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Dispersion analysis and improved F-expansion method for space–time fractional differential equations

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Abstract

In this article, an improved F-expansion method with the Riccati equation is suggested for space–time fractional differential equations for exact solutions. The fractional complex transformation is used to convert the space–time fractional differential equations into ordinary differential equations. The application of the method is described by solving space–time fractional potential Yu–Toda–Sasa–Fukuyama equation, and the solutions of the equation are successfully established in terms of the hyperbolic, trigonometric and rational types of functions. The graphical analysis describes the effect of fractional orders \(\alpha \), \(\beta \), \(\gamma \), \(\delta \) of time and space derivatives, respectively, on the wave profile of solutions. The dispersion relation is obtained using the linear analysis, and it shows that waves follow anomalous or normal dispersion depending upon space or time fractional-order values.

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Bikramjeet Kaur wishes to thank University Grants Commission (UGC), New Delhi, India, for the financial support under Grant No. (F1-17.1/2013-14/MANF-2013-14-SIK-PUN-21763). Rajesh Kumar Gupta thanks Council of Scientific and Industrial Research (CSIR), India, for the financial support under Grant No. 25(0257)/16/EMR-II.

Appendix

Appendix

1.1 Formulation of space–time fractional potential YTSF Eq. (4)

The Lagrangian [3, 24] for the potential YTSF Eq. (3) can be obtained as follows

$$\begin{aligned}&\mathcal {L}(u_{x},u_{y},u_{z},u_{t},u_{xx},u_{xz})\nonumber \\&\quad =2u_{x}u_{t} +\frac{1}{2}u_{xx}u_{xz}-\frac{3}{2}u_{y}^{2}-u_{x}^{2}u_{z}. \end{aligned}$$
(53)

Similarly, the Lagrangian for space–time fractional Eq. (4) is obtained as follows

$$\begin{aligned}&F(D_{x}^{\beta }u,D_{y}^{\gamma }u,D_{z}^{\delta }u,D_{t}^{\alpha }u, D_{x}^{2\beta }u,D_{x}^{\beta }D_{z}^{\delta }u)\nonumber \\&\quad =2D_{x}^{\beta }uD_{t}^{\alpha }u +\frac{1}{2}D_{x}^{2\beta }uD_{x}^{\beta }D_{z}^{\delta }u\nonumber \\&\qquad -\frac{3}{2}(D_{y}^{\gamma }u)^{2}-(D_{x}^{\beta }u)^{2}D_{z}^{\delta }u, \end{aligned}$$
(54)

where \(D_{x}^{\beta }\), \(D_{y}^{\gamma }\), \(D_{z}^{\delta }\), \(D_{t}^{\alpha }\), \(D_{x}^{2\beta }\) denote the fractional derivatives \(\frac{\partial ^{\beta }}{\partial {x}^{\beta }}\), \(\frac{\partial ^{\gamma }}{\partial {y}^{\gamma }}\), \(\frac{\partial ^{\delta }}{\partial {z}^{\delta }}\), \(\frac{\partial ^{\alpha }}{\partial {t}^{\alpha }}\), \(\frac{\partial ^{2\beta }}{\partial {x}^{2\beta }}\), respectively. The functional for Eq. (4) can be written as follows

$$\begin{aligned} J_{F}(u)= & {} \int _{R}(\mathrm{d}x)^{\beta }\int _{R}(\mathrm{d}y)^{\gamma }\int _{R}(\mathrm{d}z)^{\delta } \int _{R}(\mathrm{d}t)^{\alpha }\nonumber \\&\cdot F(D_{x}^{\beta }u,D_{y}^{\gamma }u,D_{z}^{\delta } u,D_{t}^{\alpha }u,D_{x}^{2\beta }u,D_{x}^{\beta }D{z}^{\delta }u), \end{aligned}$$
(55)

The variation of functional for Eq. (55) is given by using the method given in [4, 5, 24] and is obtained as follows

$$\begin{aligned} \varDelta J_{F}(u)=&\int _{R}(\mathrm{d}x)^{\beta }\int _{R}(\mathrm{d}y)^{\gamma }\int _{R}(\mathrm{d}z)^{\delta } \int _{R}(\mathrm{d}t)^{\alpha }\nonumber \\&\cdot \left[ \frac{\partial F}{\partial D_{t}^{\alpha }u} \varDelta D_{t}^{\alpha }u+\frac{\partial F}{\partial D_{x}^{\beta }u} \varDelta D_{x}^{\beta }u\right. \nonumber \\&+\frac{\partial F}{\partial D_{y}^{\gamma }u}\varDelta D_{y}^{\gamma }u +\frac{\partial F}{\partial D_{z}^{\delta }u}\varDelta D_{z}^{\delta }u\nonumber \\&\left. +\frac{\partial F}{\partial D_{x}^{2\beta }u}\varDelta D_{x}^{2\beta }u +\frac{\partial F}{\partial D_{x}^{\beta }D_{z}^{\delta }u} \varDelta D_{x}^{\beta }D_{z}^{\delta }u\right] , \end{aligned}$$
(56)

where \({\int }_{a}^{\tau }(\mathrm{d}\tau )^{j}f(\tau )=j\int _{a}^{t} \mathrm{d}(\tau )(t-\tau )^{j}f(\tau )\). Using the fractional integration by parts

$$\begin{aligned}&\int _{a}^{b}(\mathrm{d}Z)^{j}f(Z)D_{Z}^{j}g(Z)\nonumber \\&\quad =\varGamma (1+j)\left[ g(Z)f(Z)|_{a}^{b} -\int _{a}^{b}\mathrm{d}(Z)^{j}g(Z)D_{Z}^{j}f(Z)\right] ,\nonumber \\&\qquad f(Z),g(Z)\in [a,b], \end{aligned}$$
(57)

we can obtain

$$\begin{aligned} \varDelta J_{F}(u)=&\int _{R}(\mathrm{d}x)^{\beta }\int _{R}(\mathrm{d}y)^{\gamma }\int _{R}(\mathrm{d}z)^{\delta } \int _{R}(\mathrm{d}t)^{\alpha }\nonumber \\&\cdot \left[ -D_{t}^{\alpha } \left( \frac{\partial F}{\partial D_{t}^{\alpha }u}\right) -D_{x}^{\beta } \left( \frac{\partial F}{\partial D_{x}^{\beta }u}\right) \right. \nonumber \\&-D_{y}^{\gamma }\left( \frac{\partial F}{\partial D_{y}^{\gamma }u}\right) -D_{z}^{\delta }\left( \frac{\partial F}{\partial D_{z}^{\delta }u}\right) \nonumber \\&\left. +D_{x}^{2\beta }\left( \frac{\partial F}{\partial D_{x}^{2\beta }u}\right) +D_{x}^{\beta }D_{z}^{\delta }\left( \frac{\partial F}{\partial D_{x}^{\beta } D_{z}^{\delta }u}\right) \right] . \end{aligned}$$
(58)

Optimising the variation Eq. (56), \(\varDelta J_{F}(u)=0\), we can obtain the Euler–Lagrangian equation for Eq. (4) as follows

$$\begin{aligned}&-D_{t}^{\alpha }\left( \frac{\partial F}{\partial D_{t}^{\alpha }u}\right) -D_{x}^{\beta }\left( \frac{\partial F}{\partial D_{x}^{\beta }u}\right) -D_{y}^{\gamma } \left( \frac{\partial F}{\partial D_{y}^{\gamma }u}\right) \nonumber \\&\quad -D_{z}^{\delta }\left( \frac{\partial F}{\partial D_{z}^{\delta }u}\right) +D_{x}^{2\beta }\left( \frac{\partial F}{\partial D_{x}^{2\beta }u}\right) \nonumber \\&\quad +D_{x}^{\beta }D_{z}^{\delta }\left( \frac{\partial F}{\partial D_{x}^{\beta }D_{z}^{\delta }u}\right) =0. \end{aligned}$$
(59)

Substituting Eq. (54) into Eq. (59), we obtain space–time fractional potential YTSF Eq. (4) as follows

$$\begin{aligned}&-4\,\frac{\partial ^{\beta }}{\partial x^{\beta }} \left( \frac{\partial ^{\alpha }u}{\partial t^{\alpha }}\right) +\frac{\partial ^{3\beta }}{\partial x^{3\beta }} \left( \frac{\partial ^{\delta }u}{\partial z^{\delta }}\right) +3\,\frac{\partial ^{2\gamma }u}{\partial y^{2\gamma }}\nonumber \\&\quad +4\,\left( \frac{\partial ^{\beta }u}{\partial x^{\beta }}\right) \left( \frac{\partial ^{\beta }}{\partial x^{\beta }} \left( \frac{\partial ^{\delta }u}{\partial z^{\delta }}\right) \right) \nonumber \\&\quad +2\,\left( \frac{\partial ^{2\beta }u}{\partial x^{2\beta }} \right) \left( \frac{\partial ^{\delta }u}{\partial z^{\delta }}\right) =0. \end{aligned}$$
(60)

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Kaur, B., Gupta, R.K. Dispersion analysis and improved F-expansion method for space–time fractional differential equations. Nonlinear Dyn 96, 837–852 (2019). https://doi.org/10.1007/s11071-019-04825-w

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  • DOI: https://doi.org/10.1007/s11071-019-04825-w

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