Abstract
Many scientific phenomena are linked to wave problems. This paper presents an effective and suitable technique for generating approximation solutions to multidimensional problems associated with wave propagation. We adopt a new iterative strategy to reduce the numerical work with minimum time efficiency compared to existing techniques such as the variational iteration method (VIM) and homotopy analysis method (HAM) have some limitations and constraints within the development of recurrence relation. To overcome this drawback, we present a Sawi integral transform (\(\mathbb {S}\)T) for constructing a suitable recurrence relation. This recurrence relation is solved to determine the coefficients of the homotopy perturbation strategy (HPS) that leads to the convergence series of the precise solution. This strategy derives the results in algebraic form that are independent of any discretization. To demonstrate the performance of this scheme, several mathematical frameworks and visual depictions are shown.
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Introduction
Several notable advances in computational approaches have been developed for engineering and scientific applications, including geometrical description, flexible artificial materials, and acoustic wave propagation^{1,2,3}. Partial differential equations (PDEs) have a significant impact on many scientific and engineering fields, including electronics, hydrodynamics, computational motion, physical biology, the engineering of chemicals, dietary fiber, mechanics, material dynamics, and geometrical optics^{4,5,6,7}. Numerous researchers have investigated different methods to derive the analytical results for such PDEs. Utilizing a meshfree approach named the Radial basis function pseudo spectral (RBFPS) method, researchers numerically examined the solutions for both integer and fractional KdV type equations on a finite domain with periodic boundary conditions^{8,9}. Although the computations associated with these approaches are fairly straightforward and certain variables are based on the assumption of a variety of limitations. As a result, many scientists are looking for new techniques to overcome these restrictions. Numerous scientists and other researchers have offered several methods for assessing the analytical findings^{10,11,12}. Several academics and scientists have used HPS^{13,14} to solve complicated physical problems. When employing this method, the solution series converges relatively quickly in most cases. The authors^{15,16} used HPS to the oscillation challenges in nonlinearity and demonstrated its effectiveness in providing analytical findings.
The wave problem is a partial differential equation for a scalar function offering wave propagation in the motion of fluids. Wazwaz^{17} used the VIM to study linear and nonlinear problems. Ghasemi et al.^{18} computed the effective results for twodimensional nonlinear differential problem using HPS. Keskin and Oturanc^{19} proposed a new method for the analytical results of wave problems. Dehghan et al.^{20} applied HAM to derive the approximation results for PDEs. Ullah et al.^{21} proposed a homotopy optimum technique to generate algebraic findings for wave challenges. Thorwe and Bhalekar^{22} used Laplace transform method to obtain approximation solution of partial integrodifferential equations. Adwan et al.^{23} presented analytical findings for multidimensional wave challenges and validated the proposed technique. The HPS was applied for the approximate solutions of wave equations by Jleli et al.^{24}. The researchers in^{25} proposed the finite element technique and separated the wave system to derive their approximate solution. These approaches include a lot of limitations and assumptions during the estimation of problems.
The current study aims to use a new iterative technique for multidimension challenges by combining \(\mathbb {S}\)T and HPS. In the present work, we eliminate these drawbacks and constraints by offering a novel iterative method for these multidimensional wave issues. An iteration series with approximate findings that are close to the precise outcomes is produced by this new strategy. This technique performs more effectively and produces more appealing outcomes for the present challenges. The following is a description of this work: the concept of Sawi integral transform is given in “Fundamental concepts”. In “Formulation of new iterative strategy”, we build our new strategy to achieve the multidimension model findings. The convergence theorem has been laid out in “Convergence of new iterative strategy”. In “Numerical applications”, a few numerical examples are examined to demonstrate the power of new technique and we offer the conclusion at the end of “Conclusion remarks and future work”.
Fundamental concepts
In this portion, we give few fundamental features of \(\mathbb {S}\)T that are helpful in the development of our new strategy.
Sawi transform
Definition 2.1
Let \(\vartheta \) be a function of \(\eta \ge 0\). Then, \(\mathbb {S}\)T is^{26,27}
in which \(\mathbb {S}\) represents the symbol of \(\mathbb {S}\)T. Now
where \(Q(\theta )\) shows the function of \(\vartheta (\eta )\). The \(\mathbb {S}\)T of \(\vartheta (\eta )\) for \(\eta \ge 0\) exist if \(\vartheta (\eta )\) tends to exponentially ordered and piecewise continuous. The existence of \(\mathbb {S}\)T for \(\vartheta (\eta )\) is basically predicated on the two requirements mentioned.
Proposition 1
Now, we define the basic propositions of \(\mathbb {S}\)T. Therefore, let \(\mathbb {S}\{\vartheta _{1}(\eta )\}=Q_{1}(\theta )\) and \(\mathbb {S}\{\vartheta _{2}(\eta )\}=Q_{2}(\theta )\)^{28,29}, thus
Proposition 2
Now, for the differential characteristics of \(\mathbb {S}\)T, we consider \(\mathbb {S}\{\vartheta (\eta )\}=Q(\theta )\), the differential characteristics are defined as^{30}
Formulation of new iterative strategy
This section examines the approximate solutions of 1D, 2D, and 3D wave problems by using new iterative strategy (NIS). This approach can be used to solve differential equations based on initial conditions. We stated that the construction of this approach does not depend on integrating and other suppositions. Let a differential equation like that
subjected to initial conditions
where \(f(\vartheta )\) denotes the nonlinear element, \(f(x_{1},\eta )\) is known component of arbitrary constants \(a_{1}\) and \(a_{2}\), and \(\vartheta (x_{1},\eta )\) is a uniform function. Moreover, we may express Eq. (4) like this:
A function of a real variable can be transformed into an expression of a complex variable using an integral transformation known as the Sawi transform in mathematics. This transformation has several uses in the fields of science and technology because it serves as a tool to deal with differential problems.
Apply \(\mathbb {S}\)T on Eq. (6), we get
Using the formula as defined in Eq. (3), it yields
Thus, \(Q(\theta )\) is derived as
On inverse \(\mathbb {S}\)T on Eq. (7), we get
Use the condition (5), we obtain
This Eq. (8) is known as the development of NIS of Eq. (4).
Let HPS be introduced as
where as the nonlinear variable \(f(\vartheta )\) is stated as
Hence, we are able to generate \(H_{n}'s\) polynomial as
Use Eqs. (9)–(11) in Eq. (8) and evaluate the similar components of p, it yields
Following this procedure, which results in
Hence, Eq. (12) provides a closedform approximation to the differential problem.
Convergence of new iterative strategy
Theorem 4.1
Let \([a,b]\times [0,T]\) be the rectangular interval on which the Banach space \(B\equiv C([a,b]\times [0,T])\) is defined. Then Eq. (12) \(\vartheta (x_{1},\eta )=\sum _{i=0}^{\infty }\vartheta _{i}(x_{1},\eta )\) is convergent series, if \(\vartheta _{0}\in B\) is bounded and \(\left\ \vartheta _{i+1}\right\ \le \left\ \vartheta _{i}\right\ , \forall \vartheta _{i} \in B\), and for \(0<\delta <1\).
Proof
Taking the series \(\left\{ F_r\right\} \) as a partial sum of Eq. (12), we obtain
Next, we establish that \(\left\{ F_r\right\} _{r=0}^{\infty }\) is a Cauchy sequence in B in order to validate this theorem. Therefore,
Hence, for any pair \(r, n \in N\), where \(r>n\), we have
where \(\beta =\frac{\left( 1\delta ^{rn}\right) }{(1\delta )} \delta ^{n+1}\). Since \(\vartheta _0(x_{1}, \eta )\) is bounded, therefore \(\left\ \vartheta _0(x_{1}, \eta )\right\ <\infty \). As n grows and \(n \rightarrow \infty \) leads to \(\beta \rightarrow 0\) for \(0<\delta <1\), so
Consequently, \(\left\{ F_r\right\} _{r=0}^{\infty }\) in B is a Cauchy sequence. It follows that the series solution of Eq. (12) is convergent. \(\square \)
Theorem 4.2
If \(\sum _{k=0}^n \vartheta _k(x_{1}, \eta )\) represents the approximate series solution of Eq. (4), then maximal absolute error can be determined by
in which \(\delta \) is a digit which means \(\dfrac{\left\ \vartheta _{i+1}\right\ }{\left\ \vartheta _i\right\ } \le \delta \).
Proof
Using Eq. (15) from Theorem (4.1), we obtain
Here, \(\left\{ F_r\right\} _{r=0}^{\infty } \rightarrow \vartheta (x_{1}, \eta )\) as \(r \rightarrow \infty \) and from Eq. (13), we get \(F_n=\sum _{k=0}^n \vartheta _k(x_{1}, \eta )\),
Now, \((1\delta ^{rn})<1\), since \(0<\delta <1\)
\(\square \)
Hence, the proof.
Numerical applications
We provide some numerical tests for showing the precision and reliability of NIS. We can observe that, as compared to other approaches, this method is substantially easier to apply in obtaining the convergence series. We illustrate the physical nature of the resulting plot distribution with graphical structures. Furthermore, a visual depiction of the error distribution demonstrated the near correspondence between the NIS outcomes and the precise results. We can compute the absolute error estimates by evaluating the exact solutions with the NIS values.
Example 1
Consider the one dimensional wave equation
subjected to initial
and boundary conditions
Apply \(\mathbb {S}\)T on Eq. (21), we get
Using the formula as defined in Eq. (3), it yields
Thus, \(Q(\theta )\) reveals as
On inverse \(\mathbb {S}\)T, we have
Thus HPS yields such as
By assessing comparable components of p, we arrive at
Likewise, we can consider the approximation series in such a way that
which can approaches to
Figure 1 shows periodic soliton waves in two diagrams: Fig. 1a 3D surface plot for analytical results of \(\vartheta (x_{1},\eta )\) and Fig. 1b shows 3D surface plot for precise results of \(\vartheta (x_{1}, \eta )\) for onedimensional wave equation at \(10\le x_{1} \le 10\) and \(0\le \eta \le 0.01\). The effective agreement among analytical and the precise results at \(0\le x_{1} \le 5\) along \(\eta =0.1\) is shown in Fig. 2, which further validates the strong agreement of NIS for example (5.1). We can precisely propagate any surface to reflect the pertinent natural physical processes, according to this technique. The error distribution among analytical and precise results for \(\vartheta (x_{1}, \eta )\) along \(x_{1}\)space at different values is shown in Table 1. This contraction demonstrates the effectiveness of proposed technique in finding the closedform results for the wave problems.
Example 2
Consider the twodimensional wave equation
subjected to initial
and boundary conditions
Apply \(\mathbb {S}\)T on Eq. (27), we get
Using the formula as defined in Eq. (3), it yields
Thus, \(Q(\theta )\) reveals as
On inverse \(\mathbb {S}\)T, we have
Thus HPS yields such as
By assessing comparable components of p, we arrive at
Likewise, we can consider the approximation series in such a way that
which can approaches to
Figure 3 shows periodic soliton waves in two diagrams: Fig. 3a: 3D surface plot for analytical results and Fig. 3b: 3D surface plot for precise results of \(\vartheta (x_{1},y_{1},\eta )\) for twodimensional wave equation at \(5\le x_{1} \le 5\), \(0\le \eta \le 0.01\) along \(y_{1}=0.5\). The effective agreement among analytical and the precise results at \(0\le x_{1} \le 5\), \(y_{1}=0.1\) along \(\eta =0.1\) is shown in Fig. 4, which further validates the strong agreement of NIS for example (5.2). We can precisely propagate any surface to reflect the pertinent natural physical processes, according to this technique. The error distribution among analytical and precise results for \(\vartheta (x_{1},y_{1},\eta )\) along \(x_{1}\)space at different values is shown in Table 2. This contraction demonstrates the effectiveness of proposed technique in finding the closedform results for the wave problems.
Example 3
Consider the threedimensional wave equation
subjected to initial
and boundary conditions
Apply \(\mathbb {S}\)T on Eq. (33), we get
Using the formula as defined in Eq. (3), it yields
Thus, \(Q(\theta )\) reveals as
On inverse \(\mathbb {S}\)T, we have
Thus HPS yields such as
By assessing comparable components of p, we arrive at
Likewise, we can consider the approximation series in such a way that
which can approaches to
Figure 5 shows two diagrams: Fig. 5a: 3D surface plot for analytical results and Fig. 5b: 3D surface plot for precise results of \(\vartheta (x_{1},y_{1},z_{1},\eta )\) for twodimensional wave equation at \(5\le x_{1} \le 10\) and \(0\le \eta \le 0.01\) with \(y_{1}=0.5\) and \(z_{1}=0.5\). The effective agreement among analytical and the precise results at \(0\le x_{1} \le 10\), \(y_{1}=0.5\), \(z_{1}=0.5\) along \(\eta =0.5\) is shown in Fig. 6, which further validates the strong agreement of NIS for example (5.3). We can precisely propagate any surface to reflect the pertinent natural physical processes, according to this technique. The error distribution among analytical and precise results for \(\vartheta (x_{1},y_{1},z_{1},\eta )\) along \(x_{1}\)space at different values is shown in Table 3. This contraction demonstrates the effectiveness of proposed technique in finding the closedform results for the wave problems.
Conclusion remarks and future work
In this article, we successfully applied the new iterative strategy for the approximate results of multidimensional wave problems. This technique uses the recurrence relation to produce the findings of the analysis. The findings obtained from numerical examples show that our technique is simple to implement and has a greater rate of convergence than existing approaches. The Sawi integral transform has the ability to control the global error, which makes it a suitable method for solving problems with rapidly changing solutions. The method is relatively easy to implement, especially for problems with periodic solutions. The 3D figures in the illustrated problems show the periodic soliton waves in the deep well. The physical behavior of the problems is depicted by the 3D graphical representations, and the visual inaccuracy between the exact outcomes and the produced results is represented by the 2D plot distribution. This method requires accurate initial guesses for the solution, which can be challenging in some cases. In terms of its effectiveness and efficiency, the Sawi integral transform is a relatively new method and has not been widely studied or compared to other numerical methods for solving PDEs. However, in the cases where it has been applied, it has shown promising results, with relatively high accuracy and efficiency compared to other methods. This composition of Sawi transform and the homotopy perturbation strategy gives the solution of multidimensional problems which is very useful in wave propagation. This novel iterative technique can also be used to solve other physical chemistry, engineering, and medical research challenges, such as calculating the growth rate of tumors, calculating the total quantity of infecting cells, calculating the amount of viral particles in blood during HIV1 diseases, analyzing the impact of humidity on skew plate vibration, and calculating the amount of chemicals involved in chemical chain reactions in the future.
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Acknowledgements
This research was supported by the Chunhui Project of the Chinese Ministry of Education (202201245).
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J.L.: methodology, writingoriginal draft. M.N.: investigation, M.S.O.: software. Y.A.: supervision, funding project. This paper has been read and approved by all authors.
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Liu, J., Nadeem, M., Osman, M.S. et al. Study of multidimensional problems arising in wave propagation using a hybrid scheme. Sci Rep 14, 5839 (2024). https://doi.org/10.1038/s41598024564775
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DOI: https://doi.org/10.1038/s41598024564775
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