equidistant partition , as above with and also the known weights for interval and for interval , equality (3.1) can be written as
[TABLE]
where , and depend upon the used method of Newton–Cotes for estimating the integrals in (3.1). From (3.2), we have
[TABLE]
For partition , we consider a perturbed system of nonlinear equations obtained by neglecting the truncation error of (3.1) as follows
[TABLE]
The perturbed system of nonlinear equations (3.4) can be rewritten as
[TABLE]
where and with
[TABLE]
For partition and the known weights , we define an inner product in by
[TABLE]
This inner product induces the norm
[TABLE]
The following proposition gives us the property of the mapping .
Proposition 3.1**.**
Let the assumption (iii) be satisfied. Then
[TABLE]
for some positive integer .
Proof.
Define the operator as
[TABLE]
First, we prove that (3.6) is hold for . From (iii), we have
[TABLE]
for all . From this and Cauchy–Schwarz inequality, we obtain
[TABLE]
and hence
[TABLE]
Then we have
[TABLE]
This implies that
[TABLE]
for all . We may assume without loss of generality that
[TABLE]
for all with .
By taking equidistant partition , as above with and also the known weights for interval , we have
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integrals in (3.8). From (3.9), we have
[TABLE]
Therefore, by taking equidistant partition , as above with and also the known weights for interval , the inequality (3.10) can be written as
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integral in (3.10). Hence, for sufficiently large , we have
[TABLE]
for all . That means
[TABLE]
Hence
[TABLE]
Consequently, (3.6) is hold for .
We now prove that (3.6) is hold for . From (3.7), we have
[TABLE]
and hence
[TABLE]
Then we have
[TABLE]
It follows that
[TABLE]
for all . We may assume without loss of generality that
[TABLE]
for all with .
By taking equidistant partition , as above with and also the known weights for interval , we have
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integrals in (3.12). From (3.13), we have
[TABLE]
Therefore, by taking equidistant partition , as above with and also the known weights for interval , the inequality (3.14) can be written as
[TABLE]
where depend upon the used method of Newton–Cotes for estimating the integrals in (3.14). By taking equidistant partition , as above with and also the known weights for interval , the inequality (3.15) can be written as
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integrals in (3.15). Therefore, for sufficiently large , we have
[TABLE]
for all . That means
[TABLE]
Thus
[TABLE]
Consequently, (3.6) is hold for .
Next, we shall prove that (3.6) is hold for some integer . Continuing the process from (3.7) and (3.11), we obtain
[TABLE]
and hence
[TABLE]
By induction, we can show that
[TABLE]
Combining now (3.16) and (3.17), we get
[TABLE]
Hence
[TABLE]
[TABLE]
for all . In a similar way as above, we can show that
[TABLE]
This implies that
[TABLE]
Consequently, (3.6) is hold for some integer . This completes the proof of the proposition. ∎
We now give some properties of the mapping in the following proposition.
Proposition 3.2**.**
Let the assumptions (iv) and (v) be satisfied. Then
[TABLE]
and
[TABLE]
Proof.
From (iv), we have
[TABLE]
for all . From this and Cauchy–Schwarz inequality, we obtain
[TABLE]
and hence
[TABLE]
We may assume without loss of generality that
[TABLE]
for all with .
By taking equidistant partition , as above with and also the known weights for interval , we have
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integrals in (3.20). From (3.21), we have
[TABLE]
Therefore, by taking equidistant partition , as above with and also the known weights for interval , the inequality (3.22) can be written as
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integral in (3.22). Hence, for sufficiently large , we have
[TABLE]
for all . That means
[TABLE]
Hence
[TABLE]
which proves (3.18).
Let us prove (3.19). From (v), by taking equidistant partition , as above with , and also the known weights for interval , we obtain
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integral. Therefore, by taking equidistant partition , as above with and also the known weights for interval , the inequality (3.23) can be written as
[TABLE]
where and depend upon the used method of Newton–Cotes for estimating the integral in (3.23). Hence, for sufficiently large , we have
[TABLE]
for all with . That means
[TABLE]
This completes the proof of the proposition. ∎
In order to prove our main results, we need the following theorems.
Theorem 3.3**.**
Assume is a nonempty closed set in a Banach space and is continuous. Suppose that is a contractive operator for some positive integer . Then has a unique fixed point in . Moreover, the iteration process
[TABLE]
converges to the fixed point .
Proof.
Theorem 3.4**.**
Let the assumptions of Theorem 3.3 be satisfied and let be constructed by iteration process (3.24). Then for , the following estimates hold
[TABLE]
where is the contraction coefficient of the operator is the residual of .
Proof.
For proof see [12]. ∎
Now, we shall give the existence and uniqueness of the solution of the perturbed system of nonlinear equations (3.5).
Theorem 3.5**.**
Let the assumptions (i)–(v) be satisfied. Then the perturbed system of nonlinear equations (3.5) has a unique solution for any .
Proof.
We shall carry out a change of variable
[TABLE]
By Proposition 3.2, the mapping is monotone and Lipschitz - continuous with Lipschitz coefficient equal to . Therefore, by Theorem 2.4, the system of equations (3.26) has a unique solution for any , i.e., the mapping is determined in the whole space . By virtue of the monotonicity of the mapping , the mapping is Lipschitz - continuous with Lipschitz coefficient equal to . Indeed, for all we have
[TABLE]
After changing the variable (3.26), the perturbed system of nonlinear equations (3.5) will take the following form
[TABLE]
Define the mapping as
[TABLE]
Then the system of equations (3.27) can be rewritten as
[TABLE]
It follows from (3.28) that for all and for some positive integer ,
[TABLE]
By virtue of Proposition 3.1 and Lipschitz continuity of the mapping , we have
[TABLE]
Since when is sufficiently large, we see that is a contractive mapping with contraction coefficient equal to . By Theorem 3.3, the mapping has a unique fixed point , i.e., the system of equations (3.27) has a unique solution for any . Consequently, the perturbed system of nonlinear equations (3.5) has a unique solution for any . This completes the proof of the theorem. ∎
In the following proposition, we shall estimate , where with (note that is an analytical solution of (1.1)) and is the exact solution of the perturbed system of nonlinear equations (3.5).
Proposition 3.6**.**
Let the assumptions (i)–(v) be satisfied. Then
[TABLE]
where when is chosen sufficiently large.
Proof.
[TABLE]
which means
[TABLE]
Then we have
[TABLE]
and hence
[TABLE]
It follows that
[TABLE]
Since in every Newton–Cotes formula , we obtain
[TABLE]
By virtue of the contraction of the mapping and the monotonicity of the mapping , we have
[TABLE]
where and . Consequently,
[TABLE]
This completes the proof of the proposition. ∎
The inequality (3.30) leads to the following corollary.
Corollary 3.7**.**
* vanishes when .*
Next, we shall construct the iterative algorithm to find approximate solutions of the perturbed system of nonlinear equations (3.5). To solve the perturbed system of nonlinear equations (3.5), we first have to solve the system of equations (3.27) and after that we solve the system of equations (3.26). In the proof of Theorem 3.5, we have proved that the system of equations (3.27) has a unique solution by using the contraction mapping principle and the system of equations (3.26) has a unique solution for any by using parameter continuation method. The approximate solutions of the system of equations (3.27) are obtained by using the standard iteration process
[TABLE]
For the initial approximation we take . At the same time at each step of above iteration process when calculating the value , we have to solve the system of equations of the form (3.26), as
[TABLE]
Substituting for in the iteration processes (2.5a)–(2.5d), the approximate solutions of the system of equations (3.32) are obtained by using the following iteration processes
[TABLE]
Therefore the approximate solutions of the perturbed system of nonlinear equations (3.5) can be found by the following iteration processes
[TABLE]
Now we estimate the error of approximate solutions of the perturbed system of nonlinear equations (3.5). Assume that the number of steps in each iteration scheme of the iteration processes (3.34a)–(3.34e) is the same and equals . Let be approximate solutions of the perturbed system of nonlinear equations (3.5). Note that depends on , hence we denote . We have the following result.
Theorem 3.8**.**
Let the assumptions of Theorem 3.5 be satisfied. Then the sequence of approximate solutions constructed by iteration processes (3.34a)–(3.34e) converges to the exact solution of the perturbed system of nonlinear equations (3.5). Moreover, the following estimates hold
[TABLE]
where is the smallest natural number such that , is a natural number such that and is chosen sufficiently large such that , is the residual of , .
Proof.
For simplicity, we assume that and , where denotes the zero element in . Indeed, if or , we can define two mappings by
[TABLE]
then and the perturbed system of nonlinear equations (3.5) is equivalent to
[TABLE]
where . It follows from (3.36) that for all
[TABLE]
Therefore the Propositions 3.1 and 3.2 can be applied to the mappings and , respectively. Consequently, Theorem 3.5 can be applied to the perturbed system of nonlinear equations (3.37).
We split the proof into two steps.
Step . We estimate the error of approximate solutions of the system of equations (3.27). Firstly, we estimate the errors in calculating the values . Since , it follows that . Thus . At the same time at each step of the iteration process (3.31) when calculating the value , we will use the iteration processes (3.33a)–(3.33d). Let and be the approximate and exact values of , respectively. It follows from Theorem 2.5 that the values are calculated with the error
[TABLE]
where is the smallest natural number such that .
Since is a convergence sequence, it follows that is bounded for all positive integer . We now determine the supremum of ( is the number of steps in each iteration scheme). For any we have
[TABLE]
On the other hand, we have
[TABLE]
Let be natural number such that . Then for any , we have
[TABLE]
Thus
[TABLE]
It follows from (3.39) and (3.40) that
[TABLE]
for any . Hence the values are calculated with the error
[TABLE]
where
[TABLE]
By Proposition 3.1, we have
[TABLE]
Therefore the values are calculated with the error not more than .
Next, we shall estimate the error of an iteration process in the calculation of . By Theorem 3.4, the error of an iteration process in the calculation of equals , where is chosen sufficiently large such that , is the residual of . We have
[TABLE]
where . Let us prove that
[TABLE]
Obviously, . For some non-negative integer , we have
[TABLE]
and
[TABLE]
Hence
[TABLE]
We have
[TABLE]
Since , it follows that
[TABLE]
On the other hand, we have
[TABLE]
for any positive integer . Hence
[TABLE]
Combining (3.44) and (3.45), we get
[TABLE]
Thus
[TABLE]
which implies that for some non-negative integer . Therefore (3.43) is proved. It follows from (3.42) and (3.43) that
[TABLE]
for every integer . Hence the error of an iteration process in the calculation of equals .
Consequently, the error of approximate solutions of the system of equations (3.27) gives the estimate
[TABLE]
Step . We estimate the error of approximate solutions of the system of equations (3.26)
[TABLE]
Since the mapping is Lipschitz - continuous with Lipschitz coefficient equal to , the substitution of the error into the right - hand side of the system of equations (3.26) causes an error of not more than in the corresponding solution . The error of an iteration process in the calculation of equals . Consequently,
[TABLE]
By (3.41), we obtain (3.35). This completes the proof of the theorem. ∎
Remark 3.9**.**
Let be integer part of , i.e., . We have for every . Since , it follows that . From this and (3.35), we have
[TABLE]
where
[TABLE]
Let . From (3.46), we get
[TABLE]
It follows from (3.47) that for any given , we can find the number of iteration such that .
Remark 3.10**.**
We shall now estimate the complexity of the proposed iterative algorithm (3.34a)–(3.34e). The iteration processes (3.34a)–(3.34e) can be written as the following symbolic notation
[TABLE]
The procedure for calculating each value in the specified element is called an elementary operation. We shall call the number of elementary operations necessary to implement algorithm (3.34a)–(3.34e) is the volume of the calculations . From the symbolic notation (3.48) it follows that .
4. Illustrative example
In this section, to illustrate our above results an example is presented. The computations associated with the example were performed using Maple 12 on personal computer.
Example 4.1**.**
Consider the nonlinear Volterra–Fredholm integral equation
[TABLE]
The analytical solution of this integral equation is on . It is easy to verify that the functions and satisfy the conditions (i)-(v) of the Theorem 3.5 with . For approximating the left-hand integrals, we use composite midpoint rule and take a partition with the discretization parameter . We take . It follows from (3.47) that . Taking . Since , we have . Taking , the number of iterations needed is (the number of steps in each iteration scheme is the same and equals ). Table 1 presents approximate solutions obtained by using the iteration processes (3.34a)–(3.34e) with and , also exact solutions are given for comparison.
5. Conclusions
In this paper, a new numerical method has been proposed to solve nonlinear Volterra–Fredholm integral equations. With this method, the perturbed system of nonlinear equations obtained by discretization is solved by an iterative method, which is based on a hybrid of the method of contractive mapping and parameter continuation method. Lastly, an illustrative example is given to demonstrate the effectiveness and convenience of the proposed method.
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