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Explicit Jacobi elliptic exact solutions for nonlinear partial fractional differential equations
- Khaled A Gepreel^{1, 2}Email author
https://doi.org/10.1186/1687-1847-2014-286
© Gepreel; licensee Springer. 2014
- Received: 20 August 2014
- Accepted: 24 October 2014
- Published: 4 November 2014
Abstract
In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. An algebraic method is improved to construct uniformly a series of exact solutions for some nonlinear time-space fractional partial differential equations. We construct successfully a series of some exact solutions including the elliptic doubly periodic solutions with the aid of computerized symbolic computation software package such as Maple or Mathematica. This method is efficient and powerful in solving a wide classes of nonlinear partial fractional differential equations. The Jacobi elliptic doubly periodic solutions are generated by the trigonometric exact solutions and the hyperbolic exact solutions when the modulus $m\to 0$ and $m\to 1$, respectively.
Keywords
- fractional calculus
- fractional complex transformation
- modified Riemann-Liouville derivative
- Jacobi elliptic functions
- nonlinear fractional differential equations
1 Introduction
- (i)First, we have the space-time fractional derivative nonlinear Korteweg-de Vries (KdV) equation [21]$\frac{{\partial}^{\alpha}u}{\partial {t}^{\alpha}}+au\frac{{\partial}^{\beta}u}{\partial {x}^{\beta}}+\frac{{\partial}^{3\beta}u}{\partial {x}^{3\beta}}=0,\phantom{\rule{1em}{0ex}}t>0,0<\alpha ,\beta \le 1,$(1)
- (ii)Then we have the space-time fractional derivative nonlinear fractional Zakharov-Kunzetsov-Benjamin-Bona-Mahony (ZKBBM) equation [23],$\frac{{\partial}^{\alpha}u}{\partial {t}^{\alpha}}+\frac{{\partial}^{\beta}u}{\partial {x}^{\beta}}-2au\frac{{\partial}^{\beta}u}{\partial {x}^{\beta}}-b\frac{{\partial}^{\alpha}}{\partial {t}^{\alpha}}\left(\frac{{\partial}^{2\beta}u}{\partial {x}^{2\beta}}\right)=0,\phantom{\rule{1em}{0ex}}t>0,0<\alpha ,\beta \le 1,$(2)
where a, b are arbitrary constants. When $\alpha ,\beta \to 1$, this system has been investigated by Benjamin et al. [23] for the first time, as an alternative model to the KdV equation for long waves and it plays an important role in the modeling of nonlinear dispersive systems. The Benjamin-Bona-Mahony equation is applicable to the study of drift waves in a plasma or Rossby waves in rotating fluids.
2 Preliminaries
There are many types of the fractional derivatives such as the Kolwankar-Gangal local fractional derivative [24], Chen’s fractal derivative [25], Cresson’s derivative [26], and Jumarie’s modified Riemann-Liouville derivative [27, 28].
In this section, we give some basic definitions of fractional calculus theory which will be used in this work.
where $f:R\to R$, $x\mapsto f(x)$ denotes a continuous (but not necessarily first order differentiable) function.
where B is the beta function. The function $f(x)$ should be differentiable with respect to $x(t)$ and $x(t)$ is fractional differentiable in (6). The above results are employed in the following sections. The Leibniz rule is given (7) for modified Riemann-Liouville derivative which is modified by Wu in [30].
The modified Riemann-Liouville derivative has been successfully applied in probability calculus [31], fractional Laplace problems [32], the fractional variational approach with several variables [33], the fractional variational iteration method [34], the fractional variational approach with natural boundary conditions [35], and the fractional Lie group method [36].
3 The improved extended proposed algebraic method for nonlinear partial fractional differential equations
where u is an unknown function, F is a polynomial in u and its partial fractional derivatives in which the highest order fractional derivatives and the nonlinear terms are involved.
We give the main steps of the modified extended proposed algebraic method for nonlinear partial fractional differential equations.
If possible, we should integrate (10) term by term one or more times.
where ${e}_{0}$, ${e}_{1}$, and ${e}_{2}$ are arbitrary constants.
Step 3. The positive integer ‘m’ can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Eq. (10).
Step 4. We must substitute (11) into (10) and using (12), collect all terms with the same order of $\varphi (\xi )$ together, then equating each coefficient of the resulting polynomial to be zero. This yields a set of algebraic equations for ${\alpha}_{i}$ ($i=0,\pm 1,\dots ,\pm m$), ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, M, and N. We then solve this system of algebraic equation with the help of Maple software package to determine ${\alpha}_{i}$ ($i=0,\pm 1,\dots ,\pm m$), ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, M, and N.
The exact solutions of the Jacobi elliptic differential equation ( 12 ) when ${\mathit{e}}_{\mathbf{0}}$ , ${\mathit{e}}_{\mathbf{1}}$ and ${\mathit{e}}_{\mathbf{2}}$ take special values
${\mathit{e}}_{\mathbf{0}}$ | ${\mathit{e}}_{\mathbf{1}}$ | ${\mathit{e}}_{\mathbf{2}}$ | ϕ(ξ) |
---|---|---|---|
1 | $-(1+{m}^{2})$ | ${m}^{2}$ | sn(ξ) or cd(ξ) |
$1-{m}^{2}$ | $2{m}^{2}-1$ | $-{m}^{2}$ | cn(ξ) |
${m}^{2}-1$ | $2-{m}^{2}$ | −1 | dn(ξ) |
${m}^{2}$ | $-({m}^{2}+1)$ | 1 | ns(ξ) or dc(ξ) |
$-{m}^{2}$ | $2{m}^{2}-1$ | $1-{m}^{2}$ | nc(ξ) |
−1 | $2-{m}^{2}$ | ${m}^{2}-1$ | nd(ξ) |
$1-{m}^{2}$ | $2-{m}^{2}$ | 1 | cs(ξ) |
1 | $2-{m}^{2}$ | $1-{m}^{2}$ | sc(ξ) |
1 | $2{m}^{2}-1$ | ${m}^{2}({m}^{2}-1)$ | sd(ξ) |
${m}^{2}({m}^{2}-1)$ | $2{m}^{2}-1$ | 1 | ds(ξ) |
$\frac{1}{4}$ | $\frac{1}{2}(1-2{m}^{2})$ | $\frac{1}{4}$ | ns(ξ)±cs(ξ) |
$\frac{1}{4}(1-{m}^{2})$ | $\frac{1}{2}(1+{m}^{2})$ | $\frac{1}{4}(1-{m}^{2})$ | nc(ξ)±sc(ξ) |
$\frac{{m}^{2}}{4}$ | $\frac{1}{2}({m}^{2}-2)$ | $\frac{{m}^{2}}{4}$ | sn(ξ)±icn(ξ) |
$\frac{{m}^{2}}{4}$ | $\frac{1}{2}({m}^{2}-2)$ | $\frac{1}{4}$ | ns(ξ)±ds(ξ) |
$\frac{{m}^{2}}{4}$ | $\frac{1}{2}({m}^{2}-2)$ | $\frac{{m}^{2}}{4}$ | $\sqrt{1-{m}^{2}}sd(\xi )\pm cd(\xi )$ |
$\frac{1}{4}$ | $\frac{1}{2}(1-{m}^{2})$ | $\frac{1}{4}$ | $mcd(\xi )\pm i\sqrt{1-{m}^{2}}nd(\xi )$ |
$\frac{1}{4}$ | $\frac{1}{2}(1-2{m}^{2})$ | $\frac{1}{4}$ | msn(ξ)±idn(ξ) |
$\frac{1}{4}$ | $\frac{1}{2}(1-{m}^{2})$ | $\frac{1}{4}$ | $\sqrt{1-{m}^{2}}sc(\xi )\pm dc(\xi )$ |
$\frac{1}{4}({m}^{2}-1)$ | $\frac{1}{2}(1+{m}^{2})$ | $\frac{1}{4}({m}^{2}-1)$ | msd(ξ)±nd(ξ) |
$\frac{1}{4}$ | $\frac{1}{2}({m}^{2}-2)$ | $\frac{{m}^{2}}{4}$ | sn(ξ)/(1 ± dn(ξ)) |
There are other cases which are omitted here for convenience; see [37].
Step 6. Since the general solutions of (12) are discussed in Table 1, then substituting ${\alpha}_{i}$ ($i=0,\pm 1,\dots ,\pm m$), ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, M, and N and the general solutions of (12) into (11), we obtain more new Jacobi elliptic exact solutions for the nonlinear partial fractional derivative equation (8).
4 Applications
In this section, we construct some new Jacobi elliptic exact solutions of some nonlinear partial fractional differential equations via the time-space fractional nonlinear KdV equation and the time-space fractional nonlinear Zakharov-Kunzetsov-Benjamin-Bona-Mahomy equation using the modified extended proposed algebraic method which has been paid attention to by many authors.
4.1 Example 1: Jacobi elliptic solutions for nonlinear fractional KdV equation
where ${\alpha}_{0}$, ${\alpha}_{1}$, ${\alpha}_{2}$, ${a}_{3}$, ${a}_{4}$, L, and K are arbitrary constants to be determined later. Substituting (14) and (12) into (13), collecting all terms of $\varphi (\xi )$, and then setting each coefficient $\varphi (\xi )$ to be zero, we get a system of algebraic equations. With the aid of Maple or Mathematica we can solve this system of algebraic equations to obtain the following cases of solutions:
where ${C}_{1}$, L, K, ${e}_{0}$, ${e}_{1}$, and ${e}_{2}$ are arbitrary constants.
where ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, and ${C}_{2}$ are arbitrary constants.
According to the general solutions of (12) which are discussed in Table 1, we have the following families of exact solutions:
where ${C}_{2}=\frac{2{C}_{1}^{2}[-576L{m}^{2}{K}^{4}-4\text{,}608{K}^{9}{m}^{2}(1+{m}^{2})+{L}^{3}-48{K}^{6}{(1+{m}^{2})}^{2}L+128{K}^{9}{(1+{m}^{2})}^{3}]}{3{[192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}]}^{2}}$.
where ${C}_{2}$ = $\frac{1}{3{[-192{K}^{6}{m}^{2}(1-{m}^{2})-{L}^{2}+16{K}^{6}{(2{m}^{2}-1)}^{2}]}^{2}}${$2{C}_{1}^{2}(576L{m}^{2}{K}^{4}(2{m}^{2}-1)$ − $4\text{,}608{K}^{9}{m}^{2}(2{m}^{2}-1)(1-{m}^{2})$ + ${L}^{3}$ − $48{K}^{6}{(2{m}^{2}-1)}^{2}L$ − $128{K}^{9}{(2{m}^{2}-1)}^{3})$}.
where ${C}_{2}=\frac{2{C}_{1}^{2}(576L{K}^{4}({m}^{2}-1)-4\text{,}608{K}^{9}({m}^{2}-1)(2-{m}^{2})+{L}^{3}-48{K}^{6}{(2-{m}^{2})}^{2}L-128{K}^{9}{(2-{m}^{2})}^{3})}{3{[192{K}^{6}(1-{m}^{2})-{L}^{2}+16{K}^{6}{(2-{m}^{2})}^{2}]}^{2}}$.
where ${C}_{2}=\frac{2{C}_{1}^{2}(-36L{K}^{4}{(1-{m}^{2})}^{2}+144{K}^{9}{(1-{m}^{2})}^{2}(1+{m}^{2})+{L}^{3}-12{K}^{6}{(1+{m}^{2})}^{2}L-16{K}^{9}{(1+{m}^{2})}^{3})}{3{[12{K}^{6}{(1-{m}^{2})}^{2}-{L}^{2}+4{K}^{6}{(1+{m}^{2})}^{2}]}^{2}}$.
where ${C}_{2}=\frac{2{C}_{1}^{2}(-36L{K}^{4}{m}^{4}+144{K}^{9}{m}^{4}({m}^{2}-2)+{L}^{3}-12{K}^{6}{({m}^{2}-2)}^{2}L-16{K}^{9}{({m}^{2}-2)}^{3})}{3{[12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{({m}^{2}-2)}^{2}]}^{2}}$.
Similarly, we can write down the other families of exact solutions of (1) which are omitted for convenience.
4.2 Example 2: Jacobi elliptic solutions for nonlinear fractional ZKBBM equation
where ${\alpha}_{0}$, ${\alpha}_{1}$, ${\alpha}_{2}$, ${a}_{3}$, ${a}_{4}$, L, and K are arbitrary constants to be determined later. Substituting (25) and (12) into (24), collecting all the terms of powers of $\varphi (\xi )$ and setting each coefficient $\varphi (\xi )$ to zero, we get a system of algebraic equations. With the aid of Maple or Mathematica we can solve this system of algebraic equations to obtain the following sets of solutions:
where b, L, K, ${e}_{0}$, ${e}_{1}$, and ${e}_{2}$ are arbitrary nonzero constants. There are many other cases which are omitted for convenience.
According to the general solutions of (12) which are discussed in Table 1, we have the following families of Jacobi elliptic exact solutions to the nonlinear ZKBBM equation:
where ${C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+192{b}^{2}{K}^{4}{L}^{2}{m}^{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{(1+{m}^{2})}^{2}\}$.
where ${C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK-192{b}^{2}{K}^{4}{L}^{2}(1-{m}^{2}){m}^{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{(2{m}^{2}-1)}^{2}\}$.
where ${C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK-192{b}^{2}{K}^{4}{L}^{2}({m}^{2}-1)-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{(2-{m}^{2})}^{2}\}$.
where ${C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+12{b}^{2}{K}^{4}{L}^{2}{(1-{m}^{2})}^{2}-{L}^{2}+4{b}^{2}{K}^{4}{L}^{2}{(1+{m}^{2})}^{2}\}$.
where ${C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+12{b}^{2}{K}^{4}{L}^{2}{m}^{4}-{L}^{2}+4{b}^{2}{K}^{4}{L}^{2}{({m}^{2}-2)}^{2}\}$.
Similarly, we can write down the other families of exact solutions of (2) which are omitted for convenience.
5 Conclusion
In this article we constructed the Jacobi elliptic exact solutions for the nonlinear partial fractional differential equations with the help of the complex fractional transformation and the improved extended proposed algebraic method. This method is effective and powerful for finding the Jacobi elliptic solutions for nonlinear fractional differential equations. Jacobi elliptic solutions are generalized to the hyperbolic exact solutions and trigonometric exact solutions when the modulus $m\to 1$ and $m\to 0$, respectively.
Author’s contributions
The author read and approved the final manuscript.
Declarations
Acknowledgements
The author would like to thank the editors and the anonymous referees for their careful reading of the paper.
Authors’ Affiliations
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