Abstract
In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.
1. Introduction
Most of the phenomena in mathematical physics, biological mathematics, and applied mathematics are modeled in the form of differential equations. Solutions to such differential equations are needed to analyze and predict changes in a physical system. In most cases, it is impossible to calculate an exact solution. Therefore, to solve these problems, various numerical and seminumerical methods have been modeled.
Liao presented one of the first analytical methods for nonlinear problems which do not require a small parameter [1]. This method has been applied to many situations in various fields of science and technology [2–4]. In late 90s, Prof. He introduced the homotopy perturbation method (HPM) for highly nonlinear equations [5]. This technique matures into a full fledged theory for nonlinear problems with the efforts of many researchers, notably Ji-Huan He and his students. This method combines homotopy concepts with perturbation theory and does not rely on small or large parameter like other traditional perturbation techniques [6]. Wide range of problems has been solved using HPM [7–10].
Various modifications of HPM have also been proposed by different researchers to tackle more complex problems. Darvishi et al. combine HPM with adomian polynomials to sine-Gorden type equations in [11]. Biazar and Eslami proposed a modification of HPM based on Taylor series expansion of the kernel and source term [12]. HPM with rank upgrading technique for the superior nonlinear oscillation is proposed in [13]. Qayyum et al. used coupling of HPM with Laplace transform for squeezing flows in [14]. Bota and Caruntu applied HPM with the least square method to fluid flow problems in [15]. For enhanced results, Le-He extension of HPM is proposed in [16]. Ji et al. further applied Le-He extension of HPM to nonlinear packaging system in [17]. Ain et al. introduced Li-He modified extension of HPM for micro-electro-mechanical systems in [18].
In this article, HPM is combined with LS optimizer along with some refine inital guesses to obtain fast convergent semianalytical solutions of linear and nonlinear BVPs. This scheme takes few iterations (cycles) to achieve accurate solution, and hence, it has less computational cost with improved accuracy.
2. Basic Idea of LSHPM
Let us consider the following differential equations along with boundary conditions.where and represent the linear and nonlinear parts, represents the boundary operator, is the boundary of the domain, represents an unknown value, and is a known function. We construct a homotopy such that which satisfieswhere is the embedding parameter and is the initial guess of equation (1) that satisfies the boundary conditions:
Therefore, as changes from 0 to 1, the solution approaches from to . We expand into a Taylor series expansion about to obtain an approximate solution:
Setting , the approximate HPM solution of (1) would be
We reassign the convergence controlling parameters as coefficients of series (6) and substitute supposed solution back in (1) to get the residual function:
Now, we compute the sum of square of residuals:
After computing , we find the optimal values of from system of equations obtained from . Then, putting these optimal values back into will provide refined solution.
3. Application of LSHPM
Problem 1. Second-order linear ODE:subject to boundary conditions,
The exact solution of this problem is . First, we construct a homotopy as
Zeroth-order problem is
The solution to (12) is
First-order problem is
The solution to (14) is
Second-order problem is
The solution to (16) is
Third-order problem is
The solution to (18) is
By combining (13), (15), (17), and (19) will give approximate solution of (9) as follows:
It follows that (20) consist of , and hence, the required solution is
By applying the boundary conditions given in (27), we have
By putting in gives
Now, replacing with in (27)
Next, we minimize to get optimal values of . Here,
The third-order approximation solution is
The results are presented in Table 1.
Problem 2. Let us consider the following second-order nonhomogeneous nonlinear ODE [20]:subject to boundary conditions,
This problem was studied by Momani [20] by applying the differential transform method, and later, Mohamad [19] by applying four different numerical methods for this problem. The exact solution to this problem is .
The first step is to construct a homotopy:
Then, we linearize the problem as follows.
Zeroth-order problem is
The solution to (30) is
First-order problem is
The solution to (32) is
Second-order problem is
The solution to (34) is
By combining (31), (33), and (35) will give an approximate series solution of (27) as
(36) consists of ; hence, the required solution will take the form
By applying boundary conditions from (27) gives
By putting in gives
Now, replacing with in (27), we get the following residual function:
Next, we minimize to get optimal values of .
Here,
Hence, second-order approximate solution is
The results are presented in Table 2.
Problem 3. Third-order linear ODE [21]:subject to boundary conditions
The exact solution to this problem is . After applying LSHPM proceedure, second-order approximate solution is
The results are presented in Table 3.
Problem 4. Third-order nonlinear ODE [24]:subject to boundary conditions
The exact solution to this problem is . The first-order approximate solution using LSHPM is
The results are presented in Table 4.
Problem 5. Fourth-order linear ODE [25]:subject to boundary conditions
The exact solution to this problem is . The first-order approximate solution using LSHPM is
The results are presented in Table 5.
Problem 6. Fourth-order nonlinear ODE [26]:subject to boundary conditions where is the Reynolds number, and we have fixed it to . The second-order approximate solution of this problem using LSHPM is
The results are presented in Table 6.
Problem 7. Fifth-order linear ODE [26]:subject to boundary conditions
The exact solution of this problem is . The second-order approximate solution using LSHPM is
The results are presented in Table 7.
Problem 8. Fifth-order nonlinear ODE [29]:along with boundary conditions
The exact solution to this problem is . The zeroth-order approximate solution using LSHPM is
The results are presented in Table 8.
Problem 9. Sixth-order linear ODE [14]:subject to boundary conditions
The exact solution to this problem is . The first-order approximate solution using LSHPM is
The results are presented in Table 9.
Problem 10. Sixth-order nonlinear ODE [14]:subject to boundary conditions
The exact solution to this problem is . The zeroth-order approximate solution using LSHPM is
The results are presented in Table 10.
Problem 11. Seventh-order linear ODE [32]:subject to boundary conditions
The exact solution to this problem is . The first-order approximate solution using LSHPM is
The results are presented in Table 11.
Problem 12. Seventh-order nonlinear ODE [32]:subject to boundary conditions,
The exact solution to this problem is . The zeroth-order approximate solution using LSHPM is
The results are presented in Table 12.
4. Results and Discussion
In this article, a new modification of HPM has been introduced by hybriding HPM with LS optimizer. The proposed scheme has been tested agaist various order linear and nonlinear BVPs. The validity of LSHPM has been checked by comparing exact and approximate solutions. For testing the efficiency LSHPM, problems are also solved with HPM and results are compared with LSHPM. This can easily be observed in Tables 1–12. These tables signify the efficiency of LSHPM in terms of high accuracy with less computational cost. The convergence of LSHPM can also be observed in Figures 1–12. These figures show that LSHPM is more consistent as compared other mentioned schemes.
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5. Conclusion
In present article, an efficient and reliable modification of HPM is introduced by mixing HPM with the LS optimizer. The proposed scheme is tested against different order linear and nonlinear BVPs. The obtained solutions are compared with HPM and other numerical schemes available in the literature. Analysis of results shows that LSHPM is more consistent in terms of accuracy with less computational cost and can be used in different areas of science and technology.
Data Availability
All the data are available within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.