Fractional Subequation Method for Cahn-Hilliard and Klein-Gordon Equations

Abstract

The fractional subequation method is applied to solve Cahn-Hilliard and Klein-Gordon equations of fractional order. The accuracy and efficiency of the scheme are discussed for these illustrative examples.

1. Introduction

Fractional calculus deals with fractional integrals and derivatives of any order [1–8]. Numbers of very interesting and novel applications of fractional partial differential equations (FPDEs) in physics, chemistry, engineering, finance, biology, hydrology, signal processing, viscoelastic materials, fractional variational principles, and so forth, developed mainly in the last few decades [1–15], have led recently to an intensive effort to find accurate and stable numerical methods that are also straightforward to be implemented.

Also, the exact solutions of most of the FPDEs cannot be found easily; thus analytical and numerical methods must be used. Some of the numerical methods for solving fractional differential equations (FDE) and FPDEs were discussed in (see [7, 16–23] and the references therein).

By taking into account the results from [24], a new direct method titled fractional subequation method to search for explicit solutions of FPDEs was proposed [25]. We notice that the method relies on the homogeneous balance principle [26], Jumarie’s modified Riemann-Liouville derivative [27, 28], and the symbolic computation. With the help of this method, some exact solutions of nonlinear time fractional biological population model as well as the (4+1)-dimensional space-time fractional Fokas equation were reported [25]. Recently, the improved fractional subequation method was proposed, and it was used to solve the following two FPDEs in fluid mechanics [29].

In this paper, we suggest the fractional subequation method and utilize this method to solve the following two FPDEs.(a) The space-time fractional Cahn-Hilliard equation in the form 𝐷^𝛼_𝑡𝑢−𝛾𝐷^𝛼_𝑥𝑢−6𝑢𝐷^𝛼_𝑥𝑢²−3𝑢²−1𝐷_𝑥^2𝛼𝑢+𝐷_𝑥^4𝛼𝑢=0, (1)?where 0<𝛼≤1 and 𝑢 are the functions of (𝑥,𝑡). For the case corresponding to 𝛼=1, this equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation, and phase ordering dynamics. On the other hand it becomes important in material sciences [30, 31]. However we notice that this equation is very difficult to be solved and several articles investigated it (see, e.g., [32] and the references therein). (b)The nonlinear fractional Klein-Gordon equation [33] with quadratic nonlinearity reads as 𝐷^2𝛼_𝑡𝑡𝑢−𝐷^2𝛼_𝑥𝑥𝑢+𝛾𝑢−𝛽𝑢²=0,𝛾,𝛽≠0. (2)We notice that the nonlinear fractional Klein-Gordon equation describes many types of nonlinearities. On the other hand the Klein-Gordon equation plays a significant role in several real world applications, for example, the solid state physics, nonlinear optics, and quantum field theory.

The paper suggests a fractional subequation method to find the exact analytical solutions of nonlinear fractional partial differential equations with the Jumarie’s modified Riemann-Liouville derivative of order 𝛼 which is defined as [27] 𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()∫1−𝛼^𝑥0(𝑥−𝜉)^−𝛼−1𝑓(𝜉)(0,1−𝑓)𝛼<0,Γ()𝑑1−𝛼∫𝑑𝑥^𝑥0(𝑥−𝜉)^−𝛼𝑓(𝜉)(0,𝑓−𝑓)0<𝛼<1,_{(𝛼−𝑛)}(𝑥)^(𝑛),𝑛≤𝛼<𝑛+1,𝑛≥1. (3) As pointed out by Kolwankar and Gangal [34], even though the variable 𝑡 is taking all real positive values the actual evolution takes place only for values of 𝑡 in the fractal set 𝐶. We take 𝜒(𝑡)=1 which is a flag function. We conclude that, from the viewpoint of the Kolwankar-Gangal’s local fractional derivative, the parameter 𝛼 is the fractal dimension of time. Thus, the approximate solution is generated by some distribute function defined over the fractal sets in some closed interval [0,1]. They are continuous but not differentiable functions with respect to 𝑡.

The organization of the manuscript is as follows. In Section 2, we briefly explain the fractional subequation method for solving fractional partial differential equations. In Section 3, we extend the application of the proposed method to two nonlinear equations. Finally, Section 4 is devoted to our conclusions.

2. The Method

The fundamental ingredients of the fractional subequation method for solving fractional partial differential equations are described in [29]. The starting point is to consider a given nonlinear fractional partial differential equation in 𝑢(𝑥,𝑡)𝑝𝑢,𝑢_𝑥,𝑢_𝑡,𝐷^𝛼_𝑡𝑢,𝐷^𝛼_𝑥𝑢,…=0,0<𝛼<1, (4) where 𝐷^𝛼_𝑡𝑢 and 𝐷^𝛼_𝑥𝑢 are Jumarie’s modified Riemann-Liouville derivatives of 𝑢, 𝑢=𝑢(𝑥,𝑡) is an unknown function, and 𝑃 is a polynomial in 𝑢 and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.

To specify 𝑢 explicitly, we use in this paper the proposal in four basic steps proposed in [29, 35]; namely, we reduce, by using the traveling wave transformation, the given nonlinear FPDE to a nonlinear fractional differential equation (FDE). After that we assume that the reduced equation obtained previously admits the following solution 𝑢(𝜉)=^𝑛_𝑖=0𝑎_𝑖𝜑^𝑖, (5) where 𝑎_𝑖 (𝑖=0,1,…,𝑛−1,𝑛) are constants to be found, 𝑛 denotes a positive integer determined by balancing the highest order derivatives with the highest nonlinear terms in (4) or the modified one (see [35] for more details), and the new variable 𝜑=𝜑(𝜉) fulfilling the fractional Riccati equation: 𝐷^𝛼_𝜉𝜑=𝜎+𝜑²,0<𝛼≤1. (6) The next step is to substitute (5) along with (6) into the modified version of the equation and to use the properties of Jumarie’s modified Riemann-Liouville derivative, in order to get a polynomial in 𝜑(𝜉). Requesting all coefficients of 𝜑^𝑘 (𝑘=0,1,2,…) to be zero, we end up to a set of overdetermined nonlinear algebraic equations for 𝑐,??𝑘,??𝑎_𝑖??(𝑖=0,1,…,𝑛−1,𝑛).

Finally, assuming that 𝑐,??𝑘,??𝑎_𝑖??(𝑖=0,1,…,𝑛−1,𝑛) are obtained by solving the algebraic equations in the previous step, and substituting these constants and the solutions of (6) into (5), we get the explicit solutions of (4).

3. Main Results

In this section, we apply the method presented in Section 2 for solving the FPDEs (1) and (2), respectively.

Example 1. We consider the space-time fractional Cahn-Hilliard equation as 𝐷^𝛼_𝑡𝑢−𝛾𝐷^𝛼_𝑥𝑢−6𝑢𝐷^𝛼_𝑥𝑢²−3𝑢²−1𝐷_𝑥^2𝛼𝑢+𝐷_𝑥^4𝛼𝑢=0. (7)

Making use of the travelling wave transformation 𝑢=𝑢(𝜉),𝜉=𝑘𝑥+𝑐𝑡. (8) Equation (7) is reduced into a nonlinear FDE easy to solve, namely, 𝑐^𝛼𝐷^𝛼_𝜉𝑢−𝛾𝑘^𝛼𝐷^𝛼_𝜉𝑢−6𝑢𝑘^𝛼𝐷^𝛼_𝜉𝑢²−3𝑢²−1𝑘^2𝛼𝐷_𝜉^2𝛼𝑢+𝑘^4𝛼𝐷_𝜉^4𝛼𝑢=0. (9) Next we suppose that (9) has a solution in the form given below 𝑢=^𝑛_𝑖=0𝑎_𝑖𝜑^𝑖, (10) where 𝜑 obeys the subequation (6).

By balancing the highest order derivative terms and nonlinear terms in (9), gives the value of??𝑛=1, we substitute (10), along with (6), into (9), and then setting the coefficients of 𝜑^𝑗(𝑗=0,1,…,5) to zero, we finally end up with a system of algebraic equations, namely, 𝑎₁𝜎𝑐^𝛼−6𝑎₀𝑎²₁𝜎²𝑘^2𝛼−𝑎₁𝛾𝜎𝑘^𝛼=0,−6𝑎³₁𝜎²𝑘^2𝛼−16𝑎₁𝜎²𝑘^4𝛼−6𝑎²₀𝑎₁𝜎𝑘^2𝛼+2𝑎₁𝜎𝑘^2𝛼𝑎=0,₁𝑐^𝛼−24𝑎₀𝑎²₁𝜎𝑘^2𝛼−𝑎₁𝛾𝑘^𝛼=0,−18𝑎³₁𝜎𝑘^2𝛼−40𝑎₁𝜎𝑘^4𝛼−6𝑎²₀𝑎₁𝑘^2𝛼+2𝑎₁𝑘^2𝛼=0,−18𝑎₀𝑎²₁𝑘^2𝛼=0,−12𝑎³₁𝑘^2𝛼−24𝑎₁𝑘^4𝛼=0. (11) Solving the set of algebraic equations yields 𝑎₀=0,𝑎₁=±𝑖2𝑘^2𝛼, (12) where 𝑘^2𝛼=1/2𝜎, ??𝑐^𝛼=𝑘^𝛼 and 𝜎 denotes an arbitrary constant.

By using (8)–(12) after some tedious calculations, the exact solutions of (7), namely, generalized hyperbolic function solutions (see [24] for their definitions) and generalized trigonometric function solutions are obtained as 𝑢=±𝑖2𝑘^2𝛼√−𝜎tanh_𝛼√−𝜎(𝑘𝑥+𝑐𝑡),𝜎<0,±𝑖2𝑘^2𝛼√−𝜎coth_𝛼√()±−𝜎𝑘𝑥+𝑐𝑡,𝜎<0,2𝑘^2𝛼√𝜎tan_𝛼√𝜎(𝑘𝑥+𝑐𝑡),𝜎>0,±𝑖2𝑘^2𝛼√𝜎cot_𝛼√𝜎(𝑘𝑥+𝑐𝑡),𝜎>0. (13) We stress on the fact that when 𝛼→1 these obtained exact solutions give the ones of the standard form equation of the space-time fractional Cahn-Hilliard equation (7).

Example 2. The next step is to investigate the fractional nonlinear Klein-Gordon equation in the following form: 𝐷^2𝛼_𝑡𝑡𝑢−𝐷^2𝛼_𝑥𝑥𝑢+𝛾𝑢−𝛽𝑢²=0. (14)

To solve (14), we perform the traveling wave transformation 𝑢=𝑢(𝜉),𝜉=𝑘𝑥+𝑐𝑡; (15) therefore (14) is reduced to the following nonlinear fractional ODE, namely, 𝑐^2𝛼𝐷_𝜉^2𝛼𝑢−𝑘^2𝛼𝐷^2𝛼𝑢+𝛾𝑢−𝛽𝑢²=0. (16) Next, we assume that (16) admits a solution in the form 𝑢=^𝑛_𝑖=0𝑎_𝑖𝜑^𝑖. (17) At this stage we apply the same technique as in the case of the previous example. Namely, by balancing the highest order derivative terms and nonlinear terms in (16), then substituting (17), with??𝑛=2, with (6) into (16), we finally obtain the corresponding system of algebraic equations as −𝑎²₀𝛽+𝑎₀𝛾+2𝑎₂𝜎²𝑐^2𝛼−2𝑎₂𝜎²𝑘^2𝛼𝑎=0,₁𝛾−2𝑎₀𝑎₁𝛽+2𝑎₁𝜎𝑐^2𝛼−2𝑎₁𝜎𝑘^2𝛼=0,−𝑎²₁𝛽+𝑎₂𝛾−2𝑎₀𝑎₂𝛽+8𝑎₂𝜎𝑐^2𝛼−8𝑎₂𝜎𝑘^2𝛼=0,−2𝑎₁𝑎₂𝛽+2𝑎₁𝑐^2𝛼−2𝑎₁𝑘^2𝛼=0,−𝑎²₂𝛽+6𝑎₂𝑐^2𝛼−6𝑎₂𝑘^2𝛼=0. (18) After using the Mathematica to solve (18) the following solutions are reported: 𝑎₀=𝛾+8𝜎𝑐^2𝛼−8𝜎𝑘^2𝛼2𝛽,𝑎₁𝑎=0,₂6𝑘=−^2𝛼−𝑐^2𝛼𝛽, (19) where 𝜎 denotes an arbitrary constant. Finally, from (15)–(19) we obtain the following generalized hyperbolic function solutions, generalized trigonometric function solutions, and the rational solution of (14) as 𝑢=𝛾+8𝜎𝑐^2𝛼−8𝜎𝑘^2𝛼+𝑘2𝛽6𝜎^2𝛼−𝑐^2𝛼𝛽×tanh²_𝛼√−𝜎𝜉,𝜎<0,𝛾+8𝜎𝑐^2𝛼−8𝜎𝑘^2𝛼+𝑘2𝛽6𝜎^2𝛼−𝑐^2𝛼𝛽×coth²_𝛼√−𝜎𝜉,𝜎<0,𝛾+8𝜎𝑐^2𝛼−8𝜎𝑘^2𝛼−𝑘2𝛽6𝜎^2𝛼−𝑐^2𝛼𝛽×tan_𝛼√𝜎𝜉,𝜎>0,𝛾+8𝜎𝑐^2𝛼−8𝜎𝑘^2𝛼−𝑘2𝛽6𝜎^2𝛼−𝑐^2𝛼𝛽×cot_𝛼√𝛾𝜎𝜉,𝜎>0,−6𝑘2𝛽^2𝛼−𝑐^2𝛼Γ²(1+𝛼)𝛽𝜉^𝛼+𝜔²,𝜎=0, (20) where 𝜉=𝑘𝑥+𝑐𝑡.

As 𝛼→1 (20) the results obtained above become the ones of (14).

4. Conclusions

In this paper, a fractional subequation method is used to construct the exact analytical solutions of the space-time fractional Cahn-Hilliard (1) and the fractional nonlinear Klein-Gordon equation (2). These solutions include the generalized hyperbolic function solutions, generalized trigonometric function solutions, and rational function solutions, which may be very useful to further understand the mechanisms of the complicated nonlinear physical phenomena and FPDEs. Also, this method help us to find all exact solutions of the Fan subequations involving all possible parameters, it is concise and efficient. Mathematica has been used for computations and programming in this work.