Abstract
We propose and apply coupling of the variational iteration method (VIM) and homotopy perturbation method (HPM) to solve nonlinear mixed Volterra-Fredholm integrodifferential equations (VFIDE). In this approach, we use a new formula called variational homotopy perturbation method (VHPM) and variational accelerated homotopy perturbation method (VAHPM). This approach is based on the form of He’s polynomials and on a new form of He’s polynomials. We discuss the convergence of the technique. Some numerical examples are introduced to verify the efficiency of this technique.
1. Introduction
In recent years, there has been a clear interest in integrodifferential equations which are a combination of differential and Volterra-Fredholm integral equations. Integrodifferential equations play an important role in many branches of linear and nonlinear functional analyses and their applications. The mentioned integrodifferential equations are usually difficult to solve analytically, so approximation strategies are required to obtain the solution of the linear and nonlinear integrodifferential equations [1].
Many researchers studied and discussed the linear VFIDE [2]. Al-Jubory [3] introduced some approximation methods to solve Volterra-Fredholm integral and integrodifferential equations. Dadkhah et al. in [4] used a numerical solution of nonlinear VFIDE by using Legendre wavelets. Rabbani and Kiasoltani [5] studied the solving of a nonlinear system of VFIDE by using the discrete collocation method. Gherjalar and Mohammadikia [6] solved integral and integrodifferential equations by using the B-splines function. In this work, we used the HPM and VIM to solve the two-dimensional nonlinear VFIDE as follows:with initial conditionswhere . The functions , and , are analytic functions on , and functions are given.
For VIM and HPM, which were proposed by He in [7, 8], the solution is considered as the summation of an infinite series which is assumed to be convergent to the exact solution. In recent years, HPM has been applied with great success, so relations and algorithms have been deduced and continuously improved to obtain an accurate solution for a large variety of linear and nonlinear problems. For instance, He in [7] used a strategy to solve some integrodifferential equations where he chose an initial approximate solution in the form of an exact solution with unknown constants.
In this paper, a new approach based on VIM with HPM is introduced to solve the two-dimensional nonlinear VFIDE.
2. The HPM
In this section, we will present the HPM. We consider a general integral equationwhere is an integral operator. Define a convex homotopy bywhere is a functional operator with solution . Then,and the process of changing from to is just that of changing from to . In topology, this is called deformation, and and are called homotopies.
According to the HPM, we can use the embedding parameter as a “small parameter” and assume that the solution of (4) can be written as a power series in :When , the approximate solution of (3) is obtained withSeries (7) is convergent for most cases; however, the rate of convergence depends upon the nonlinear operator [9].
3. The HPM for Solving Nonlinear Mixed VFIDE
In what follows, we display an outline for utilizing the HPM for solving the nonlinear VFIDE. Equation (1) can be written as follows:where is the multiple integration operator given as follows: So, (8) takes the formsince
To illustrate the HPM, for nonlinear mixed VFIDE, let us consider (8):By the HPM, we can expand into the formWhen , the approximate solution is obtained with and in sum, according to [10], He’s HPM considers the nonlinear term aswhere ’s are the so-called He’s polynomials [10], which can be calculated by using the formula
Substituting (13) and (15) into (12) and equating the terms with identical powers of , we have
The nonlinear terms and is a derivative operator) are usually represented by an infinite series of the so-called He’s polynomials as follows:
The components , can be computed by using the recursive relations (17).
4. A New Formula to He’s Polynomials
He’s polynomials are not unique; another formula of He’s polynomials , called accelerated He’s polynomials, is represented by ; in [11], the author proved thatin which can be written in the new mathematical formwhere the partial sum and . Substituting (13) and into (12) and equating the terms with identical powers of , we obtain the following accelerated recursive formula:
For example, if , the first four polynomials using formulas (16) and (20) are computed to be as follows.
Using formula (16),Using formula (20),
Clearly, the first four polynomials computed using the suggested formula (20) include the first four polynomials computed using formula (16) in addition to other terms that should appear in using formula (16). Thus, the solution that was obtained using formula (20) enforces many terms to the calculation processes earlier, yielding faster convergence.
5. The VIM
Consider the differential equationwhere is a linear operator, is a nonlinear operator, and is a given continuous function. The VIM presents a correction functional for (24) in the form where is a Lagrange multiplier [8, 9] which can be identified optimally via variational theory, is the th approximate solution, and denotes a restricted variation (i.e., ).
6. Adapting VIM with HPM for Solving (1) and (2)
This modified version of VHPM is obtained by the coupling of VIM with HPM. First, by using formula (16), we obtainwhich is called VHPM.
Second, by using formula (20),which is called VAHPM.
The following is the algorithm for calculating : Step1: input nonlinear term and that is the order of He’s polynomials, endpoint , initial conditions , free term, and . Step2: set . Step3: let . Step4: For , do (a) th-order derivative of both sides of the equality with respect to : (b) let of the above equality and determine by solving the equation with respect to . End do. Step5: put initial conditions. Step6: for , do Step7: calculate by applying (26), end do. Step8: set as the approximate of the exact solution.
7. Convergence Analysis
In this section, the sufficient condition that guarantees the existence of a unique solution is introduced in Theorem 1, convergence of the methods is proved in Theorems 2 and 3, and finally the maximum absolute error of the truncated series ( is estimated in Theorem 4.
Considering (10), we setWe can write (10) as
We assume is bounded for all in and
Also, we suppose the nonlinear terms and are Lipschitz continuous with
Hence, we set
Theorem 1. Two-dimensional nonlinear VFIDE has a unique solution whenever .
Proof. Let and be two different solutions of (30). Then,from which we get . Since , therefore . Therefore, and this completes the proof.
Theorem 2. The series solution of (1) using HPM convergence when and .
Proof. Denote with the Banach space of all continuous functions on with the norm for all in . Define the sequence of partial sums , and let and be arbitrary partial sums with . We are going to prove that is a Cauchy sequence in this Banach space:From (20), we have So, Let ; then,So, Since , we have ; then,But (since is bounded), so, as , then . We conclude that is a Cauchy sequence in , and therefore the series is convergent and the proof is complete.
Theorem 3. When using VIM for solving two-dimensional nonlinear VFIDE where and , then converges.
Proof. One hasBy subtracting relation (41) from (42),Hence, we setThen,Therefore, Since , then . So, the series converges and the proof is complete.
Theorem 4. The maximum absolute truncation error of the series to (1) is estimated to be
Proof. From Theorem 2 and inequality (40), we have As , then and we have So, Finally, the maximum absolute truncation error in the interval is
8. Numerical Examples
Example 1. Consider the nonlinear integrodifferential equation wherewith the initial conditions which has exact solution . This example is solved by using the variational iteration method with He’s polynomials VHPM (see (26)) and VAHPM (see (27)) expressing the nonlinear terms of and , respectively, in Table 1.
Example 2. Consider the nonlinear integrodifferential equationwherewith the boundary conditionsand the initial conditions which has exact solution . This example is solved by using VHPM (see (26)) and VAHPM (see (27)) expressing the nonlinear terms of and , respectively, in Table 2.
9. Conclusion
In this paper, we applied VHPM and VAHPM to solve nonlinear mixed VFIDE. The proposed VAHPM converges faster than the VHPM. Based on the proposed formula (27) with accelerated He’s polynomials formula (20), the convergence of the technique is proved. The presented technique is very easy to implement and it reduces the computation size.
Conflicts of Interest
The authors declare that they have no conflicts of interest.