Abstract
This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.
1. Introduction
The sine-Gordon (SG) equation is a nonlinear hyperbolic PDE, which was originally considered in the nineteenth century in the course of study of surfaces of constant negative curvature often used to describe and simulate the physical phenomena in a variety of fields of engineering and science, such as nonlinear waves, propagation of fluxons, and dislocation of metals [1–4]. Because sine-Gordon equation leads to different types of soliton solutions, it has been receiving an enormous amount of attention. Soliton solution travels without experiencing any deformation through the medium, even when it collides with another soliton. The solitons, identified in many wave and particle systems, are of importance in the theory of nonlinear differential equations. As one of the crucial equations in nonlinear science, the sine-Gordon equation has been constantly investigated and solved numerically and analytically in recent years [5–8]. For the one-dimensional sine-Gordon equation, Maitama and Hamza [5] introduced analytical method called the Natural Decomposition Method (NDM) for solving nonlinear Sine-Gordon equation. Fayadh and Faraj [9] also applied combined Laplace transform method and VIM to get the approximate solution of the one-dimensional sine-Gordon equation.
For the two-dimensional sine-Gordon equation, Su [6] obtained numerical solution of two-dimensional nonlinear sine-Gordon equation using localized method of approximate particular solutions. In [10], the author developed and analyzed an energy-conserving local discontinuous Galerkin method for the two-dimensional SGE on Cartesian grids. Duan et al. [11] proposed a numerical model based on lattice Boltzmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, and the method was extended to solve the other two-dimensional wave equations, such as nonlinear hyperbolic telegraph equation as indicated in [12]
In 2020, [13] developed a local Kriging meshless solution to the nonlinear ()-dimensional sine-Gordon equation. The meshless shape function is constructed by Kriging interpolation method to have Kronecker delta function property for the two-dimensional field function, which leads to convenient implementation of imposing essential boundary conditions.
In paper [14], the authors constructed kink wave solutions and traveling wave solutions of the ()-dimensional sine-Gordon equation from the well-known AKNS system. For more related research about solving the sine-Gordon equations, readers may refer to [15–20].
The Laplace transform method (LTM) is one of the integral transform methods that have been intensively used to solve linear and nonlinear equations [21]. The Laplace transform method is used frequently in engineering and physics; the output of a linear time invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter is easier to solve because of its algebraic form. The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering [22–25]. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform.
Recently, the concept of single Laplace transform is extended to double Laplace transform to solve some kind of differential equations and fractional differential equations such as linear/nonlinear space-time fractional telegraph equations, functional, integral, and partial differential equations [26–28]. Dhunde and Waghmare [29] applied the double Laplace transform method for solving a one-dimensional boundary value problems. Through this method, the boundary value problem is solved without converting it into ordinary differential equation; therefore, there is no need to find complete solution of ordinary differential equation. This is the biggest advantage of the proposed method.
Furthermore, different scholars extended the double Laplace transform method to triple Laplace transform (TLT) to solve two-dimensional nonlinear partial differential equations arising in various natural phenomena. In [30], the authors used triple Laplace transform method to the solution of fractional-order partial differential equations by using Caputo fractional derivative. Through [31–34], the triple Laplace transform method was applied to obtain the solution of fractional-order telegraph equation in two dimensions, linear Volterra integro-differential equations in three dimensions, third-order Mboctara equations, and the proof of some of its properties like linearity property, change of scale property, first shifting property, and convolution theorem property, and differential property and triple integral property are given.
Similar to the Laplace transform method, an iteration method (IM) is a fascinating task in an applied scientific branches to find the solution of nonlinear differential equation. The iterative procedure of the proposed method leads to a series, which can be summed up to find an analytical formula, or it can form a suitable approximation [35]. The error of the approximation can be controlled by properly truncating the series [36]. More surprisingly, an IM has showed effective and more rapid convergent series solution (see [37]).
The purpose of this paper, is to apply triple Laplace transform (TLT) and iterative method (IM) developed in [38] to find the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to appropriate initial and boundary conditions. Dhunde and Waghmare in [37, 39] applied double Laplace transform iteration method (TLTIM) to solve nonlinear Klein-Gordon and telegraph equations. By this method, the noise terms disappear in the iteration process, and single iteration gives the exact solution.
In the present study, we are interested in the following two-dimensional sine-Gordon equation [40, 41]. subject to the initial conditions and boundary conditions (Cauchy type BCs) within the problem domain of for . In Equation (1), the parameter is the damping factor for the dissipative term. When , this equation will be reduced to an undamped sine-Gordon equation and when to the damped one. The function represents Josephson current density, and the functions and are specified wave modes and velocity.
The remaining parts of this paper is structured as follows. In Section 2, we begin with some basic definitions, properties, and theorems of triple Laplace transform method. Section 3 illustrates the details of the new iterative method and its convergence. Section 4 presents the description of the model; how the approximate analytical solutions of the given SGE equations is obtained using triple Laplace transform method coupled with iterative method. In Section 5, we apply the proposed method to four illustrative examples in order to show its liability, convergence, and efficiency. Finally, concluding remarks are given in Section 6.
2. Definitions and Properties of Triple Laplace Transform Method
In this section, we give some essential definitions, properties, and theorems of triple Laplace transform of partial differential equation, which should be used in the present study.
Definition 1 (see [31]). Let be a function of three variables , and defined in the positive -plane. The triple Laplace transform of the function is defined by whenever the integral exist. Here, denote, and , and are complex numbers. From this definition, we deduce that:
Definition 2 (see [34]). The inverse triple Laplace transform of is given by the complex triple integral formula where denotes , and must be an analytic function for all , and in the region defined by the inequality , and .
Property 3 (see [34]). The triple Laplace transform of second-order partial derivatives are given by
Furthermore, the triple Laplace transform of first-order partial derivatives are given by
2.1. Existence and Uniqueness of the Triple Laplace Transform
Theorem 4 (Existence). Let be a continuous function on the interval which is of exponential order, that is, for some Consider Under this condition, the triple transform, exists for all and is in actuality infinitely differentiable with respect to
Theorem 5 (Uniqueness). Let and be continuous functions defined for and having Laplace transforms, and , respectively. If , then .
For the proof, see [34].
2.2. Some Properties of Triple Laplace Transform
Property 6 (Linearity Property of TLT). If and are two functions of , and such that and , then where and are constants.
Property 7 (Change of Scale Property). Let then where , and are nonzero constants.
For the proof, see [31, 33, 34].
Property 8 (First Shifting Property). If then
For the proof, see [31, 33, 34].
Property 9 (Second Shifting Property). If then where is the Heaviside unit step function defined by
Property 10. If then , for all , and .
For the proof, see [31].
3. The New Iterative Method
Consider the following functional equation [38]: where is a nonlinear operator in a Banach space such that , and is a known function.
, and is assumed to be the solution of Equation (15) having the series form:
The nonlinear operation can then be decomposed as
Using Equations (16) and (17), Equation (15) is equivalent to
From Equation (18), we define the following recurrence relation:
Thus, and hence,
Therefore, the term approximate solution of Equation (16) is given by
Theorem 11. If is a continuously differentiable functional in a neighborhood of and for all , then the series is absolutely convergent [35].
4. Description of the Method
Steps to be followed to apply triple Laplace Transform properties as in Table 1 coupled with new iterative method are as follows:
Step 1. Applying triple Laplace transform on both sides of Equation (1) and using Property 3, we get where is the source functioning Equation (1).
Step 2. Now, employing double Laplace transform to Equations (2) and (3), we have respectively.
By substituting Equations (24) and (25) into Equation (23) and simplifying, we obtain
Step 3. By implementing the triple inverse Laplace transformation of Equation (26), we obtain
Step 4. Assume that is the solution of Equation (1).
Then, substituting Equation (28) into Equation (27), we obtain
Step 5. By implementing the new iterative method, the nonlinear term in Equation (29) is decomposed as
Using Equation (30), Equation (29) is equivalent to
Step 6. Using triple Laplace transform coupled with new iterative method, we introduce the recursive relations and get
Step 7. The solution of Equations (1)–(3) in series form is given by
5. Illustrative Examples
In order to show the validity and effectiveness of the method under consideration, some examples are presented here.
Example 1. Consider the two-dimensional NLSGE with , on the domain . That is, with initial conditions and boundary conditions
Solution: Applying properties of triple Laplace transform on both sides of Equation (34), we get
Applying double Laplace transform to Equations (35) and (36), we obtain respectively.
By substituting Equations (38) and (39) into Equation (37), we get
Simplifying (40) gives us
Applying inverse triple Laplace transform to Equation (41), we get
Now, applying the new iterative method to Equation (42), we obtain the components of the solution as follows:
Now, we define the recurrence relation from Equation (45) for as follows: In the same way, we obtain and so on.
Therefore, the solution of Example 1 by using Equation (33) is
Let us now test the convergence of the obtained series solution. From Equation (42), we have
Thus, for all , we have
Therefore, . where represents the partial derivatives or or
Then,
Therefore, .
Therefore, ,
where represents all the second-order partial derivatives of.
Similarly, by principle of Mathematical induction, we have , for all .
As the condition of Theorem 11 are satisfied, the series solution is absolutely convergent, and hence, the solution obtained by the new iterative method is convergent on the domain of interest.
Example 2. Consider Equation (1) with , on the domain . That is, with initial conditions and boundary conditions
Solution: Applying properties of triple Laplace transform on both sides of Equation (54), we obtain
Applying double Laplace transform to Equations (55) and (56), we obtain respectively.
By substituting Equations (58) and (59) in to Equation (57) and simplifying, we get
Applying inverse triple Laplace transform to Equation (60), we get
Now, applying the new iterative method to Equation (61), we obtain the components of the solution as follows:
Substituting Equation (16) into Equation (61), we obtain the components of the solution as follows: and so on.
Therefore, the solution of Example 2 by using equation (33) is as indicated in Baccouch [10] and Deresse et al. [41].
Next, we test the convergence of the obtained series solution. From Equation (61) we have,
Thus, for all , we have
Therefore, .
Similarly, .
Therefore, .
Similarly,
Therefore, ,
where represents all the second-order partial derivatives of .
Similarly, by principle of Mathematical induction, we have , for all .
As the condition of Theorem 11 are satisfied, the series solution is absolutely convergent, and hence, the solution obtained by the new iterative method is convergent on the domain of interest.
Example 3. Consider Equation (1) on the domain with , and , That is, with initial conditions and boundary conditions
Solution: Applying properties of triple Laplace transform on both sides of Equation (69), we get
Applying double Laplace transform to Equations (70) and (71), we obtain respectively.
By substituting Equations (73) and (74) into Equation (72) and simplifying, we get
Applying inverse triple Laplace transform to Equation (75), we obtain
Now, applying the new iterative method to Equation (76), we obtain the components of the solution as follows: and so on.
Therefore, the solution of Example 3 by using Equation (33) is
This is exactly the same as the result obtained by Kang et al. [16] and Deresse et al. [41].
We now test the convergence of the obtained series solution of Example 3.
From Equation (61), we have and
Thus, for all , we have
Therefore, =.
Similarly, .
Therefore,
Similarly,
Therefore, ,
where represents all the second order partial derivatives of.
Similarly, by principle of Mathematical induction, we have , for all .
As the condition of Theorem 11 are satisfied, the solution series is absolutely convergent, and hence, the solution obtained by the new iterative method is convergent on the domain of interest.
Example 4. Consider Equation (1) with , on the domain . That is, with initial conditions and boundary conditions
Solution: Applying properties of triple Laplace transform on both sides of Equation (85), we obtain
Applying double Laplace transform to Equations (86) and (87), we obtain and respectively.
By substituting Equations (89) and (90) into Equation (88) and simplifying, we get
Applying inverse triple Laplace transform to Equation (91), we obtain
Now, we apply the iteration process.
Substituting Equation (16) into Equation (92), we obtain the components of the solution as follows: and so on.
Therefore, the solution of Example 4 by using Equation (33) is
This result shows an excellent agreement with the one obtained in [12, 40, 41].
Now, we test the convergence of the obtained series solution. From Equation (61), we have
Thus, for all , we have
Therefore, .
Similarly, .
Therefore, =.
Similarly,
Therefore, ,
where represents all the second order partial derivatives of.
Similarly, by principle of Mathematical induction, we have , for all .
As the condition of Theorem 11 are satisfied, the series solution is absolutely convergent, and hence, the solution obtained by the new iterative method is convergent on the domain of interest.
6. Conclusion
In this paper, triple Laplace transform coupled with iterative method is applied to obtain exact solution of two-dimensional nonlinear Sine-Gordon equation subject to initial and boundary conditions. Four illustrative examples are presented to show the validity of the method under consideration. The solutions of Examples 1, 2, 3, and 4 obtained by the proposed method are in an excellent agreement with the same problem that has been considered in [6, 10–12, 16, 40, 41], and further, nontrivial problems that are solved using earlier methods become trivial in the sense that the decomposition consists of only one term, i.e.,. From this study, we concluded that triple Laplace transform coupled with iterative method finds quite practical analytical results with less computational work.
Data Availability
There are no data used in this research study.
Conflicts of Interest
The authors declare that there is no conflict of interest.
Authors’ Contributions
The authors equally contributed and approved the final manuscript for submission.
Acknowledgments
The authors are grateful to thank Jimma University, College of Natural Sciences, and Department of Mathematics, for providing the necessary resources during conducting this research.