Abstract
The time-fractional version of the generalized Pochhammer–Chree equation is analyzed. In this paper, the equation is converted into an ordinary differential equation by applying certain real transformation, then the discrimination of polynomials system is used to find exact solutions depending on the fractional order derivative. The obtained solutions are graphically illustrated for different values of the fractional order derivative keeping the other parameters fixed.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
Most physics and engineering real-life problems can be perfectly described using fractional-order systems, these are the dynamical systems that can be modeled using time fractional differential equations [1,2,3]. For this, and many other reasons, studying the time fractional differential equations have attracted the attention of many researchers. In fact, finding the exact solutions of these equations became an active research topic, and many contributions have been made [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. In this paper, we consider the model of elastic waves, these are the disturbances that propagate in different media under the influence of elastic forces [28]. The standard example of elastic wave is a long rope or rubber tube held at one end. Another example is the pervasion of vibrations in a semi-rod structure onto a moving substrate which is found in many engineering applications, from the fabrication of nanotube to the extension of submarine pipes, and in many other technical and biological processes [29,30,31,32]. The mathematical model of elastic rods wave is the generalized Pochhammer–Chree equation [33, 34], and the conformable time fractional version of such equation can be written as
where \(a_1,a_2\) and \(a_3\) are arbitrary real constants, \(a_3\ne 0\), and \(D^{\alpha }\) is an operator of order \(\alpha\) representing the conformable fractional derivatives. The classical case where \(\alpha =1\) has been considered in several works for different values of the parameter n, and a number of solutions have been found for special cases [35,36,37,38,39,40,41,42,43,44,45,46]. In the current work, for the case of \(n=1\) and \(\alpha \in (0,1]\), we find the exact solutions of the Eq. (1) using the complete discrimination system for a polynomial [47, 48]. The complete discrimination system method was a primary tool in solving some differential equations, and in conducting qualitative analysis for others [49,50,51].
In “Traveling wave reduction of the conformable time fractional Pochhammer–Chree equation” section, the case of \(n=1\) for the Eq. (1) has been reduced to an ordinary equation using the traveling wave substitution. The direct integral of the obtained equation involves a quartic polynomial whose roots will be classified using the complete discrimination system. Section “Exact wave solutions” contains a complete analysis for all possible solutions for the intended equation. To make the article self-contained, a basic introduction to the conformable fractional derivatives is provided in “Appendix”. For more detailed information about fractional calculus, the reader may refer to [52,53,54].
Traveling wave reduction of the conformable time fractional Pochhammer–Chree equation
We are interested in constructing a traveling wave solutions for Eq. (1) in which \(n=1\), \(\alpha \in (0,1]\), which will takes the form
We apply the wave transformation
where \(\omega\) is the speed of the wave and \(\zeta\) is the wave variable. Inserting Eq. (3) into Eq. (2) and taking into account the properties of conformable derivitative listed in “Appendix”, we get
where \(^\prime\) refers to the derivative with respect to \(\zeta\). By integrating both sides of Eq. (4) with respect to \(\zeta\) twice and setting the first integration constant equal to zero, we have
where m is an integration constant. Taking into account \(\psi ^{\prime \prime }=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\psi }(\psi {^\prime }^2)\) and integrating both sides of the last equation with respect to \(\zeta\), we get
where g is the integration constant. To simplify the notations, we set
Substituting the last equation into Eq. (6), we obtain
where
Separating the variables in Eq. (8), we get the differential form
where
Our goal is to find the solutions of Eq. (2) by finding the solution of the equation in (10), and using the formula in Eq. (7). To integrate both sides of Eq. (10), the range of the parameters needs to be specified. The reason is that distinct values of the parameters imply different solutions. Many tools are utilized to find these ranges of parameters such as bifurcation theory [55,56,57,58,59] and the complete discrimination system for a polynomial [47]. We use the complete discrimination system for a polynomial to find the ranges of parameters for \(P_4(\phi )\). The complete discrimination system is a natural generalization of the discriminant \(\Delta =b^2-4ac\) for the quadratic polynomial \(ax^2+bx+c\), but it becomes difficult to calculate for the higher degree polynomials. The complete discrimination system for the quartic polynomial in Eq. (11) is given in [47, 48] and has the following form:
Note that for physical problems, the real propagation is required, consequently, we are going to find the permitted regions of real propagation, or equivalently, we determine certain intervals of \(\phi\) which guarantee that \(\mu P_4(\phi )\) is positive. In next section, we consider the nine cases determined by distinct types of the roots for the polynomial in Eq. (11). We will use the following basic fact for the roots of quartic polynomial without mentioning. If \(x^4+ p_3x^3+p_2x^2+p_1x+p_0\) is a polynomial whose roots are \(r_i ,i=1,2,3,4\), then \(p_3=-\sum \nolimits _{i} r_{i}\), \(p_2=\sum \nolimits _{i<j} r_{i}r_{j}\), \(p_1=-\sum \nolimits _{i<j<k} r_{i}r_{j}r_{k}\), and \(p_0=r_{1}r_{2}r_{3}r_{4}\). For computations involving elliptic integrals refer to [60].
Exact wave solutions
In this section, we aim to construct some traveling wave solutions and study the effect of the fractional order on these solutions. Based on complete discrimination system for a polynomial \(P_4(\phi )\), we consider the following cases:
Case 1: If \(D_{2}=D_{3}=D_{4}=0\), then the polynomial \(P_{4}(\phi )\) has the zero as a one real root repeated four times, and can be written as \(P_{4}(\phi )=\phi ^{4}\ge 0\) for all \(\phi\). Therefore, \(\mu <0\) gives a complex solution, and we only consider the case where \(\mu >0\) to solve the equation in Eq. (10) which will be in the following form
Let \(-\infty<\phi <0\), and \(\phi (\zeta _{0})=-\infty\), by integrating both sides of the above equation, we get
Similarly, the case where \(0<\phi <\infty\), and \(\phi (\zeta _{0})=\infty\), will generate the same solution in Eq. (14). Thus, the solution for Eq. (2) is
The solution (15) is a new solution for Eq. (2). Figure 1a, b outlines the 3D- graphic representation of the the solution (15) when \(\alpha =0.4\) and \(\alpha =0.7\). Figure 1c shows the width of the solution decreases when the fractional order increases. Furthermore, when \(\alpha\) tends to one, the solution (15) becomes also a solution for the integer order time derivative version of Eq. (2).
Case 2: If \(D_{3}=D_{4}\), \(E_{2}<0\) and \(D_{2}>0\), then the polynomial in Eq (11) has two real roots \(\phi _{1}\) and \(\phi _{2}\) where \(\phi _{2}=-3\phi _{1}\). Therefore, \(P_{4}(\phi )=(\phi -\phi _{1})^{3}(\phi +3\phi _{1})\), where \(\phi _{1}\) is postulated to be positive. We consider the following sub-cases based on \(\mu\)
-
If \(\mu \in (0,\infty )\), then the permitted region of real propagation is \(\phi \in (-\infty ,-3\phi _{1}) \cup (\phi _{1},\infty )\).
-
If \(\phi \in (-\infty ,-3\phi _{1})\), then we integrate both sides of Eq. (10), assuming that \(\phi (\zeta _{0})=-\infty\) to get
$$\begin{aligned}&\int _{-\infty }^{\phi }\frac{\mathrm{d}\phi }{(\phi _{1}-\phi )\sqrt{(\phi -\phi _{1})(\phi +3\phi _{1})}}=\frac{1}{2\phi _{1}}\left( \sqrt{\frac{\phi +3\phi _{1}}{\phi -\phi _{1}}}-1\right) \nonumber \\&\quad =\sqrt{\mu }\int _{\zeta _{0}}^{\zeta } \mathrm{d}\zeta \end{aligned}$$(16)The last equation implies
$$\begin{aligned} \phi =\phi _{1}-\frac{\phi _{1}}{\phi _{1}\sqrt{\mu }(\zeta -\zeta _{0}) +1 }+\frac{1}{\sqrt{\mu }(\zeta -\zeta _{0})}. \end{aligned}$$(17) -
If \(\phi \in (\phi _{1},\infty )\), then by assuming that \(\phi (\zeta _{0})=\infty\), the solution of Eq. (10) can be computed similarly and will generate the same solution above. Thus, the solutions of Eq. (2) include
$$\begin{aligned} \psi =\phi _{1}-\frac{\phi _{1}}{\phi _{1}\sqrt{\mu }(\zeta -\zeta _{0}) +1}+\frac{1}{\sqrt{\mu }(\zeta -\zeta _{0})}-\frac{a_{2}}{3a_{3}}. \end{aligned}$$(18)The solutions are graphically clarified for different values of the fractional order \(\alpha\) in Fig. 2.
-
-
If \(\mu \in (-\infty ,0)\), we obtain the real propagation when \(-3\phi _{1}<\phi <\phi _{1}\). So, we postulate \(\phi (\zeta _{0})=-3\phi _{1}\), and integrate both side of Eq. (10) to get
$$\begin{aligned}&\int _{-3\phi _{1}}^{\phi }\frac{\mathrm{d}\phi }{(\phi _{1}-\phi )\sqrt{(\phi _{1}-\phi )(\phi +3\phi _{1})}}=\frac{-1}{2\phi _{1}} \sqrt{\frac{\phi +3\phi _{1}}{\phi _{1}-\phi }}\nonumber \\&\quad =\sqrt{-\mu }\int _{\zeta _{0}}^{\zeta } \mathrm{d}\zeta \end{aligned}$$(19)Therefore,
$$\begin{aligned} \phi =\phi _{1}-\frac{4\phi _{1}}{1+ 4\mu \phi _{1}^{2}(\zeta -\zeta _{0})^{2}} \end{aligned}$$(20)Hence, the Eq. (2) has a solution in the form
$$\begin{aligned} \psi =\phi _{1}-\frac{4\phi _{1}}{1+ 4\mu \phi _{1}^{2}(\zeta -\zeta _{0})^{2}}-\frac{a_{2}}{3a_{3}} \end{aligned}$$(21)
The solution in (21) is a novel singular solution for Eq. (2). Figure 2a, b are 3D-graphic representation for the solution (21) when \(\alpha =0.4\) and \(\alpha =0.7\), respectively. Figure 2 clarifies the width of the solution decreases as the fractional order increases. Also, when \(\alpha\) approaches to one, we obtain a solution for Eq. (2) with integer time derivative.
Case 3: If \(D_{3}=D_{4}=0,E_{2}>0,D_{2}>0\), then the polynomial \(P_{4}(\phi )\) has two real zeros which are doubled. Moreover, each one of them is the negative of the other which implies that \(P_{4}(\phi )\) can be expressed using one root as
Since \(P_{4}(\phi )\) is non negative, then for \(\mu <0,\) the expression \(\mu P_{4}(\phi )\) is always negative for all \(\phi\), and gives complex solutions for Eq. (10), so we must neglect it. We only consider the case \(\mu >0\) where the real propagation accrues if \(\phi \in {\mathbb {R}}{\setminus }\left\{ \pm \phi _{1}\right\}\) and the equation in (10) will have the following form
We assume that \(\phi _{1}\) is positive and we study the following cases:
-
If \(\phi < -\phi _{1}\), and \(\phi (\zeta _{0})=-\infty\), then Eq. (23) implies that
$$\begin{aligned} \int _{-\infty }^{\phi }\frac{\mathrm{d}\phi }{\phi ^{2}-\phi _{1}^{2}}=\frac{-1}{\phi _{1}}\mathrm {arcoth}\left( \frac{\phi }{\phi _{1}}\right) =\sqrt{\mu }\left( \zeta -\zeta _{0}\right) \end{aligned}$$(24)which gives
$$\begin{aligned} \phi =-\phi _{1}\coth (\phi _{1}\sqrt{\mu }(\zeta -\zeta _{0})). \end{aligned}$$(25)Hence, we obtain a new solution for Eq. (2) in the form
$$\begin{aligned} \psi =-\phi _{1}\coth (\phi _{1}\sqrt{\mu }(\zeta -\zeta _{0}))-\frac{a_2}{3a_3}. \end{aligned}$$(26)Notice, when \(\alpha \rightarrow 1\), this solution will converted into a well known solution for Eq. (2) with \(\alpha \rightarrow 1\) [61].
-
If \(\phi >\phi _{1}\), the corresponding solution can obtained from Eq. (26) by replacing \(\phi _{1}\) by \(-\phi _{1}\), which generate the same solution.
-
With similar computation, if \(\phi \in (-\phi _{1},\phi _{1}),\) then \(\phi ^{2}< \phi _{1}^2\). So, by letting \(\phi (\zeta _{0})=0\), and integrating Eq. (23) from 0 to \(\phi\), we get that
$$\begin{aligned} \frac{-1}{\phi _{1}} \mathrm {arctanh}\left( \frac{\phi }{\phi _{1}}\right) =\sqrt{\mu }(\zeta -\zeta _{0}) \end{aligned}$$(27)That is,
$$\begin{aligned} \phi =-\phi _{1}\tanh (\phi _{1}\sqrt{\mu }(\zeta -\zeta _{0})) \end{aligned}$$(28)Hence, the case under consideration will generate the following solution for Eq. (2)
$$\begin{aligned} \psi =-\phi _{1}\tanh (\phi _{1}\sqrt{\mu }(\zeta -\zeta _{0}))-\frac{a_{2}}{3a_{3}}, \end{aligned}$$(29)The solution in (29) is a new solution for Eq. (2). Figure 3a, b illustrate the 3D-graphic representation of the solution (29), like a kink solution. Figure 3c shows the width of the solution increase as \(\alpha\) increase. When \(\alpha \rightarrow 1\), we obtain a well-known solution for equation integer time derivative version of Eq. (2) [61].
Case 4: This case is characterized by \(D_{2}>0,D_{3}>0, D_{4}>0\). These conditions guarantee the existence of four real roots for the polynomial \(P_4(\phi )\). Assuming that three of them are \(\phi _{1},\phi _{2},\phi _{3}\), the fourth one must be \(-(\phi _{1}+\phi _{2}+\phi _{3})\). Hence, we write \(P_{4}(\phi )=(\phi -\phi _{1})(\phi -\phi _{2})(\phi -\phi _{3})(\phi +\phi _{1}+\phi _{2}+\phi _{3})\), where \(0<\phi _{1}<\phi _{2}<\phi _{3}\). We consider the following two sub-cases:
-
If \(\mu \in (0,\infty )\), then there is a real propagation only if \(\phi \in (-\infty ,-(\phi _{1}+\phi _{2}+\phi _{3}))\cup (\phi _{1},\phi _{2})\cup (\phi _{3},\infty )\), where the Eq. (10) has the form
$$\begin{aligned} \frac{\mathrm{d}\phi }{\sqrt{(\phi -\phi _{1})(\phi -\phi _{2})(\phi -\phi _{3})(\phi +\phi _{1}+\phi _{2}+\phi _{3})}}=\sqrt{\mu }\zeta \end{aligned}$$(30)-
- If we elect \(\phi \in (-\infty ,-(\phi _{1}+\phi _{2}+\phi _{3}))\), postulate \(\phi (\zeta _{0})=-(\phi _{1}+\phi _{2}+\phi _{3})\), and integrate both sides of Eq. (30), we get
$$\begin{aligned} \phi =\phi _{1}+\frac{(\phi _{3}-\phi _{1})(2\phi _{1}+\phi _{2}+\phi _{3})}{\phi _{1}-\phi _{3}+(\phi _{1}+\phi _{2}+2\phi _{3})\mathrm {sn^{2}}(\Omega (\zeta -\zeta _{0}),k)} \end{aligned}$$(31)where
$$\begin{aligned} \Omega= & {} \frac{1}{2}\sqrt{\mu (\phi _{3}-\phi _{1})(\phi _{1}+2\phi _{2}+\phi _{3})},\nonumber \\ k= & {} \sqrt{\frac{(\phi _{2}-\phi _{1})(\phi _{1}+\phi _{2}+2\phi _{3})}{(\phi _{3}-\phi _{1})(\phi _{1}+2\phi _{2}+\phi _{3})}} \end{aligned}$$(32)Hence, we get a novel periodic solution for Eq. (2) in the form
$$\begin{aligned} \psi =\phi _{1}+\frac{(\phi _{3}-\phi _{1})(2\phi _{1}+\phi _{2}+\phi _{3})}{\phi _{1}-\phi _{3}+(\phi _{1}+\phi _{2}+2\phi _{3})\mathrm {sn^{2}}(\Omega (\zeta -\zeta _{0}),k)}-\frac{a_{2}}{3a_{3}}. \end{aligned}$$(33)Figure 4a, b outline the periodicity of the solution (33) for different values of the fractional order \(\alpha\), but the amplitude and the width of the solution are affected. Figure 4c shows the width of the solution increases as the fractional order increases, but the amplitude is approximately unchanged. We also examine the degeneracy of the solution (33). If \(\phi _3=\phi _2\), the modulus of the elliptic function, k, will be reduced to one. Hence, the solution (33) degenerates to
$$\begin{aligned} \psi =\phi _1-\frac{2(\phi _2^2-\phi _1^2)}{(\phi _1-\phi _2)+(\phi _1+3\phi _2)\mathrm {tanh}^2(\Omega (\zeta -\zeta _0)} \end{aligned}$$(34)which is also a new solution. If \(\phi _2=\phi _1\), the modules of the elliptic function becomes zero and hence, the solution (33) degenerates to
$$\begin{aligned} \psi =\phi _1+\frac{(\phi _3-\phi _1)(3\phi _1+\phi _3)}{(\phi _1-\phi _3)+2(\phi _1+\phi _3)\mathrm {sin}^2\Omega (\zeta -\zeta _0)} \end{aligned}$$(35)which is also a new solution for Eq. (2). Notice, the two solutions (34) and (35) will transform to a well-known solutions for the time integer derivative of Eq. (2) [61].
-
- If we choose \(\phi \in (\phi _{1},\phi _{2})\) and set \(\phi (\zeta _{0})=\phi _{1}\), then the integral from \(\phi _{1}\) to \(\phi\) will generate the following solution
$$\begin{aligned} \phi= & {} -(\phi _{1}+\phi _{2}+\phi _{3})\nonumber \\&\quad +\frac{(\phi _{1}+2\phi _{2}+\phi _{3})(2\phi _{1}+\phi _{2}+\phi _{3})}{\phi _{1}+2\phi _{2}+\phi _{3}+(\phi _{1}-\phi _{2})\mathrm {sn^{2}}(\Omega (\zeta -\zeta _{0}),k)} \end{aligned}$$(36)Hence, we obtain a novel solution for Eq. (2) in the form
$$\begin{aligned} \psi= & {} -(\phi _{1}+\phi _{2}+\phi _{3})\nonumber \\&\quad +\frac{(\phi _{1}+2\phi _{2}+\phi _{3})(2\phi _{1}+\phi _{2}+\phi _{3})}{\phi _{1}+2\phi _{2}+\phi _{3}+(\phi _{1}-\phi _{2})\mathrm {sn^{2}}(\Omega (\zeta -\zeta _{0}),k)}-\frac{a_{2}}{3a_{3}}. \end{aligned}$$(37)Figure 5a, b shows that the solution (37) is periodic for distinct values of the fractional order \(\alpha\), but its width and its amplitude are influenced. Figure 5c clarifies the width of the solution (37) decreases as the fractional order \(\alpha\) increases while the amplitude is approximately unaltered. Let us now study the degeneracy of the solution (37). When \(\phi _3=\phi _2\), the modules of the elliptic function, k, becomes one, and the solution (37) degenerates to
$$\begin{aligned} \psi =-\phi _1-2\phi _2+\frac{2(\phi _1+3\phi _2)(\phi _1+\phi _2)}{\phi _1+3\phi _2+(\phi _1-\phi _2)\mathrm {tanh}^2\Omega (\zeta -\zeta _0)}, \end{aligned}$$(38)which is also a new solution for Eq. (2). When \(\phi _2=\phi _1\), the modules of the elliptic function, k, equals to zero and the solution (37) is reduced to \(\psi =\phi _1\) which is a trivial solution for Eq. (2). Notice, when \(\alpha \rightarrow 1\), the solution (38) will be converted to a well known solution for the time integer derivative for Eq. (2).
-
- If we choose \(\phi \in (\phi _{3},\infty )\) with assumption \(\phi (\zeta _{0})=\phi _{3}\), and integrate both sides of Eq. (30), we obtain a new solution for Eq. (2)
$$\begin{aligned} \phi =\phi _{2}-\frac{(\phi _{2}-\phi _{3})(\phi _{1}+2\phi _{2}+\phi _{3})}{(\phi _{1}+2\phi _{2}+\phi _{3})-(\phi _{1}+\phi _{2}+2\phi _{3})\mathrm {sn^{2}}(\Omega (\zeta -\zeta _{0}),k)} \end{aligned}$$(39)where k and \(\Omega\) are as defined in Eq. (32) above. Hence, we obtain a new solution for Eq. (2) in the form
$$\begin{aligned} \psi= & {} \phi _{2}-\frac{(\phi _{2}-\phi _{3})(\phi _{1}+2\phi _{2}+\phi _{3})}{(\phi _{1}+2\phi _{2} +\phi _{3})-(\phi _{1}+\phi _{2}+2\phi _{3})\mathrm {sn^{2}}(\Omega (\zeta -\zeta _{0}),k)}\nonumber \\&\quad -\frac{a_2}{3a_3} \end{aligned}$$(40)Similarly, we can study the degeneracy of the solution (40). Notice, when \(\alpha \rightarrow 1\), the solution (40) is also a new solution of the time integer order derivative for Eq. (2).
-
-
If \(\mu \in (-\infty ,0)\), then there is a real propagation only if \(\phi \in (-\phi _{1}-\phi _{2}-\phi _{3},\phi _{1})\cup (\phi _{2},\phi _{3})\), where the Eq. (10) will be the form
$$\begin{aligned} \frac{\mathrm{d}\phi }{\sqrt{(\phi _{1}-\phi )(\phi _{2}-\phi )(\phi _{3}-\phi )(\phi +\phi _{1}+\phi _{2}+\phi _{3})}}=\sqrt{-\mu }\zeta \end{aligned}$$(41)-
If we elect \(\phi \in (-\phi _{1}-\phi _{2}-\phi _{3},\phi _{1})\) with assumption \(\phi (\zeta _{0})=-\phi _{1}-\phi _{2}-\phi _{3}\), and integrate both sides of Eq. (41), then Eq. (10) possesses a new solution in the form
$$\begin{aligned} \phi =\phi _{3}+\frac{(\phi _{1}-\phi _{3})(\phi _{1}+\phi _{2}+2\phi _{3})}{(\phi _{3}-\phi _{1})+(2\phi _{1}+\phi _{2}+\phi _{3})\mathrm {sn^{2}}(\Omega _{1}(\zeta -\zeta _{0}),k_{1})} \end{aligned}$$(42)where
$$\begin{aligned}&\Omega _{1}=\frac{1}{2}\sqrt{-\mu (\phi _{3}-\phi _{1})(\phi _{1}+2\phi _{2}+\phi _{3})}, k_1\nonumber \\&\quad =\sqrt{\frac{(\phi _{3}-\phi _{2})(2\phi _{1}+\phi _{2}+\phi _{3})}{(\phi _{3}-\phi _{1})(\phi _{1}+2\phi _{2}+\phi _{3})}} \end{aligned}$$(43)Hence, Eq. (2) has a new wave solution in the form
$$\begin{aligned} \psi = \phi _{3}+\frac{(\phi _{1}-\phi _{3})(\phi _{1}+\phi _{2}+2\phi _{3})}{(\phi _{3}-\phi _{1})+(2\phi _{1}+\phi _{2}+\phi _{3})\mathrm {sn^{2}}(\Omega _{1}(\zeta -\zeta _{0}),k_{1})}-\frac{a_2}{3a_3}. \end{aligned}$$(44)If \(\phi _2=\phi _1\), the modules \(k_1=1\). Therefore, the solution (44) degenerates to
$$\begin{aligned} \psi =\phi _3+\frac{2(\phi _1^2-\phi _3^2)}{\phi _3-\phi _1+(3\phi _1+\phi _3)\mathrm {tanh}^2\Omega _1(\zeta -\zeta _0)} \end{aligned}$$(45)which is also a new solution for Eq. (2). When \(\phi _3=\phi _2\), the modules \(k_1=0\), and the solution (44) degenerates to
$$\begin{aligned} \psi =\phi _2+\frac{(\phi _1-\phi _2)(\phi _1+3\phi _2)}{\phi _2-\phi _1+2(\phi _1+\phi _2)\mathrm {sin}^2(\Omega _1(\zeta -\zeta _0)} \end{aligned}$$(46)which is also a novel solution for Eq. (2). Notice, when \(\alpha \rightarrow 1\), the solution (44) reduces to a new wave solution for the Eq. (2) with \(\alpha =1\).
-
With similar computations, we can select \(\phi \in (\phi _{2},\phi _{3})\) with \(\phi (\zeta _{0})=\phi _{2}\), and integrate both side of Eq. (2) to get the following novel solution of Eq. (10)
$$\begin{aligned} \psi =\phi _{1}-\frac{(\phi _{1}-\phi _{2})(\phi _{1}-\phi _{3})}{\phi _{1}-\phi _{3}+(\phi _{3}-\phi _{2})\mathrm {sn^{2}}(\Omega _{1}(\zeta -\zeta _{0}),k_{1})}-\frac{a_2}{3a_3} \end{aligned}$$(47)where \(k_{1}\) and \(\Omega _{1}\) are as defined in Eq. (43) above.
Notice that the solution (47) degenerates to a trivial solution for Eq. (2) whether \(k_1=1(\phi _1=\phi _2)\), or \(k_1=0(\phi _3=\phi _2)\). When \(\alpha \rightarrow 1\), the solution (47) becomes also a new solution for Eq. (2) with \(\alpha \rightarrow 1\).
-
Case 5: This case is determined by the constrains \(D_{4}=0\) and \(D_{2}D_{3}<0\). Based on these conditions, \(P_{4}(\phi )\) is written as \(P_{4}(\phi )=(\phi -\phi _{1})^{2}(\phi -\phi _{2})(\phi -\overline{\phi }_{2})\) where the bar indicates the complex conjugate and \(\phi _{1}=-\mathrm {Re} \phi _{2}\). Since \((\phi -\phi _{2})(\phi -\overline{\phi }_{2})=(\phi -\mathrm {Re}\phi _{2})^2+\mathrm {Im}\phi _{2}^2\ge 0\), then \(P_{4}(\phi )\ge 0\), and the conditions for real propagation are \(\mu \in (0,\infty )\) and \(\phi \in {\mathbb {R}}\), in which case the Eq. (10) becomes
Notice that the case in which \(\mu \in (-\infty ,0)\) is excluded since it does not give real propagation. Thus, for \(\mu \in (0,\infty )\), we select \(\phi\) such that \(\phi <\phi _{1}\), assume \(\phi (\zeta _{0})=-\infty\), and integrate both sides of the above equation using the substitution \(\phi =\phi _1-\frac{1}{y}\), the Eq. (2) has a new solution in the form
where \(\epsilon =\zeta _{0}-\frac{1}{\sqrt{\mu (4\phi _{1}^{2}+\mathrm {Im^{2}}\phi _{2})}}{\sinh ^{-1}(-2\phi _{1}/\mathrm {Im}\phi _{2})}\).
If \(\alpha \rightarrow 1\), the solution (49) is a well known solution for time integer derivative for Eq. (2). Similarly, we can find the solution when \(\phi >\phi _{1}\).
Case 6: This case is determined by \(D_{2}D_{3}\le 0\), and \(D_{4}>0\). These conditions guarantee the existence of two complex conjugate roots, namely \(\phi _{1}, \overline{\phi }_{1}, \phi _{2},\overline{\phi }_{2}\) for \(P_{4}(\phi )\). This means, \(P_{4}(\phi )\) takes the form \(P_{4}(\phi )=(\phi -\phi _{1})(\phi -\overline{\phi }_{1})(\phi -\phi _{2})(\phi -\overline{\phi }_{2})\), where \(\mathrm {Re}\phi _{1}=-\mathrm {Re} \phi _{2}\). Based on \((\phi -\phi _{k})(\phi -\overline{\phi }_{k})=(\phi -\mathrm {Re}\phi _{k})^2+\mathrm {Im}\phi _{k}^2\) for \(k=1,2\), then \(P_{4}(\phi )\ge 0\). Hence, there is a real propagation for Eq. (2) if \(\mu \in (0,\infty ), \phi \in {\mathbb {R}}\), where the equation in Eq. (10) becomes
Let
and
By evaluating the integrals of both side of Eq. (50), we get
where \(k_{2}=\sqrt{\frac{4AB}{\left( A+B\right) ^2}}\).
Therefore, we obtain a new solution for Eq. (2) in the form
It is obvious that when \(\alpha \rightarrow 1\), the solution (54) is also a new solution for the Eq. (2) with \(\alpha \rightarrow 1\). Now, let us investigate the degeneracy of the solution (54). It is easy to show that \(k_2=1\) if either one of the complex roots \(\phi _1\) or \(\phi _2\) is real, i.e., \(\mathrm {Im}\phi _1=0\) or \(\mathrm {Im}\phi _2=0\). Therefore, the solution (54) reduces to \(\psi =\mathrm {Re}\phi _1-\frac{a_2}{3a_3}\) which is trivial solution for Eq. (2).
Case 7: This case is characterized by \(D_{4}<0\) and \(D_{2}D_{3}\le 0\), which implies that \(P_{4}(\phi )\) has two real roots and two complex conjugate roots. That is,
where \(\phi _{1}<\phi _{2}\) and \(\mathrm {Re}\phi _{3}=-\frac{1}{2}(\phi _{1}+\phi _{2})\). Let \(A_{1}^{2}=\frac{1}{4}(\phi _{1}+3\phi _{1})^2+(\mathrm {Im}\phi _{3})^2\), \(B_{1}^{2}=\frac{1}{4}(3\phi _{1}+\phi _{2})^2+(\mathrm {Im}\phi _{3})^2\). Thus, we consider the following two cases
-
If \(\mu \in (0,\infty )\), then the choice \(\phi \in (-\infty ,\phi _{1}) \cup (\phi _{2},\infty )\) gives a real propagation where the Eq. (10) has the following form
$$\begin{aligned} \frac{\mathrm{d}\phi }{\sqrt{(\phi -\phi _{1})(\phi -\phi _{2})(\phi -\phi _{3})(\phi -\overline{\phi _{3}})}}=\sqrt{\mu } \mathrm{d}\zeta \end{aligned}$$(56)-
If \(\phi \in (\phi _{2},\infty )\), we choose \(\phi (\zeta _{0})=\phi _{2}\) and integrate both sides of Eq. (56), we get
$$\begin{aligned} \phi =\frac{\phi _1A_1+\phi _2B_1}{A_1+B_1}-\frac{2A_1B_1(\phi _1-\phi _2)}{(A_1+B_1)[(A_1+B_1)\mathrm {cn}(\sqrt{\mu A_1B_1}(\zeta -\zeta _0),k_3)-A_1+B_1]} \end{aligned}$$(57)where \(k_{3}=\sqrt{\frac{(A_{1}+B_{1})^{2}-(\phi _{2}-\phi _{1})^{2}}{4A_{1}B_{1}}}\) and \(A_1\),\(B_1\) as defined above.
Consequently, Eq. (2) has a new solution in the form
$$\begin{aligned} \psi =\frac{\phi _1A_1+\phi _2B_1}{A_1+B_1}-\frac{2A_1B_1(\phi _1-\phi _2)}{(A_1+B_1)[(A_1+B_1)\mathrm {cn}(\sqrt{\mu A_1B_1}(\zeta -\zeta _0),k_3)-A_1+B_1]}-\frac{a_2}{3a_3}. \end{aligned}$$(58)It is easy to show that if \(A_1+B_1=\phi _2-\phi _1\), then \(k_3=0\) and the solution (58) degenerates to
$$\begin{aligned} \psi =\phi _2-A_1+\frac{2A_1(A_1-\phi _2+\phi _1)}{(\phi _1-\phi _2)[\mathrm {cos}\sqrt{\mu _1A_1B_1}(\zeta -\zeta _0)]} \end{aligned}$$(59)which also is a new solution for Eq. (2). Similarly, \(k_3=1\) when \(A_1-B_1=\phi _2-\phi _1\) and the solution (58) degenerates to
$$\begin{aligned} \psi =\frac{\phi _2^2+(A_1-\phi _1)\phi _2+\phi _1A_1}{2A_1+\phi _2-\phi _1}-\frac{2A_1(A_1+\phi _2-\phi _1)(\phi _1-\phi _2)}{(2A_1+\phi _2-\phi _1)^2\mathrm {sech}\sqrt{\mu _1A_1B_1}(\zeta -\zeta _0)+(2A_1+\phi _2-\phi _1)(\phi _2-\phi _1)} \end{aligned}$$(60)which is also a novel solution for Eq. (2). Furthermore, when \(\alpha \rightarrow 1\), the solution (58) reduces to a new solution for integer time derivative of Eq. (2).
-
If \(\phi \in (-\infty ,\phi _{1})\), then by choosing \(\phi (\zeta _{0})=-\infty\), we get
$$\begin{aligned} \phi =\frac{\phi _{1}B_{1}-\phi _{2}A_{1}+ (\phi _{1}B_{1}+\phi _{2}A_{1})\mathrm {cn}(\sqrt{\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{3})}{(B_{1}-A_{1})+(A_{1}+B_{1})\mathrm {cn}(\sqrt{\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{3})} \end{aligned}$$(61)Thus, we get a novel solution for Eq. (2) in the form
$$\begin{aligned} \psi =\frac{\phi _{1}B_{1}-\phi _{2}A_{1}+ (\phi _{1}B_{1}+\phi _{2}A_{1})\mathrm {cn}(\sqrt{\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{3})}{(B_{1}-A_{1})+(A_{1}+B_{1})\mathrm {cn}(\sqrt{\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{3})}-\frac{a_2}{3a_3}. \end{aligned}$$(62)
-
-
If \(\mu \in (-\infty ,0)\), then the allowed interval for real propagation is \((\phi _{1},\phi _{2})\) in which case the Eq. (10) has the form of
$$\begin{aligned} \frac{\mathrm{d}\phi }{\sqrt{(\phi -\phi _{1})(\phi _{2}-\phi )(\phi -\phi _{3})(\phi -\overline{\phi _{3}})}}=\sqrt{-\mu } \mathrm{d}\zeta \end{aligned}$$(63)By choosing \(\phi \in (\phi _{1},\phi _{2})\) with the assumption \(\phi (\zeta _{0})=\phi _{1}\), and integrating both side of Eq. (63), we obtain
$$\begin{aligned} \phi =\frac{\phi _{2}B_{1}+\phi _{1}A_{1}-(\phi _{1}A_{1}+\phi _{2}B_{1})\mathrm {cn}(\sqrt{-\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{4})}{B_{1}+A_{1}-(B_{1}-A_{1})\mathrm {cn}(\sqrt{-\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{4})} \end{aligned}$$(64)where \(k_{4}=\sqrt{\frac{(\phi _{2}-\phi _{1})^{2}-(A_{1}-B_{1})^{2}}{4A_{1}B_{1}}}\) and \(A_1\), \(B_1\) as defined above
Consequently, A new solution for Eq. (2) is written as
$$\begin{aligned} \psi =\frac{\phi _{2}B_{1}+\phi _{1}A_{1}-(\phi _{1}A_{1}+\phi _{2}B_{1})\mathrm {cn}(\sqrt{-\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{4})}{B_{1}+A_{1}-(B_{1}-A_{1})\mathrm {cn}(\sqrt{-\mu A_{1}B_{1}}(\zeta -\zeta _{0}),k_{4})}-\frac{a_2}{3a_3}. \end{aligned}$$(65)
Case 8: If \(D_{3}>0\) \(D_{2}>0\), and \(D_{4}=0\) then \(P_{4}(\phi )\) has four zeros which one of them is doubled and the others are simple. That is,
where \(\phi _{1}<\phi _{2}<0<\phi _{3}\) and \(\phi _{3}=-(2\phi _{1}+\phi _{2})\). We consider the following cases:
-
If \(\mu \in (0,\infty )\), then the real propagation occurs if \(\phi \in (-\infty ,\phi _{1})\cup (\phi _{1},\phi _{2})\cup (\phi _{3},\infty )\), in which case the Eq. (10) becomes
$$\begin{aligned} \frac{\mathrm{d}\phi }{\left| \phi -\phi _{1}\right| \sqrt{(\phi -\phi _{2})(\phi +2\phi _{1}+\phi _{2})}}=\sqrt{\mu } \mathrm{d}\zeta \end{aligned}$$(67)By choosing \(\phi \in ( -2\phi _{1}-\phi _{2},\infty )\) and assuming that \(\phi (\zeta _{0})=-(2\phi _{1}+\phi _{2})\), we have
$$\begin{aligned} \phi =\phi _1+\frac{(3\phi _1+\phi _2)(\phi _2-\phi _1)}{\phi _1+\phi _2+2\phi _1\mathrm {cos}\sqrt{\mu (\phi _2-\phi _1)(3\phi _1+\phi _2)}(\xi -\epsilon )}. \end{aligned}$$(68)Hence, Eq. (2) has a new solution in the form
$$\begin{aligned} \psi =\phi _1+\frac{(3\phi _1+\phi _2)(\phi _2-\phi _1)}{\phi _1+\phi _2+2\phi _1\mathrm {cos}\sqrt{\mu (\phi _2-\phi _1)(3\phi _1+\phi _2)}(\xi -\epsilon )}-\frac{a_{2}}{3a_{3}}. \end{aligned}$$(69)Similarly, we can find the solution for \(\phi \in (-\infty ,\phi _{1})\) \(\cup (\phi _{1},\phi _{2})\).
-
If \(\mu \in (-\infty ,0)\), then the allowed interval for real propagation is \((\phi _{2},\phi _{3})\). By choosing \(\phi \in (\phi _{2},\phi _{3})\) with the assumption \(\phi (\zeta _{0})=\phi _{2}\), and where Eq. (10) becomes
$$\begin{aligned} \frac{\mathrm{d}\phi }{\left( \phi -\phi _{1}\right) \sqrt{(\phi -\phi _{2})(-\phi +2\phi _{1}+\phi _{2})}}=\sqrt{-\mu } \mathrm{d}\zeta \end{aligned}$$(70)By integrating both side of Eq. (70), we obtain
$$\begin{aligned} \phi =\phi _{1}+\frac{(\phi _{2}-\phi _{1})(3\phi _{1}+\phi _{2})}{2\phi _{1}+(\phi _{1}+\phi _{2})\mathrm {cosh}\sqrt{-\mu (3\phi _{1}+\phi _{2})(\phi _{1}-\phi _{2})}(\zeta -\epsilon )} \end{aligned}$$(71)Thus, the Eq. (2) has a new solution in the form
$$\begin{aligned} \psi =\phi _{1}+\frac{(\phi _{2}-\phi _{1})(3\phi _{1}+\phi _{2})}{2\phi _{1}+(\phi _{1}+\phi _{2})\mathrm {cosh}\sqrt{-\mu (3\phi _{1}+\phi _{2})(\phi _{1}-\phi _{2})}(\zeta -\epsilon )}-\frac{a_{2}}{3a_{3}}. \end{aligned}$$(72)
Case 9: If \(D_{2}<0\) and \(D_{3}=D_{4}=0\), then \(P_{4}(\phi )\) has two doubled imaginary roots along with their conjugates. i.e., \(P_{4}(\phi )=(\phi -\phi _{1})^{2}(\phi -\overline{\phi _{1}})^{2}\ge 0\). The real propagation occurs only if \(\mu >0\). Therefore, Eq. (10) has the following form
We postulate \(\phi (\zeta _{0})=0\) and integrate both sides of the last equation to obtain
Hence, Eq. (2) has a novel solution in the form
Figure 6a, b outlines the 3D graphic representation for diverse values of \(\alpha\). Figure 6c illustrates the width of the solution (75) increases as the fractional order \(\alpha\) increases. Notice, when \(\alpha \rightarrow 1\), the solution (75) is reduced to a well known solution for the integer order derivative [61].
Conclusions
This work has endeavored to study the problem for constructing wave solutions for conformable time fractional version of the generalized Pochhammer–Chree equation. A certain transformation has been applied to transform the equation under consideration into an a second order ordinary differential equation which is integrated once to give the differential form (10). The key step to integrate this differential form is knowing the types of the roots of the the polynomial \(P_4(\phi )\). The complete discrimination system of this polynomial has been employed and has implied about nine cases. For each case, we determined the intervals of real propagation and integrated the differential form of Eq. (10) along these intervals. There are several intervals of real propagation corresponding to each type of the zeros of \(P_4(\phi )\) which have been enabled us to construct more than one solution for the equation under consideration. Finally, we have illustrated some of these solutions graphically for different values of the fractional order derivatives. Some of these solutions will be reduced to new wave solutions for the generalized Pochhammer- Chree equation when the fractional order derivative approaches one. We also investigate the degeneracy of some solutions involving Jacobi-elliptic functions.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equation, 1st edn. Elsevier Science, Amsterdam (2006); ISSN (Series): 0304-0208
Kubica, A., Ryszewska, K., Yamamoto, M.: Time-Fractional Differential Equations A Theoretical Introduction. Springer Nature Singapore Pte Ltd, Singapore (2020); ISSN 2191-8198
Tarasov, V.: Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2010); ISSN 1867-8440
Bekhouche, F., Komashynska, I.: Traveling wave solutions for the space-time fractional (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff equation via two different methods. Int. J. Math. Comput. Sci. 16(4), 1729–1744 (2021)
Pandeyl, R., Mishrai, H.: The numerical solution of time fractional Kuramoto–Sivashinsky equations via homotopy analysis fractional Sumudu transform method. Math. Eng. Sci. Aerosp. 12(3), 863–882 (2021)
Thabet, H., Kendre, S., Peters, J., Kaplan, M.: Solitary wave solutions and traveling wave solutions for systems of time-fractional nonlinear wave equations via an analytical approach. Comput. Appl. Math. 39, 144 (2020)
Jena, R., Chakraverty, S.: Residual power series method for solving time-fractional model of vibration equation of large membranes. J. Appl. Comput. Mech. 5(4), 603–615 (2019)
Li, C., Kosti’c, M., Li, M., Piskarev, S.: On a class of time-fractional differential equations. Int. J. Theory Appl. 2012(4), 639-668 (2012)
Dehestania, H., Ordokhania, Y., Razzaghib, M.: Numerical solution of variable-order time fractional weakly singular partial integro-differential equations with error estimation. Math. Model. Anal. 25(4), 680–701 (2020)
Cerdik Yaslan, H.: Numerical solution of the nonlinear conformable space-time fractional partial differential equations. Indian J. Pure Appl. Math. 52, 407–419 (2021)
Phoosree, S., Chinviriyasit, S.: New analytic solutions of some fourth-order nonlinear space-time fractional partial differential equations by \(\frac{{G^{\prime}}}{{G}}\) -expansion method. Songklanakarin J. Sci. Technol. 43(3), 795–801 (2021)
Hosseini, K., Bekir, A., Kaplan, M., Guner, O.: On a new technique for solving the nonlinear conformable time-fractional differential equations. Opt. Quantum Electron. 49, 343 (2017)
Khan, N., Ahmed, S.: Finite difference method with metaheuristic orientation for exploration of time fractional partial differential equations. Int. J. Appl. Comput. Math. 7, 121 (2021)
Topsakal, M., Ta şcan, F.: Exact travelling wave solutions for space-time fractional Klein-Gordon equation and (2+1)-dimensional time-fractional Zoomeron equation via auxiliary equation method. Appl. Math. Nonlinear Sci. 5(1), 437–446 (2020)
Eskandari, E., Taghizadeh, N.: Exact solutions of two nonlinear space-time fractional differential equations by application of Exp-function method. Appl. Appl. Math. 15(2), 970–977 (2020)
Sadri, K., Hosseini, K., Baleanu, D., Ahmadian, A., Salahshour, S.: Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel. Adv. Differ. Equ. 2021, 348 (2021)
Akbulut, A., Kaplan, M.: Auxiliary equation method for time-fractional differential equations with conformable derivative. Comput. Math. Appl. 75, 876–882 (2018)
Darvishi, M.T., Najafi, M.: Propagation of sech-type solutions for conformable fractional nonlinear Schrodinger models. Comput. Methods Sci. Eng. 2(2), 35 (2020)
Darvishi, M.T., Najafi, M., Wazwaz, A.M.: Some optical soliton solutions of space-time conformable fractional Schrödinger-type models. Phys. Scr. 96(6), 065213 (2021)
Darvishi, M.T., Najafi, M., Wazwaz, A.M.: Conformable space-time fractional nonlinear (1+ 1)-dimensional Schrödinger-type models and their traveling wave solutions. Chaos Solitons Fractals 150, 111187 (2021)
Darvishi, M.T., Najafi, M., Shin, B.C.: Conformable fractional sense of foam drainage equation and construction of its solutions. J. Korean Soc. Ind. Appl. Math. 25(3), 132–148 (2021)
Kumar, D., Paul, G.C., Seadawy, A.R., Darvishi, M.T.: A variety of novel closed-form soliton solutions to the family of Boussinesq-like equations with different types. J. Ocean Eng. Sci. (2021). https://doi.org/10.1016/j.joes.2021.10.007
Ali, M., Alquran, M., Jaradat, I.: Explicit and approximate solutions for the conformable-caputo time-fractional diffusive predator-prey model. Int. J. Appl. Comput. Math. 7(3), 1–11 (2021)
Bekhouche, F., Alquran, M., Komashynska, I.: Explicit rational solutions for time-space fractional nonlinear equation describing the propagation of bidirectional waves in low-pass electrical lines. Rom. J. Phys. 66, 7–8 (2021)
Alquran, M., Yousef, F., Alquran, F., Sulaiman, T.A., Yusuf, A.: Dual-wave solutions for the quadratic-cubic conformable-Caputo time-fractional Klein–Fock–Gordon equation. Math. Comput. Simul. 185, 62–76 (2021)
Alquran, M.: Optical bidirectional wave-solutions to new two-mode extension of the coupled KdV-Schrodinger equations. Opt. Quant. Electron. 53(10), 1–9 (2021)
Alquran, M.: Physical properties for bidirectional wave solutions to a generalized fifth-order equation with third-order time-dispersion term. Results Phys. 28, 104577 (2021)
Pelissier, M., Hoeber, H., van de Coevering, N., Jones, I.: Classics of Elastic Wave Theory. Society of Exploration Geophysicists, Tulsa (2007)
Beltran-Carbajal, F.: Advances in Vibration Engineering and Structural Dynamics. Intechopen, London (2012); ISBN: 978-953-51-0845-0
Hussain, C.: Handbook of Nanomaterials for Industrial Applications. Elsevier Inc (2018); ISBN: 978-0-12-813351-4
Zill, D.: A First Course in Differential Equations with Modeling Applications. Brooks/Cole, USA (2013); ISBN-13: 978-1111827052
Swigon, D., Coleman, B., Tobias, I.: The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes. Biophys. J. 47, 2515–2530 (1998)
Chree, C.: Longitudinal vibrations of a circular bar. Q. J. Math. 21, 287–298 (1886)
Pochhammer, L.: Biegung des kreiscylinders-fortpflanzungsgeschwindigkeit kleiner schwingungen in einem kreiscylinder. Journal fr die reine und angewandte Mathematik 81, 326–336 (1876)
Bagolubasky, I.: Some examples of inelastic soliton interaction. Comput. Phys. Commun. 13, 149–155 (1977)
Clarkson, P., LeVaque, R., Saxton, R.: Solitary wave interactions in elastic rods. Stud. Appl. Math. 1986(75), 95–122 (1986)
Parker, A.: On exact solutions of the regularized long wave equation: a direct approach to partially integrable equations. J. Math. Phys. 36, 3498–3505 (1995)
Shawagfeh, N., Kaya, D.: Series solution to the Pochhammer–Chree equation and comparison with exact solutions. Comput. Math. Appl. 47, 1915–1920 (2004)
Zhang, W., Wenxiu, M.: Explicit solitary wave solutions to generalized Pochhammer–Chree equation. J. Appl. Math. Mech. 20, 666–674 (1999)
Feng, Z.: On explicit exact solutions for the Lienard equation and its applications. Phys. Lett. A 293, 50–56 (2002)
Parand, K., Rad, J.: Some solitary wave solutions of generalized Pochhammer–Chree equation via Exp-function method. Int. J. Math. Comput. Sci. 4(7), 991-996 (2010)
Rani, A., Khan, N., Ayub, K., Khan, M., Ul-Hassan, M., Ahmed, B., Ashraf, M.: Solitary wave solution of nonlinear PDEs arising in mathematical physics. Open Phys. 17, 381–389 (2019)
Mohebbi, A.: Solitary wave solutions of the nonlinear generalized Pochhammer–Chree and regularized long wave equations. Nonlinear Dyn. 70, 2463–2474 (2012)
Ilyashenko, A., Kuznetsov, S.: Longitudinal Pochhammer–Chree waves in mild auxetics and non-auxetics. J. Mech. 35(3), 327–334 (2019)
Achab, A.E.L.: On the integrability of the generalized Pochhammer–Chree (PC) equations. Phys. A Stat. Mech. Appl. 545, 123576 (2020)
Wazwaz, M.: The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the Pochhammer–Chree equations. Appl. Math. Comput. 195, 24–33 (2008)
Yang, L., Hou, X.R., Zeng, Z.B.: A complete discrimination system for polynomials. Sci. China Ser. E 39(6), 628–646 (1996)
Cheng-Shi, L.: Exact travelling wave solutions for (1+ 1)-dimensional dispersive long wave equation. Chin. Phys. 14(9), 1710 (2005)
Liu, C.: Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations. Comput. Phys. Commun. 181, 317–324 (2010)
Kai, Y., Chen, S., Zheng, B., Zhang, K., Yang, N., Xu, W.: Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial method. Chaos Solitons Fractals 141, 110314 (2020)
Yang, N., Xu, W., Zhang, K., Zheng, B.: Exact solutions to the space-time fractional shallow water wave equation via the complete discrimination system for polynomial method. Results Phys. 20, 103728 (2021)
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Syst. Appl. 10(2), 109–137 (2015)
Rahmat, M.: A new definition of conformable fractional derivative on arbitrary time scales. Adv. Differ. Equ., 354 (2019). https://doi.org/10.1186/s13662-019-2294-y
Elbrolosy, M.E., Elmandouh, A.A.: Dynamical behaviour of nondissipative double dispersive microstrain wave in the microstructured solids. Eur. Phys. J. Plus 136(9), 1–20 (2021)
AL Nuwairan, M., Elmandouh, A.: Qualitative analysis and wave propagation of the nonlinear model for low-pass electrical transmission lines. Phys. Scr. 96, 095214 (2021)
Elmandouh, A.A.: Integrability, qualitative analysis and the dynamics of wave solutions for Biswas-Milovic equation. Eur. Phys. J. Plus 136(6), 1–17 (2021)
Elmandouh, A.A.: Bifurcation and new traveling wave solutions for the 2D Ginzburg-Landau equation. Eur. Phys. J. Plus 135(8), 1–13 (2020)
Elbrolosy, M.E., Elmandouh, A.A.: Bifurcation and new traveling wave solutions for (2 + 1)-dimensional nonlinear Nizhnik–Novikov–Veselov dynamical equation. Eur. Phys. J. Plus 135(6), 533 (2020)
Byrd, P.F., Fridman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1971); ISBN: 0-387-05318-2
Li, J., Zhang, L.: Bifurcations of traveling wave solutions in generalized Pochhammer–Chree equation. Chaos Solitons Fractals 14(4), 581–593 (2002)
Acknowledgements
The author acknowledge the Deanship of Scientific Research at King Faisal University for the financial support.
Funding
This work was supported through the Annual Funding track by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Project No. AN000518].
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that the research was conducted in the absence of any conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Conformable derivatives
Appendix: Conformable derivatives
As we know the fractional calculus is more suitable in describing the real world problems appearing in engineering and physical science. Recently, scholars study the fractional calculus and introduced new operators such as Caputo, Riemann Liouville and conformable fractional operator. The usage of the conformable fractional operator overcomes some restrictions of the different fractional operator’s properties such as the chain rule, the derivative of the quotient of two functions, product of two functions, and mean value theorem. Thus, it becomes more interesting in describing many physical problems.
Definition 1
[52] Let \(g:(0,\infty )\rightarrow {\mathbb {R}}\) be a function, then the conformable fractional derivative of order \(\alpha\) is defined as
for all \(t>0\) and \(0<\alpha \le 1\).
We present some significant properties of the conformable derivatives. Let the two functions \(g_1, g_2\) are \(\alpha -\) conformable differential for \(t>0\) and a, b are two constants. We have the following properties
-
1.
\(T_\alpha (ag_1+bg_2)=aT_\alpha (g_1)+bT_\alpha (g_2)\),
-
2.
\(T_\alpha (t^\rho )=\rho t^{\rho -\alpha }\), for all \(\rho \in {\mathbb {R}}\),
-
3.
\(T_\alpha (g_1g_2)=g_1T_\alpha (g_2)+g_2T_\alpha (g_1)\).
-
4.
\(T_\alpha \left( \frac{g_1}{g_2}\right) =\frac{1}{g_{2}^{2}}\left( g_2T_\alpha (g_1)-g_1T_\alpha (g_1)\right)\)
-
5.
\(T_\alpha (g)(t)=t^{1-\alpha }\frac{\mathrm{d}g}{\mathrm{d}t}(t)\).
-
6.
If \(g:(0,\infty ) \rightarrow {\mathbb {R}}\) is a map which is differentiable and \(\alpha -\) differentiable and f is another function which is defined in the range of g, then \(T_{\alpha }(g\circ f)=t^{1-\alpha }f^\prime (t)g^\prime (f(t))\).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
AL Nuwairan, M. The exact solutions of the conformable time fractional version of the generalized Pochhammer–Chree equation. Math Sci 17, 305–316 (2023). https://doi.org/10.1007/s40096-022-00471-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40096-022-00471-3
Keywords
- Time fractional differential equation
- Generalized Pochhammer–Chree equation
- Elastic wave
- Exact solutions
- The complete discrimination system of a polynomial