Abstract
Wave problem arises in various phenomena of science and engineering. This study proposes an efficient and appropriate scheme to produce the approximate solutions of multidimensional problems arising in wave propagation. We use a new iterative strategy (NIS) to minimize the computational cost and less time than other approaches studied in the literature. Since the variational iteration method (VIM) has some assumptions and restrictions of variables during the construction of the recurrence relation, we present a Sawi integral transform for the construction of the recurrence to overcome this difficulty. The series solution for this recurrence relation is determined using the homotopy perturbation strategy (HPS) in the form of convergence that provides the precise solution after a few iterations. This NIS does not require any discretization in the findings of results and derives the algebraic equations. Some numerical models and graphical representations are illustrated to verify the performance of this scheme.
1. Introduction
Significant improvements in computational techniques have been made recently for engineering and scientific applications such as characterization of geomaterials, soil-structure interaction, elastic metamaterials, etc., as well as the simulation of seismic wave propagation. The simulation’s temporal and spatial scales, constitutive laws, interface and boundary conditions, and numerical schemes must all be carefully chosen depending on the application of interest [1–3].
The study of partial differential equations (PDEs) has an important contribution in numerous branches of science and engineering such as electronics, hydrology, computational dynamics, physical chemistry, chemical engineering, optical fiber, mechanics, dynamics of substances, and the geometric optical [4–6]. Various researchers have studied the analytical schemes to obtain the approximate solution of these PDEs. Even though the computations for these methods are quite simple, some variables are premised on the hypothesis of numerous types of restrictions. Therefore, many scientists are searching for novel approaches to solve this kind of limitations. Many experts and scientists have presented several approaches to evaluate the analytical results [7–9]. In the past, a number of experts and researchers applied the homotopy perturbation technique (HPS) [10, 11] to a variety of physical issues because this method consistently reduces the complex problem to a simple solution. The solution series converges using this strategy quite quickly in the majority of instances. The authors [12, 13] implemented the idea of the homotopy perturbation method for the nonlinear oscillation problems and showed that this scheme provided the analytical results very efficiently.
The wave problem is a PDE for a scalar function that governs the wave propagation phenomena in fluid dynamics. Wazwaz employed the VIM to research linear and nonlinear difficulties [14]. The homotopy perturbation method was utilized by Ghasemi et al. [15] to calculate the numerical solutions of the two-dimensional nonlinear differential equation. For the approximate solution of wave problems, Keskin and Oturanc [16] proposed a new approach. Ullah et al. [17] suggested a homotopy optimal approach to derive the analytic results of wave problems. Adwan et al. [18] provided the computational results of multidimensional wave problems and demonstrated the validity of the suggested method. Jleli et al. [19] utilized the HPS for the approximate results of wave problems. The authors [20] presented the finite element approach and discretized the two-dimensional wave model to obtain the analytical results. These methods contain numerous restrictions and presumptions when it comes to estimating the solution. We provide a new iterative approach for these multidimensional wave problems and tackle these limitations and restrictions in our current research.
The aim of the current study is to apply a new iterative strategy (NIS) with the combination of the Sawi integral transform and the HPS for multidimensional problems. This NIS generates an iteration series that provides approximation results near the exact results. This scheme shows a better performance and yields attractive results for the illustrated problems. This article is studied as follows: Section 2 provides the idea of Sawi integral transform with the convergence theorem. We construct a scheme of NIS for obtaining the results of multidimensional models in Section 3. Some numerical applications are considered to show the capability of NIS in Section 4, and lastly, we present the conclusion in Section 5.
2. Fundamental Concepts
In this section, we present some essential properties of Sawi transform to understand the concept of their formulation. These properties are very helpful in utilizing the numerical problems of this paper.
2.1. Sawi Transform
Definition 1. Consider be a function of , then Sawi transform (T) is [21, 22]where is denoted as a symbol of T. Then, we havewhere represents the transform function of . The Sawi transform of the function for exist if is piecewise continuous and of exponential order. The mentioned two conditions are the only sufficient conditions for the existence of Sawi transforms of the function .
Proposition 2. We define the fundamental properties of T such as let and , then [23, 24]
Proposition 3. Let us define the differential properties of T such as if , the differential properties are defined as [25]
Theorem 4. Assume that and be two Banach spaces such that is a nonlinear operator. Therefore, . The Banach contraction states that is denoted as a unique fixed point , i.e., , so we haveand assume that , where then, we have
Proof. In the light of mathematical induction at , we haveLet the result be accurate for , thusTherefore,Hence, using , we havewhich shows .
: Since and as a .
Therefore, we have .
3. Formulation of NIS
In this section, we consider 1D, 2D, and 3D wave problems for the use of the new iterative strategy (NIS). This scheme is applicable to solve the differential problems using the initial conditions. We made a point where the development of this method is independent from integration and any other presumptions. Consider a differential problem such aswith initial conditionswhere is a uniform function, represents the nonlinear components, and is the source term of arbitrary constants and . We can also write equation (11) such as
In mathematics, the Sawi transform is an integral transform that converts a function of a real variable to a function of a complex variable. This transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication.
Employing T on equation (13), it yields
Employing the propositions as defined in equation (4), we get
Thus, is derived as
Using inverse T on equation (16), we getusing initial conditions, we get
This equation (18) is the formulation of NIS of equation (11).
Now, we assume HPS such asand nonlinear components are defined aswe can produce the as
Using equations (19)–(21) in (18) and evaluating the similar components of , it yields
On proceeding this process, which yields
Thus, equation (23) is a rough estimate of the differential problem’s solution.
4. Numerical Applications
To demonstrate the accuracy and dependability of NIS, we offer some numerical samples. We see that, compared to earlier schemes, this method is significantly easier to implement and much simpler to generate the series of convergence. Through graphical structures, we demonstrate the physical character of the resultant plot distribution. Additionally, a graphic representation of the error distribution highlighted how close the NIS results are to the accurate solutions. We calculate the values of absolute errors by the difference of the exact solutions with the NIS results.
4.1. Example 1
Imagine a wave equation in one dimension:in the initial circumstance:
Apply T on equation (24), we get
Employing the propositions as defined in equation (4), we get
Thus, is derived as
Using inverse T, it yields
Thus, HPS yields such as
Evaluating similar components of , we obtain
Similar to this, we can take into account the approximation series such thatwhich can approach to
Figure 1 consists of two graphs; Figure 1(a) is the NIS solution of , and Figure 1(b) is the precise solution of at and for one-dimensional wave equation. Figure 2 demonstrates the error distribution between the obtained and the precise results at along and confirms the strong agreement of this scheme for problem 1. It claims that we can accurately simulate any surface to reflect the appropriate natural physical processes. Table 1 represents the absolute error between the exact solution and the NIS results. This compression shows that NIS results are very close to the exact solution and yields a fast convergence only after a few imitations.
(a)
(b)
4.2. Example 2
Imagine a wave equation in the two-dimensional form:with the initial circumstance:
Apply T on equation (34), we get
Employing the propositions as defined in equation (4), we get
Thus, is derived as
Using inverse T, it yields
Thus, HPS yields such as
Comparing the same elements of , we get
Similar to this, we can take into account the approximation series such thatwhich can approach to
Figure 3 consists of two graphs; Figure 3(a) is the NIS solution of , and Figure 3(b) is the precise solution of at , along for two-dimensional wave equation. Figure 4 demonstrates the error distribution between the obtained and the precise results at , along and confirms the strong agreement of this scheme for problem 2. It claims that we can accurately simulate any surface to reflect the appropriate natural physical processes. Table 2 represents the absolute error between the exact solution and the NIS results. This compression shows that NIS results are very close to the exact solution and yields a fast convergence only after a few imitations.
(a)
(b)
4.3. Example 3
We take into consideration the three-dimensional wave problem.with the initial conditionand boundary condition
Applying T on equation (44), we get
Employing the propositions as defined in equation (4), we get
Thus, is derived as
Using inverse T, it yields
Thus, HPS yields such as
Evaluating similar components of , we obtain
Similar to this, we can take into account the approximation series such thatwhich can approach to
Figure 5 consists of two graphs; Figure 5(a) is the NIS solution of , and Figure 5(b) is the precise solution of at and with and for three-dimensional wave equation. Figure 6 demonstrates the error distribution between the obtained and the precise results at , , along and confirms the strong agreement of this scheme for problem 3. It claims that we can accurately simulate any surface to reflect the appropriate natural physical processes. Table 3 represents the absolute error between the exact solution and the NIS results. This compression shows that NIS results are very close to the exact solution and yields a fast convergence only after a few imitations.
(a)
(b)
5. Conclusion and Future Work
This article presents the study of a new iterative scheme (NIS) using the combination of Sawi integral transform and HPS for the approximate results of multidimensional wave problems. This NIS derives the results with the consistency of analysis from the recurrence relation. The derived results from numerical examples demonstrate that our scheme is very easy to implement and the rate of convergence is higher than other approaches. The 3D graphical representations show the physical behavior of the problems, and 2D plots’ distribution represents the visual error among the obtained and the precise results. We show that NIS has the best agreement with the precise solutions to the problems. We aim to apply this scheme the nonlinear and fractional differential problems in our future work.
Data Availability
The data used to support the findings of this study are available on request from the corresponding author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.