Abstract
We reconstruct the variational iteration method that we call, parametric iteration method (PIM). The purposed method was applied for solving nonlinear Volterra integrodifferential equations (NVIDEs). The solution process is illustrated by some examples. Comparisons are made between PIM and Adomian decomposition method (ADM). Also exact solution of the 3rd example is obtained. The results show the simplicity and efficiency of PIM. Also, the convergence of this method is studied in this work.
1. Introduction
It is well known that many events in scientific fields deal with integrodifferential equations. The nonlinear Volterra integrodifferential equations play a major role in many physical processes such as nanohydrodynamic [1], dropwise condensation [2], and biologic [3]. The various numerical methods exist for solving NVIDEs, for example, variational iteration method (VIM), Adomian decomposition method [4, 5], Chebyshev polynomials [6], and Bernstein’s approximation [7]. First, Liao purposed homotopy analysis method [8] and it is applied in many scientific problems [9, 10]; then, VIM was purposed by He [11]. In this paper, we reconstruct the VIM that we call PIM. PIM was applied successfully for solving boundary value problems [12]. We consider nonlinear integrodifferential equations as follows: and the initial value for both of the two equations is as follows:
In this work, the numerical solution of (1) and (2) is possible by PIM when ,, are continuous and is continuous operator. Parametric iteration method provides solution for NVIDEs as a sequence of iterations. In this study, some examples are given and we solve them using parametric iteration method and compare the obtained results with ADM results. In all these cases, the present technique worked excellently, as it will be shown in this study.
2. The Basic Idea of the Parametric Iteration Method
In this section, we describe PIM for solving nonlinear Volterra integrodifferential equations. Then, the local convergence is discussed.
2.1. Parametric Iteration Method
The PIM provides the solution for (1) and (2) as a sequence of approximations. This method gives rapidly convergent successive approximations of the exact solution if such a solution exists; otherwise, approximations can be used for numerical purposes. We assume and are the linear and nonlinear operators on . To explain the basic idea of PIM, we consider (1) and (2) as follows: where with the property when denotes the so-called auxiliary linear operator with respect to , is a nonlinear continuous operator with respect to , is the known continuous function, and . The basic essence of this method is to construct a family of iterative processes for (1) and (2) as [13] with the initial conditions where or and is the initial guess which can be chosen arbitrarily, but the suitable selection is positively affect for the rate of convergence [13], or it can also be solved from its corresponding linear homogeneous equation or linear nonhomogeneous equation . The parameter and function denote the so-called auxiliary parameter and auxiliary function. The selection of , was described in [13]. Also, we are free to choose the auxiliary linear operator , the auxiliary parameter , the auxiliary function , and the initial approximation . Therefore, if the successive approximations are obtained by PIM in terms of the auxiliary parameter , then exact solution may be given by . According to [13], Let be the solution space and let denote the set of base functions. Hence, we can represent the solution in the series , where is a coefficient belonging to real numbers. As long as the set of base functions is determined, the auxiliary linear operator , the initial approximation , and the auxiliary function must be chosen in such a way that all solutions of the corresponding PIM equations (4) exist and it can be expressed by this set of base functions. Now, in order to avoid expensive computational works for solving (1) and (2) via PIM, it is straightforward to use the following set of base functions: that is, where are unknown coefficients to be determined and is a constant belonging to real numbers. Now, we set the auxiliary operator as follows: The initial guess is to form combination of -terms of (9); that is,
According to (11) and the initial conditions (3) and with due attention to , the coefficients will be determined. Also, we set . The selecting of is arbitrary, but the suitable selection depends on the base functions for solution [13], and we use the PIM processes to compute the approximation solutions of (1) and (2).
2.2. The Valid Region of
Assume that we gain a family of solution series in terms of the auxiliary parameter by means of PIM. We consider this solution as a function in terms of ,; then, we derive (once or more) this function with respect to in that ; that is, let be the solution of (1) or (2); then, we set therefore, will be in terms of ; now we plot curve, and according to these curves, it is easy to discover the valid region of , which corresponds to the line segments nearly parallel to the horizontal axis. This region is called valid region of which we denote by . We ensure that the solution series converge for any .
2.3. Analysis of Convergence of the Parametric Iteration Formula
In this section, we study the local convergence of approximate solution provided by PIM for solving (1). The convergence of approximate solution for (2) is similar to (1).
Initially, let and set ; therefore, we have from (5) the following parametric iteration formula:
The iterative formula (14) expressed by sequence makes a recurrence sequence . Obviously, the limit of the sequence will be the solution of (1) if the sequence is convergent.
In order to prove that the sequence is convergent, we construct a series: Noticing that the sequence will be convergent if the series is convergent.
Theorem 1. If is Lipschitz-continuous in and and where , are positive real numbers, then the series of (15) is convergent; that is, the sequence is convergent for .
Proof. According to (14), note that
where
From (14) and (17) and the assumption that where denotes the Lipschitz constant of , it follows that
In view of (21), the convergence of the series (15) can be concluded for the solution domain and with the help of some mathematical software. Therefore, the series of (15) is absolute convergence; that is, the sequence is convergent for .
3. Illustrative Examples
Now, we use PIM to solve two examples of the kind of (1) and (2) and compare the obtained results with ADM [5] to show the efficiency of PIM.
Example 1. The first example is a nonlinear Volterra integrodifferential equation of the second kind as follows:
According to PIM proceeding, we define Then, we have from (10) that . Using initial condition and gives us, that ,that is,. Now, If we set , then we obtain from (5) and (8) that Thus we integrate from both sides of (25); then the PIM equations are as follows: Some of the iterations obtained from (25) are as follows and the other iterations have also been calculated by Maple 13: The obtained result for the 3rd iteration was shown in Table 1; also the valid region of that is was presented in Figure 1.
Example 2. We consider the nonlinear Volterra integrodifferential equation of the second kind as follows: According to PIM procedure we set Then, we have from (10) , and (29) gives us that ,; that is, . Now similar to Example 1, the iteration scheme is as follows: Some of the iterations obtained from (30) are as follows:
The results of Example 2 are available in Table 2. for (27) is presented in Figure 2.
Example 3. We consider the nonlinear Volterra integrodifferential equation of the second kind as follows: Similar to Examples 1 and 2
Now, we consider the initial solution as , and (33) and (34) give us that ,; that is, . According to PIM iterative formula for this example we have
Starting with and initial solution that is , the exact solution of this example, which is , is achieved. The valid value of parameter for (36) is .
4. Results
In this section, we present the results of Examples 1 and 2 in two tables and plot the -curve to determine . All the computations have been done with Maple 13.
5. Conclusion
In this paper, we reconstruct the VIM that we call parametric iteration method, and PIM was applied to solve the nonlinear Volterra integrodifferential equations. In order to illustrate the method, we solve three examples. PIM results compared to ADM show that the former is easier in practice and more accurate for NVIDEs. For the 3rd example exact solution was achieved. Further, the convergence of PIM for solving NVIDEs in the valid region of was presented. Additionally, if we increase the number of iterations by PIM scheme, it seems that the results will have more accuracy in solutions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.