Numerical experiments for long nonlinear internal waves via Gardner equation with dual-power law nonlinearity

Ak, Turgut

doi:10.1142/s0129183119500669

Cited by 4 publications

(8 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To be able to make comparison between numerical and analytical solutions, solitary wave solution of GE is studied to see advance of wave during running time of the algorithm, which is defined analytically in some studies [1, 10] as

\begin{array}{c} u (x, t) goodbreak= S \sec h ((), k ((), x goodbreak- x_{0} goodbreak- italicct)) \\ k goodbreak= \sqrt{\frac{c}{μ_{3}}}, S goodbreak= \frac{6 c}{μ_{1} ((), 1 goodbreak+ \sqrt{1 + 6 \frac{μ_{2} c}{μ_{1}^{2}}})} . \end{array}

With parameters

μ_{1} = 1,

μ_{2} = 1

μ_{3} = 5

, and

x_{0} = 0,

initial profiles, having bell‐type shape is locate along the

x

‐axis centered at

x = 0

in the restricted region

false[- 100, 100 false]

. Three BCs are used to have solvable system of algebraic equations: two of them are on left,

u false(- 100, t false)

and

u_{x} false(- 100, t false)

and one is on the right,

u false(100, t false) = 0

.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The interaction of two positive bell shape solitaries are also studied using IC [1]:

\begin{array}{l} u (x, 0) goodbreak= S_{1} italicsech ((), k_{1} ((), x goodbreak- x_{1})) goodbreak+ S_{2} italicsech ((), k_{2} ((), x goodbreak- x_{2})) \\ k_{i} goodbreak= \sqrt{\frac{c_{i}}{μ_{3}}}, S_{i} goodbreak= \frac{6 c_{i}}{μ_{1} ((), 1 goodbreak+ \sqrt{1 + 6 \frac{μ_{2} c_{i}}{μ_{1}^{2}}})}, i goodbreak= 1, 2 . \end{array}

This IC gives two positive bell shaped solitaries of heights

0.26492

and

0.67377

positioned at

x = - 25

and

x = 25

, respectively, at the beginning, Figure 3b. Both solitaries propagate to the left along the

x

‐axis as time goes.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Performance of the QTT B-spline FEM will be evaluated for the GE. Combination of the KdV equation and modified KdV equation GE which has the following form u t + 𝜇 1 uu x + 𝜇 2 u 2 u x + 𝜇 3 u xxx = 0 ( 1 ) in which there exist evolution term u t , two nonlinear terms uu x and u 2 u x and third order dissipative term u xxx . The initial condition (IC) u(x, 0) = u 0 (x)…”

Section: Introductionmentioning

confidence: 99%

“…Quintic B-spline differential quadrature method has been used to obtain the numerical approximation of the GE [31]. Collocation method based on quintic, exponential and extended cubic B-splines are set up for solving the GE in the studies [1,14,15]. Solutions of the fractional GE with collocation method using radial basis function is given in the work [21].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

Hepson

2022

Numerical Methods Partial

View full text Add to dashboard Cite

The collocation method is established for getting solution of the Gardner equation (GE). The GE is fully integrated by way of the Crank-Nicolson method for time variable and collocation method for spatial variable. Trial function of the collocation method is set up using combination of quartic trigonometric tension (QTT) B-splines. Convergence analysis of suggested method is investigated. Performance of the QTT B-splines is searched by studying three test problems: propagation of bell shape solitary wave, interaction of two positive bell shape solitary wave and wave generation.

show abstract

\begin{array}{c} u (x, t) goodbreak= S \sec h ((), k ((), x goodbreak- x_{0} goodbreak- italicct)) \\ k goodbreak= \sqrt{\frac{c}{μ_{3}}}, S goodbreak= \frac{6 c}{μ_{1} ((), 1 goodbreak+ \sqrt{1 + 6 \frac{μ_{2} c}{μ_{1}^{2}}})} . \end{array}

With parameters

μ_{1} = 1,

μ_{2} = 1

μ_{3} = 5

, and

x_{0} = 0,

initial profiles, having bell‐type shape is locate along the

x

‐axis centered at

x = 0

in the restricted region

false[- 100, 100 false]

. Three BCs are used to have solvable system of algebraic equations: two of them are on left,

u false(- 100, t false)

and

u_{x} false(- 100, t false)

and one is on the right,

u false(100, t false) = 0

.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The interaction of two positive bell shape solitaries are also studied using IC [1]:

\begin{array}{l} u (x, 0) goodbreak= S_{1} italicsech ((), k_{1} ((), x goodbreak- x_{1})) goodbreak+ S_{2} italicsech ((), k_{2} ((), x goodbreak- x_{2})) \\ k_{i} goodbreak= \sqrt{\frac{c_{i}}{μ_{3}}}, S_{i} goodbreak= \frac{6 c_{i}}{μ_{1} ((), 1 goodbreak+ \sqrt{1 + 6 \frac{μ_{2} c_{i}}{μ_{1}^{2}}})}, i goodbreak= 1, 2 . \end{array}

This IC gives two positive bell shaped solitaries of heights

0.26492

and

0.67377

positioned at

x = - 25

and

x = 25

, respectively, at the beginning, Figure 3b. Both solitaries propagate to the left along the

x

‐axis as time goes.…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

Hepson

2022

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…The equation defines various interesting physical phenomena, such as the weakly nonlinear long waves in various physical applications (Nakoulima et al, 2004). Many analytic and numerical methods have been proposed and implemented for solution of the equation (Zhang, 1998;Kaya and Inan, 2005;Wazwaz, 2007;Biswas and Zerrad, 2008;Lu and Shi, 2010;Triki et al, 2010;Ak et al, 2018;Ak, 2019).…”

Section: Introductionmentioning

confidence: 99%

Birleştirilmiş KdV-mKdV Denkleminin Yaklaşık Analitik Çözümleri Üzerine Bir Çalışma

Ak¹,

İnan²

2020

Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi

Self Cite

View full text Add to dashboard Cite

In this paper, the investigation focuses on solitary wave solutions of the combined KdV-mKdV equation by using reduced differential transform method (RDTM). To prove validity of the proposed method, the approximate analytic solutions and exact solutions of the equation are compared via absolute errors. The obtained results are represented by graphics. The effects of the time and dispersion parameter (μ) on analytic solutions are investigated. As a result, it can be said that the applied method is quite precise and successful for similar type equations.

show abstract

KdV, Extended KdV, 5th-Order KdV, and Gardner Equations Generalized for Uneven Bottom Versus Corresponding Boussinesq’s Equations

Rozmej

Karczewska

2021

NODYCON Conference Proceedings Series

View full text Add to dashboard Cite

Numerical experiments for long nonlinear internal waves via Gardner equation with dual-power law nonlinearity

Cited by 4 publications

References 16 publications

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

Birleştirilmiş KdV-mKdV Denkleminin Yaklaşık Analitik Çözümleri Üzerine Bir Çalışma

KdV, Extended KdV, 5th-Order KdV, and Gardner Equations Generalized for Uneven Bottom Versus Corresponding Boussinesq’s Equations

Contact Info

Product

Resources

About