KdV Equation Solution by Double Laplace Adomian Decomposition Method and Its Convergence Analysis

Habib, Umar; Zeb, Salman; Shah, Kamal; Hussain, Sardar Muhammad

doi:https://doi.org/10.1155/2022/1132557

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Applications Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

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Applications and Utilization of Laplace Transform in Science and Engineering Problems

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Research Article | Open Access

Volume 2022 | Article ID 1132557 | https://doi.org/10.1155/2022/1132557

KdV Equation Solution by Double Laplace Adomian Decomposition Method and Its Convergence Analysis

Umar Habib,¹Salman Zeb,¹Kamal Shah,^1,2and Sardar Muhammad Hussain³

Academic Editor: Ardashir Mohammadzadeh

Received04 Jun 2022

Revised19 Aug 2022

Accepted26 Aug 2022

Published12 Sept 2022

Abstract

In this paper, the double Laplace Adomian decomposition method (DLADM) is utilized for the solution of the nonlinear Korteweg–de Vries (KdV) equation. The convergence analysis of the suggested approach has been carried out and some examples of the KdV equation are solved for demonstrating the proposed approach. We also compared the obtained series solutions with existing method solutions which show close agreement with each other.

1. Introduction

A differential equation (DE) is one which involves the derivative of one or more dependent variables in terms of one or more independent variables [1]. DE has a long history that started in century when Newton and Leibniz developed calculus [2]. In century the major work is done in the Leibnizian convention and its multi-component formation is expanded which resulted in PDEs. New masters emerged such as Lagrange, Laplace, and Euler who made significant contributions to astronomy, mechanics, and continuous media. In the century various types of equations, methods, and their solutions were developed such as special functions and Fourier analysis. Also in this century, with the influence of Maxwell, most developments were made in the field of electricity and magnetism, heat theory, and optics. Relativity theory was enriched and most applications were made to the field of dynamical systems and quantum mathematics in the century [3]. Differential equations are used in a variety of recent research work in scientific fields and technology including physics, engineering, and biology. For example, Goodwine [4] studied the applications of DE in aeronautical and mechanical engineering. Moreover, Sumithra [5] investigated DE and modeled the variation of physical quantities such as velocity, displacement, pressure, temperature, current, and voltage with respect to time and position. Applications of DEs in physics such as Newton’s second law of motion, the Euler–Lagrange, Hamilton’s, and Einstein’s field equations has also been discussed. DEs also has a wide range of applications in biology for example Sumithra [5] explored the Lotka–Volterra and Predator–Prey equations which shows the importance of DEs in biology. He also discussed other significant DEs like equations of population growth and radioactive decay.

Nonlinear partial differential equations (NLPDEs) are found in various physical models including plasma physics and fluid dynamics. Nonlinear system of PDEs have also received great attention and they have been occurred in a variety of processes of biology and chemistry and it has various uses in these disciplines [6]. Sine–Gordon, Klein–Gordon, KdV, and Boussinesq equations are some of the most significant examples of NLPDEs [7]. As the superposition principle does not apply to NLPDEs, theretofore numerical solutions are usually employed to solve to such kinds of equations. The KdV problem is an NLPDE of the third order idealized in 1877 by Boussinesq [8]. The equation is named after two Dutchmen Korteweg and Devries, who originally obtained the problem in 1895 [9]. The KdV problem was derived essentially from a hydromagnetic model of cold plasma [10, 11]. It is also been implied to depict numerous essential physical processes in different fields like in plasma physics, harmonic crystals [6], and shallow water waves [12] among others. It depicts shallow sea waves with a long wavelength and a tiny amplitude. Some approaches like numerical strategies, soliton solution, inverse scattering transform, and least action principle [13] are found in the literature to solve the KdV problem.

The Laplace transform is one of the approaches for investigating the analytical solution of linear differential equations. Various famous practitioners have used the aforementioned method extensively [14, 15]. Kilicman and Gadain [16] expanded the idea of the Laplace transform to double Laplace transform (DLT). In comparison to the single Laplace transform, the DLT is more useful and appropriate [17]. However, there is a scarcity of work on the DLT [18]. There are certain uses of the DLT in combination with decomposition methods to handle NLPDEs [19–23]. It is thought that a universal strategy for discovering accurate solutions to NLPDEs has not yet been discovered [6]. For approaches related to fuzzy systems we refer the readers to [24–26]. The ADM on the other hand is thought to be capable of handling NLPDEs. The ADM is used to solve linear as well as nonlinear equations. This method has various advantages over other strategies, most notably are as it avoids linearization and perturbation when solving nonlinear equations. It provides a precise and explicit answer with fewer calculations and avoids physically implausible assumptions. Several scholars have looked into the convergence of Adomian’s approach [27–30]. The method generates a series solution and converges to its exact solution if it exists.

In this work, we consider a non-linear KdV problem of third order with initial and boundary conditions in the following form:where is any positive real number, , are real numbers [31]. The solution of (1) has been obtained using DLADM and its convergence analysis has also been studied. The suggested approach has been employed on some examples for its applications and the obtained solutions are compared with existing method solutions in the literature.

2. Preliminaries

In this section, some basic concepts are described which are used in our main work.

2.1. Double Laplace Transform

DLT of a function is defined as follows:where denotes DLT and is the DLT of .

2.2. Differentiation Property

Let (x, t) be continuous and be piecewise continuous functions in and , . Let be of exponential order as , . Then, the DLT of and exists and are defined as follows:

We will utilize the following formulae for DLT of derivatives of functions:where and are first and third order partial derivatives w.r.t. and respectively, denotes double Laplace transform w.r.t. and .

2.3. Double Inverse Laplace Transform

For an analytical function where are complex numbers and such that a, b where and are real constants, then the inverse double Laplace transform = is defined as follows:

The following Table 1 contains some important functions and their Laplace transform used in this work.

For further study about DLT, see [16, 18, 32, 33].

3. Solution of KdV Equation Using DLADM

Considering (1) and utilizing the DLT on it where , while making use of (4) and (5), we obtain the following equation:where and are the DLTs of and respectively, and , , and are the LT of and , respectively.

(7) has been simplified and rearranged, and we obtain the following equation:

Using the inverse LT in (8), we obtain the following equation:

Using the ADM, the solution function and the non-linear terms are treated as follows:

The Adomian polynomial is given by the following equation:

Using (10) and (11), (9) becomes the following equation:

Utilizing the Adomian decomposition formula (9) and (12) for , we have the following equation:

(13) implies that

Comparing the terms we have the following equation:

The series solution is

4. Convergence Analysis

This section is devoted to the convergence analysis of DLADM for KdV equation. The general KdV equation is

For (Hilbert Space) define by the following equation:where with , such that

Theorem 1. DLADM is used for the nonlinear KdV equation approaches to the exact solution if the following hypotheses holds:
and hemi-continuous operator “K”For , there exist with ,

Proof. The operator form of (18) for all is given by the following equation:Similarly for all we have the following equation:Subtracting (24) from (23) and putting where then simplifying, we obtain the following equation:The inner product form of (25) isas is a differential operator in H therefore utilizing the Schwartz inequality for , we obtain the following equation:Since , henceutilizing the triangle inequality , (28) implies thatUsing the fact that for all there must exists constants say and such that , with , , (29) becomeswhich impliesSimilarly for differential in and for constant , we obtain the following equation:which impliesMaking use of (31) and (33), (26) becomesTaking , we obtain the following equation:hence holds.
To verify , for all the inner product form of (25) isFor differential operator and any , we have the following equation:utilizing and making use of triangle inequality, (37) becomesUsing that , , we obtain the following equation:Similarly for differential operator we have the following equation:and making use of (39) and (40) in (36) we obtain the following equation:Taking , (41) implieswhich shows that holds and the proof is completed.

5. Applications

In this section, we solved some examples of the KdV problem to explain the applications of the proposed method for nonlinear PDEs. Comparison of DLADM solutions with existing method solutions are provided in Tables 2 and 3.

Example 1. We consider the following problem with conditions:where is the Dirac Delta function.

Applying DLT to (43) and then utilizing (4) and (5), we obtain the following equation:

Utilizing the initial and boundary conditions and simplifying we obtain the following equation:where we have used the following equation:

Taking inverse DLT of (45), we have the following equation:

Using the ADM, from (47) we have the following equation:

By comparing terms we obtain evaluating (12) for we obtain , putting this value in (49) and simplifying, we obtain the following equation:Similarly,

As the series solution is given by the following equation:

Putting values we obtain the following solution:

5.1. Convergence Analysis

For all and , , (34) implies that

Taking , (54) implies that

Hence holds.

To verify , for all and , , (41) impliestaking , we ge the following from (56):

Therefore also holds.

Example 2. Considering the following problem

and applying DLT on the problem while using the formulae (4) and (5) and utilizing initial and boundary conditions. Simplifying, we obtain the following equation:

Using the ADM, we have the following equation:

By comparison, we have the following equation:and

Using the convolution theorem for , (62) becomes the following equation:

In the same way we can find ., and the solution in series form is then given by the following equation:

5.2. Convergence Analysis

For all and , , the result specified in (20) gives the following equation:

Taking , (65) implieshence holds.

To verify , for all and , , the result given in (41) implies

Taking , we have from (67), the following:which implies holds.

6. Conclusion

In this paper, the KdV problem solution has been obtained via DLADM. A convergence analysis of the proposed strategy is also carried out for studying its convergence analysis in order to show the accuracy of the resulting series solution. Furthermore, examples of the KdV equation were solved using DLADM and the results are compared with the SAM procedure which agree with each other. In the future, the DLADM procedure can be studied for finding solutions of nonlinear PDEs arising in different fields of science and engineering.

Data Availability

No data were required to perform this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Umar Habib et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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