Research Article | Open Access
Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method
The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrödinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.
The solutions of linear and nonlinear partial differential equations play an important role in many fields of science and engineering such as solid-state physics, nonlinear optics, plasma physics, fluid dynamics, chemical kinetics, and biology. In this work, we consider two nonlinear partial differential equations. One is the Klein-Gordon equation with power nonlinearity: with the constant or , and another is the nonlinear Schrödinger equation: with a trapping potential and . Here, , and are real constants and . For solving nonlinear partial differential equations including, above nonlinear problems, many powerful methods have been developed such as Bäcklund the transformation , Darboux’s transformation [2, 3], Tanh function , homogeneous balance , Jacobi’s elliptic method [6, 7], F-expansion method [8, 9], and auxiliary equation [10–12]. Recently, iterative-type methods such as Adomian decomposition [13–16], homotopy perturbation [17–20], and variational iteration [21–24] have been used to find accurate approximations by using symbolic mathematical packages: Mathematica, Maple, and Matlab. For solving nonlinear Schrödinger equations, many efficient discretized numerical schemes have been proposed such as split-step finite-difference method (SSFD) [25, 26], split-step Fourier pseudospectral method , pseudospectral method based on Hermit functions , and the method  in one-dimensional problems.
Here, we propose the differential transform method to solve our model problems in (1.1), (1.2). The DTM is close to the Taylor series, but it is different from the conventional high-order Taylor series in determining coefficients. The basic idea of DTM was introduced by Zhou  in solving initial value problems in electrical circuit analysis. The DTM has been employed to solve many important problems science and engineering fields and obtain highly accurate approximations [28–39]. However, it also have some difficulties due to the nonlinearity. Here, we introduce the modified version of the standard DTM, the projected DTM, which is a simple and effective method comparing with the standard DTM.
This paper is organized as follows. A detail description of the projected DTM will be given in Section 2. To our model problems, nonlinear Klein-Gordon and Schrödinger equations, both the standard DTM and the projected DTM, are applied and the corresponding algebraic equations are presented in Section 3. In Section 4, various numerical examples are demonstrated. For each illustrative example, numerical results obtained by DTM, PDTM, and other numerical method are compared. The conclusion will be made in the last section.
2. Description of the Projected Differential Transform Method
In this section, we describe the definition and some properties of the standard DTM. Moreover, we present the basic idea of the projected differential transform method. Suppose a function is analytic in the given domain and . Let us define the differential transform of at by The differential inverse transform of is defined by For , we have In other words, Some fundamental operations for the standard DTM are presented in Table 1. It has been proved that the standard DTM is an efficient tool for solving many linear and nonlinear problems [28–39]. However, there are also some difficulties in DTM. Let us consider the differential transform for which involves six summations in the Table 1. Thus, it is necessary to have a lot of computational work to calculate such differential transform for the large numbers .
In what follows, we introduce the basic idea of modified version of the DTM, the projected DTM. The DTM is based on the Taylor series for all variables. Here, we consider the Talyor series of the function with respect to the specific variable. Assume that the specific variable is the variable . Then we have the Taylor series expansion of the function at as follows:
Definition 2.1. The projected differential transform of with respect to the variable at is defined by
Definition 2.2. The projected differential inverse transform of with respect to the variable at is defined by Since the PDTM results from the Taylor series of the function with respect to the specific variable, it is expected that the corresponding algebraic equation from the given problem is much simpler than the result obtained by the standard DTM. The detail description of the corresponding algebraic equation will be followed in the next section.
3. Comparison of the Standard and Projected DTMs
In this section, we present the comparison of the standard DTM and the projected DTM for solving our model problems, the nonlinear Klein-Gordon and Schrödinger equations. As seen in the previous section, it is the key to obtain the corresponding algebraic equation of the differential transform for the given problems in DTM. For the model problems, we present the corresponding algebraic equations of the differential transform in the standard DTM and the projected DTM at . Firstly, let us consider the following one-dimensional nonlinear Klein-Gordon equation: with initial conditions where , and are known constants and the constant or . The standard DTM for the (3.1) gives the following algebraic equation: where and is the differential transform for the function . The initial conditions give , where are the differential transforms for the function , respectively.
The projected DTM with respect to variable gives the following algebraic equation: where and is the projected differential transform of with respect to the variable . The initial conditions give and .
Here, we apply the DTM to solve the following Schrödinger equation: with an initial condition . Then the standard DTM gives the following with ; is the differential transform of .
The projected DTM with respect to the variable gives the following algebraic equation: where is the differential transform for the trapping potential function and .
4. Illustrative Examples
In order to show the effectiveness of the PDTM for solving the nonlinear Klein-Gordon and Schrödinger equations, several examples are demonstrated. For all illustrative examples, we consider the projected differential transform with respect to the variable . To compare with numerical results obtained by DTM and PDTM, we define the partial sum of both methods as follows:
Example 4.1. Let us consider the nonlinear Klein-Gordon equation (3.1) with quadratic nonlinearity with constants , , and in the interval . The initial conditions are given by That is, and in (3.2).
The Projected DTM. The initial conditions yield and . Substituting and into (3.5) gives
Table 2 shows the numerical results obtained by various methods. Here, the five terms of Adomian decomposition method (ADM), the fourth iteration of variational iteration method (VIM), the partial sum of DTM, and the partial sum of PDTM are tested to compare with numerical results at various values of for each , and . For all test points , numerical approximations obtained by the PDTM agree in three decimal places.
Example 4.2. Let us consider the nonlinear Klein-Gordon equation (3.1) with cubic nonlinearity with constants and in the interval . The initial conditions are given by That is, and in (3.2). The right-hand side function in (3.1) is
The Standard DTM. From the Taylor series expansion of and , initial conditions give the following nonzero differential transforms , : Substituting (4.8) into (3.3) gives the solution in the following form:
The Projected DTM. The initial conditions yield and . Substituting and into (3.5) gives In both methods, DTM and PDTM, the exact solution can be obtained immediately from (4.9), (4.10) as Table 3 shows the and error estimates of the numerical results obtained by the radial basis function method (RBF) , DTM, and PDTM at several values of . In RBF, and are used to obtain approximate solutions. In DTM and PDTM, the partial sums and are tested. Since the DTM and PDTM are based on the Taylor series for the solution at and , respectively, it is obvious that the more closer to , the more accurate numerical approximation can be obtained. This can be shown in Table 3. Moreover, the DTM and PDTM give inaccurate approximated solutions at , but it can be easily improved by adding more terms in the partial sum. In fact, the partial sum gives in and in .
The Standard DTM. From the initial condition it is easy to obtain the following differential transforms , : Given trapping potential function yields the nonzero differential transforms , . A few values of are listed as follows: By substituting all coefficients and into (3.8), all values of can be easily obtained. A few values of are presented as follows: Thus, we have
The projected DTM. The initial condition gives and the trapping potential function yields Substituting and into (3.9) gives From (4.15) and (4.17), the DTM and PDTM yield the following closed form: which is the exact solution. Here, we compare the numerical results obtained by the spectral collocation method with preconditioning (SCMP)  and the proposed method. The test point is the Chebyshev-Gauss-Lobatto points in ; . Suppose that the exact solution , then the absolute errors of the real and imaginary parts between the exact and approximation, and , are defined by where are the approximations obtained by numerical methods. Table 4 shows the absolute error estimates and at each test point for the fixed value . In SCMP, and the partial sum of DTM and the partial sum of PDTM are tested to obtain numerical approximations. It is shown that the standard DTM gives less accurate approximations at and compared with those obtained by PDTM. This is because the standard DTM is the Taylor series expansion at and and are far away from . However, it does not occur in the PDTM at any value of because the PDTM depends on variable , not space variable .
The Standard DTM. From the initial condition we have only nonzero , :