A Numerical Approach for the Analytical Solution of Multidimensional Wave Problems

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Introduction
Signifcant improvements in computational techniques have been made recently for engineering and scientifc applications such as characterization of geomaterials, soil-structure interaction, elastic metamaterials, etc., as well as the simulation of seismic wave propagation.Te 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][2][3].
Te study of partial diferential equations (PDEs) has an important contribution in numerous branches of science and engineering such as electronics, hydrology, computational dynamics, physical chemistry, chemical engineering, optical fber, mechanics, dynamics of substances, and the geometric optical [4][5][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.Terefore, 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][8][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.Te solution series converges using this strategy quite quickly in the majority of instances.Te 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 efciently.
Te wave problem is a PDE for a scalar function that governs the wave propagation phenomena in fuid dynamics.Wazwaz employed the VIM to research linear and nonlinear difculties [14].Te homotopy perturbation method was utilized by Ghasemi et al. [15] to calculate the numerical solutions of the two-dimensional nonlinear diferential 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.Te authors [20] presented the fnite element approach and discretized the two-dimensional wave model to obtain the analytical results.Tese 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.
Te 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.Tis NIS generates an iteration series that provides approximation results near the exact results.Tis scheme shows a better performance and yields attractive results for the illustrated problems.Tis 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.

Fundamental Concepts
In this section, we present some essential properties of Sawi transform to understand the concept of their formulation.Tese properties are very helpful in utilizing the numerical problems of this paper.

Sawi Transform
Defnition 1.Consider ϑ be a function of η ≥ 0, then Sawi transform (S T) is [21,22] where S is denoted as a symbol of S T. Ten, we have where Q(θ) represents the transform function of ϑ(η).Te Sawi transform of the function ϑ(η) for η ≥ 0 exist if ϑ(η) is piecewise continuous and of exponential order.Te mentioned two conditions are the only sufcient conditions for the existence of Sawi transforms of the function ϑ(η).

Formulation of NIS
In this section, we consider 1D, 2D, and 3D wave problems for the use of the new iterative strategy (NIS).Tis scheme is applicable to solve the diferential 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 diferential problem such as with initial conditions where ϑ(x 1 , η) is a uniform function, f(ϑ) represents the nonlinear components, and f(x 1 , η) is the source term of arbitrary constants a 1 and a 2 .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.Tis transform has many applications in science and engineering because it is a tool for solving diferential equations.In particular, it transforms ordinary diferential equations into algebraic equations and convolution into multiplication.
Employing ST on equation ( 13), it yields Employing the propositions as defned in equation ( 4), we get Tus, Q(θ) is derived as Using inverse S T on equation ( 16), we get using initial conditions, we get Tis equation ( 18) is the formulation of NIS of equation (11).Now, we assume HPS such as and nonlinear components f(ϑ) are defned as we can produce the H n ′ s as Using equations ( 19)-( 21) in (18) and evaluating the similar components of p, it yields On proceeding this process, which yields Tus, equation ( 23) is a rough estimate of the diferential problem's solution.

Numerical Applications
To demonstrate the accuracy and dependability of NIS, we ofer some numerical samples.We see that, compared to earlier schemes, this method is signifcantly easier to implement and much simpler to generate the series of convergence.Trough 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 diference of the exact solutions with the NIS results.

Example 1. Imagine a wave equation in one dimension:
in the initial circumstance: Apply S T on equation ( 24), we get Employing the propositions as defned in equation ( 4), we get Tus, Q(θ) is derived as Using inverse S T, it yields Tus, HPS yields such as Evaluating similar components of p, we obtain Similar to this, we can take into account the approximation series such that which can approach to Figure 1 consists of two graphs; Figure 1(a) is the NIS solution of ϑ(x 1 , η), and Figure 1(b) is the precise solution of ϑ(x 1 , η) at − 10 ≤ x 1 ≤ 10 and 0 ≤ η ≤ 0.01 for onedimensional wave equation.Figure 2 demonstrates the error distribution between the obtained and the precise results at 0 ≤ x 1 ≤ 5 along η � 0.1 and confrms the strong agreement of this scheme for problem 1.It claims that we can accurately simulate any surface to refect the appropriate natural physical processes.Table 1 represents the absolute error between the exact solution and the NIS results.Tis compression shows that NIS results are very close to the exact solution and yields a fast convergence only after a few imitations.

Example 2. Imagine a wave equation in the twodimensional form:
with the initial circumstance: Apply S T on equation (34), we get Employing the propositions as defned in equation ( 4), we get Tus, Q(θ) is derived as Using inverse S T, it yields  Table 1: Absolute error between the NIS results and the exact results of ϑ(x 1 , η) along x-space at various points of η for Example 1. x (39) Tus, HPS yields such as Comparing the same elements of p, we get Similar to this, we can take into account the approximation series such that which can approach to Figure 3 consists of two graphs; Figure 3(a) is the NIS solution of ϑ(x 1 , y 1 , η), and Figure 3(b) is the precise solution of ϑ(x 1 , y 1 , η) at − 5 ≤ x 1 ≤ 5, 0 ≤ η ≤ 0.01 along y 1 � 0.5 for two-dimensional wave equation.Figure 4 demonstrates the error distribution between the obtained and the precise results at 0 ≤ x 1 ≤ 5, y 1 � 0.1 along η � 0.1 and confrms the strong agreement of this scheme for problem 2. It claims that we can accurately simulate any surface to refect the appropriate natural physical processes.Table 2 represents the absolute error between the exact solution and the NIS results.Tis compression shows that NIS results are very close to the exact solution and yields a fast convergence only after a few imitations.

Example 3.
We take into consideration the threedimensional wave problem.
with the initial condition and boundary condition     Table 3: Absolute error between the NIS results and the exact results of ϑ(x 1 , y 1 , z 1 , η) along x 1 -space and y 1 � z 1 � 0.5 at various points of η for Example 3. x Similar to this, we can take into account the approximation series such that  5(a) is the NIS solution of ϑ(x 1 , y 1 , z 1 , η), and Figure 5(b) is the precise solution of ϑ(x 1 , y 1 , z 1 , η) at 5 ≤ x 1 ≤ 10 and 0 ≤ η ≤ 0.01 with y 1 � 0.5 and z 1 � 0.5 for three-dimensional wave equation.Figure 6 demonstrates the error distribution between the obtained and the precise results at 0 ≤ x 1 ≤ 10, y 1 � 0.5, y 1 � 0.5 along η � 0.5 and confrms the strong agreement of this scheme for problem 3. It claims that we can accurately simulate any surface to refect the appropriate natural physical processes.Table 3 represents the absolute error between the exact solution and the NIS results.Tis compression shows that NIS results are very close to the exact solution and yields a fast convergence only after a few imitations.

Conclusion and Future Work
Tis 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.Tis NIS derives the results with the consistency of analysis from the recurrence relation.Te derived results from numerical examples demonstrate that our scheme is very easy to implement and the rate of convergence is higher than other approaches.Te 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.

Table 2 :
Absolute error between the NIS results and the exact results of ϑ(x 1 , y 1 , η) along x 1 -space and y 1 � 0.5 at various points of η for Example 2.