Abstract
In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear Klein–Gordon equations (NLKGEs) with quadratic and cubic nonlinearities subject to appropriate initial conditions. The proposed technique has the advantage of producing an analytical approximation in a convergent power series form with a reduced number of calculable terms. Two test examples from mathematical physics are discussed to illustrate the validity and efficiency of the method. In addition, numerical solutions of the test examples are presented graphically to show the reliability and accuracy of the method. Also, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.
1. Introduction
In many applications of science and engineering such as fluid dynamics, plasma physics, hydrodynamics, solid-state physics, optical fibers, acoustics, and other disciplines, the nonlinear equations appear for modeling physical phenomenon [1]. Recently, a lot of research was carried out on real-life applications modeled by PDEs [2–12]. The Klein–Gordon equation is an important class of partial differential equations and arises in relativistic quantum mechanics and field theory, which is of great importance for the high-energy physicists [13], and is used to model many different phenomena, including the propagation of dislocations in crystals and the behavior of elementary particles [14].
The n-dimensional nonlinear Klein–Gordon equation is given by partial differential equation [15]:with initial conditionswhere is a domain of , is the space variable, is the time variable, is the unknown function, is a nonlinear function of with different types of nonlinearities, is a given function, and are prescribed initial functions, and and are known constants.
In the present work, we are dealing with the approximate analytical solution of the nonlinear Klein–Gordon equation (1) with a nonlinear function having the form .
That is,subject to the initial conditionswhere and .
Remark 1. Equation (3) is called the Klein–Gordon equation with quadratic nonlinearity if and with cubic nonlinearity if .
In recent years, many researchers have paid attention to study the solutions of NLPDEs by various methods (see [16] and the references therein).
The reduced differential transform method (RDTM) was first proposed by Keskin and Oturanc [17] for solving partial differential equations (PDEs). It is an iterative procedure based on the use of the Taylor series solution of differential equations. RDTM has been successfully applied to solve various nonlinear partial differential equations [18–28].
The main aim of this study is to obtain the approximate analytical solutions for two-dimensional nonlinear Klein–Gordon equation (NLKGE) with quadratic and cubic nonlinearities, since most of the research focused on the numerical solutions for this problem. The reduced differential transform method is used for this purpose for several reasons. The first reason is that the method has not previously been applied to solve this problem. Secondly, this method can directly be applied to NLKGE. Thirdly, this method can reduce the size of the calculations and can provide an analytic approximation, in many cases exact solutions, in rapidly convergent power series form with elegantly computed terms (see [29] and the references therein). Moreover, the reduced differential transform method (RDTM) has an alternative approach of solving problems to overcome the demerit of discretization, linearization, or perturbations of well-known numerical and analytical methods such as Adomian decomposition, differential transform, homotopy perturbation, and variational iteration [29, 30].
The structure of the remaining parts of this paper is organized as follows. In Section 2, we begin with some basic definitions and describe the proposed method. In Section 3, we discuss the convergence analysis of the method. In Section 4, we apply the reduced differential transform method to solve two illustrative examples in order to show its ability and efficiency. Concluding remarks are given in Section 5.
2. Reduced Differential Transform Method
The basic definitions and operations of the two-dimensional reduced differential transform method [30–34] are introduced as follows.
Definition 1. If the function is analytic and differentiable continuously with respect to time and space in the domain of interest, thenwhere the t-dimensional spectrum function is the transformed function of .
In this study, the lower case represents the original function while the upper case stands for the transformed function.
Definition 2. The inverse differential transform of is defined asCombining equations (5) and (6), we obtainNote that the function can be written in a finite series as follows:where the tail function is negligibly small. Therefore, the exact solution of the problem is given by
Lemma 1. If , then .
Proof. By using Table 1, we haveBy Leibnitz’s theorem, this can be written as
Lemma 2. If , then .
Proof. Observe that . So, by using Table 1, we havewhere is the reduced differential transform of (Lemma 1).
Thus,To demonstrate the basic concepts of RDTM, we consider the two-dimensional nonlinear Klein–Gordon equation from equation (3) subject to the initial conditions given by equation (4).
By RDTM, from equation (3), we obtain the following recurrence relation:where and are the transformed forms of the nonlinear terms and , respectively. Similarly, by RDTM, from the initial condition (4), we getrespectively.
Substituting equation (15) into equation (14), we get the set of values . Then, the inverse reduced differential transform of these sets of values provides the n-term approximate solution asTherefore, the exact solution of the problem is given by
3. Convergence Analysis
In this section, the convergence analysis of the approximate analytical solutions which are computed from the application of RDTM [30] is presented.
Let us consider equation (3) in the following functional equation form:where is a general nonlinear operator involving both linear and nonlinear terms.
According to RDTM, equation (3) has a solution of the form
It is noted that the solutions by RDTM are equivalent to determining the sequencesby using iterative schemeassociated with the functional equation
Hence, the solution obtained by RDTM, , is equivalent to .
The sufficient condition for convergence of the series solution is given in the following theorem.
Theorem 1. Let be an operator from a from Hilbert space into . Then, the series solution converges whenever there is “” such that , and (see for proof Ref. [30]).
Theorem 2. Let be a nonlinear operator satisfying Lipschitz condition from Hilbert space into and be exact solution of the given NLKGE. If the series solution converges, then it converges to (see for proof Ref. [30]).
Definition 3. For , we defineThen, we can say that the series approximate solution converges to the exact solution when for
4. Test Examples
In this section, we consider two test examples that demonstrate the performance and efficiency of the RDTM for solving two-dimensional inhomogeneous nonlinear Klein–Gordon equations.
Example 1. Consider the two-dimensional NLKGE (3) with .
That is,with initial conditionsApplying properties of RDTM to equation (24), we obtain the following recurrence relation:whereis the transformed form of and is the transformed form of the function .
By similar scheme from (25), we obtainLetandApplying properties of RDTM on equations (29) and (30), we getandrespectively.
Thus,We then compute the set of values of .
Substituting equations (27), (31), and (32) into equation (26) with , we obtainrespectively.
So,Hence, from equation (26), we obtainAccordingly,Thus, , and hence when , equation (26) yieldsUsing similar procedure, we obtain the following results:and so on.
Substituting all these values in equation (16), we obtainand hence by equation (17), the exact solution of Example 1 is .
To make a decision on the convergence of the solution, we compute using Definition 3 and Theorem 1 as follows.
First, let us take ; then,Therefore, by Theorem 2, the solution of the given NLKGE by RDTM has a convergent solution.
Figures representing the numerical solution of Example 1 are presented below.
Figures 1(a) and 1(b), respectively, show the physical behavior of the approximate solution by RDTM of Example 1 at for equal values of and in and the associated absolute errors. From these figures, it can be observed that the numerical results by RDTM are in excellent agreement with the exact solution of the problem under consideration.
In Figure 1(c), we have plotted absolute errors by the proposed method corresponding to and . This figure represents the errors of numerical results by RDTM that could appear for the long time range.
In Figure 1(d), we have compared the approximate solutions by RDTM and exact solutions corresponding to and , at equal values of and in . This figure asserts that the numerical results obtained by RDTM are in a good agreement with the exact results, especially for . In general, Figures 1(a)–1(d) show convergence of the approximate solution obtained by RDTM to the exact solution rapidly.
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Example 2. Consider the two-dimensional NLKGE (3) with , and .
That is,with initial conditionsBy applying RDTM on equation (42), the following recursive equation is obtained:whereis the transformed form of and is the transformed form of the function .
The reduced differential transform of equation (43) isThe transformed function can be well expressed as follows.
Let andThen, the reduced differential transform of the equations (47) and (48) isandrespectively.
Thus,We now compute the set of values of .
When , from equations (45), (49), and (50), we obtainrespectively.
Thus, , and therefore, equation (44) becomesWhen , we haveAs a result, . So, equation (44) impliesIn a similar manner, we obtainand so on.
Then, the differential inverse transform of the set of values isTherefore, by equation (17), the solution of Example 2 is which is exactly the same as the result obtained by backward group preserving scheme [35].
To make a decision on the convergence of the solution, we compute using Definition 3 and Theorem 1. For this purpose, let us take ; then,So, by Theorem 2, the given NLKGE has a convergent solution.
Figures representing the numerical solution of Example 2 are presented below.
In Figures 2(a) and 2(b), the physical behavior of the approximate solution by RDTM of Example 2 at for equal values of and in and the associated absolute errors are shown, respectively. From these figures, it can be seen that the numerical results by RDTM are in excellent agreement with the exact solution of the problem under consideration.
In Figure 2(c), we have plotted absolute errors corresponding to and for equal values of and in of Example 2. This figure represents the errors of numerical results by RDTM that could appear for the long time range of the given NLKGE.
In Figure 2(d), we have compared the approximate solutions by RDTM corresponding to and with the exact solution, at equal values of and in . This figure shows that the numerical results obtained by RDTM are in excellent agreement with the exact results.
Moreover, for equal values of x and y in different intervals, the shape of the parabola changes due to the wave nature of the NLKGEs as shown in Figures 2(e) and 2(f). In general, from Figures 2(a)–2(f), it is evident that the approximate solution rapidly converges to the exact solution.
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5. Conclusion
The reduced differential transform method (RDTM) has been applied to find an approximate analytical solution of the two-dimensional nonlinear Klein–Gordon equations. The main advantage of the RDTM is that it provides the user an analytical approximation, in many cases, an exact solution, in a rapidly convergent power series form with elegantly computed terms. Two test examples are presented to show the validity of the method under consideration. The approximate solutions of Example 2 obtained by RDTM are in excellent agreement with the exact solution obtained by the backward group preserving scheme [35]. The result is also depicted in Figure 2.
Consequently, this technique can be applied to many NLPDEs arising from science and engineering fields without requiring linearization, discretization, or perturbation.
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
WGB proposed the main idea of this paper. YOM and AKG supervised her work from the first draft to revision. All authors approved the final manuscript for submission.
Acknowledgments
The authors thank Jimma University, College of Natural Sciences, and Department of Mathematics for providing the necessary resources during this research.