New exact solutions for the space-time fractional Kawahara equation

Daşçıoğlu, Ayşegül; Ünal, Sevil Çulha

doi:10.1016/j.apm.2020.07.053

Cited by 12 publications

(8 citation statements)

References 35 publications

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“…Infinitely many solutions can be found depending on K , ω , and m by utilizing these theorems and corollaries. The elementary and elliptic function solutions of Equation can be seen in Daşcıoğlu and Çulha Ünal 24 . In this paper, utilizing these elementary and elliptic function solutions, the exact solutions of Equation are obtained.…”

Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

“…In the literature, Equation has not so much solutions. Therefore, theorems and corollaries are presented to gain the new solutions of Equation 3 in Daşcıoğlu and Çulha Ünal 24 . Some of theorems and corollaries are as follows:Theorem Let F ( ξ ) be a solution of Equation .…”

Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

“…Solutions of Equation that have the same product PR can be written as a group according to their Q values. In this way, Table 1 is formed sixteen different groups as in Daşcıoğlu and Çulha Ünal 24 …”

Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

“…After that, setting its coefficients to be zero, the following equations system is obtained

k a_{1} + italicλl a_{0} a_{1} + l^{3} {Qa}_{1} + italicεl p^{2} {Qa}_{1} + italicεl r^{2} Q a_{1} = 0

2 k a_{2} + 2 italicλl a_{0} a_{2} + italicβl a_{1}^{2} + 8 l^{3} Q a_{2} + 8 italicεl p^{2} {Qa}_{2} + 8 italicεl r^{2} {Qa}_{2} = 0

italicλl a_{1} a_{2} + 2 l^{3} P a_{1} + 2 italicεl p^{2} {Pa}_{1} + 2 italicεl r^{2} {Pa}_{1} = 0

italicλl a_{2}^{2} + 12 l^{3} {Pa}_{2} + 12 italicεl p^{2} {Pa}_{2} + 12 italicεl r^{2} {Pa}_{2} = 0 .

Solving this nonlinear system, the unknown coefficients are found

a_{0} = B + 4 italicQA, a_{1} = 0, a_{2} = 12 italicPA

such that A = − ( l 2 + εp 2 + εr 2 )/ λ and B = − k /( λl ). Hence, the solution of Equation is

u = B + 4 italicQA + 12 italicPA F^{2} .

Substituting the P , Q and F given in Daşcıoğlu and Çulha Ünal…”

Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

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Exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov equation

Ünal

Daşçıoğlu

2021

Math Methods in App Sciences

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In this study, an analytic method based on the Jacobi elliptic functions has been presented to obtain the exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation. This equation is reduced to a nonlinear ordinary differential equation. The elementary and elliptic function solutions of this equation are investigated. Many exact solutions containing the rational, complex, trigonometric, and hyperbolic functions are also found. Besides, some of the solutions are demonstrated by the graphics.

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Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

“…After that, setting its coefficients to be zero, the following equations system is obtained

k a_{1} + italicλl a_{0} a_{1} + l^{3} {Qa}_{1} + italicεl p^{2} {Qa}_{1} + italicεl r^{2} Q a_{1} = 0

2 k a_{2} + 2 italicλl a_{0} a_{2} + italicβl a_{1}^{2} + 8 l^{3} Q a_{2} + 8 italicεl p^{2} {Qa}_{2} + 8 italicεl r^{2} {Qa}_{2} = 0

italicλl a_{1} a_{2} + 2 l^{3} P a_{1} + 2 italicεl p^{2} {Pa}_{1} + 2 italicεl r^{2} {Pa}_{1} = 0

italicλl a_{2}^{2} + 12 l^{3} {Pa}_{2} + 12 italicεl p^{2} {Pa}_{2} + 12 italicεl r^{2} {Pa}_{2} = 0 .

Solving this nonlinear system, the unknown coefficients are found

a_{0} = B + 4 italicQA, a_{1} = 0, a_{2} = 12 italicPA

such that A = − ( l 2 + εp 2 + εr 2 )/ λ and B = − k /( λl ). Hence, the solution of Equation is

u = B + 4 italicQA + 12 italicPA F^{2} .

Substituting the P , Q and F given in Daşcıoğlu and Çulha Ünal…”

Section: Solutions Of Time Fractional Kdv–zk Equationmentioning

confidence: 99%

See 2 more Smart Citations

Exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov equation

Ünal

Daşçıoğlu

2021

Math Methods in App Sciences

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“…Some of them are seen in Ref. [32][33][34][35][36]. The Jacobi elliptic function solutions of Eq.…”

Section: Solutions Of the Space-time Fractional Srlw Eqquationmentioning

confidence: 99%

Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation

Ünal¹,

Daşçıoğlu²,

Bayram³

2021

Mathematical Sciences and Applications E-Notes

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In this paper, by using a direct method based on the Jacobi elliptic functions, the exact solutions of the space-time fractional symmetric regularized long wave (SRLW) equation have been obtained. The elliptic function solutions of a nonlinear ordinary differential (auxiliary) equation (dF /dξ) 2 = P F 4 (ξ)+QF 2 (ξ)+ R have also been examined. Besides, the solutions have been found in general form including rational, trigonometric and hyperbolic functions. Moreover, the complex valued solutions, periodic solutions, and soliton solutions, have also been gained. Some solutions have been illustrated by the graphics.

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On the Van der Waals model on granular matters with truncated M-fractional derivative

Li,

Rezazadeh,

Sabi’u

et al. 2024

Opt Quant Electron

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In this work, exact solutions of the Van der Waals model (vdWm) are investigated with a new algebraic analytical method. The closed-form analysis of the vdW equation arising in the context of the fluidized granular matter is implemented under the effect of time-fractional M-derivative. The vdWm is a challenging problem in the modelling of molecules and materials. Noncovalent Van der Waals or dispersion forces are frequent and have an impact on the structure, dynamics, stability, and function of molecules and materials in biology, chemistry, materials science and physics. The auxiliary equation which is known as a direct analytical method is constructed for the nonlinear fractional equation. The process includes a transformation based on Weierstrass and Jacobi elliptic functions. Wave solutions of the model are analytically verified for the various cases. Then, graphical patterns are presented to show the physical explanation of the model interactions. The achieved solutions will be of high significance in the interaction of quantum-mechanical fluctuations, granular matter and other areas of vdWm applications.

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New exact solutions for the space-time fractional Kawahara equation

Cited by 12 publications

References 35 publications

Exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov equation

Exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov equation

Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation

On the Van der Waals model on granular matters with truncated M-fractional derivative

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