On the Variational Approach for Analyzing the Stability of Solutions of Evolution Equations

Abdel‐Gawad, H. I.; Osman, M. S.

doi:10.5666/kmj.2013.53.4.680

Cited by 71 publications

(28 citation statements)

References 20 publications

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“…We present some approach to study the stability of solutions of Equation Global stability: To study the global stability of Equation , we consider the integral

E false(τ_{1} false) = true \int_{0}^{x} u^{2} false(x, t, τ_{1}, x_{1}, x_{2} false) d x,

subjected to the boundary conditions BC

u false(x = X, x^{γ}, x^{β}, . . . false) = 0, u^{'} false(0, τ_{1}, t false) = 0

and

u^{''} false(0, τ_{1}, t false)

. …”

Section: Stability Approachmentioning

confidence: 99%

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

2020

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The generalized fractional Caputo‐operator is discussed. The reduction of partial differential equations retarded or advanced with delays is obtained. The applications to the space‐time‐fractional KdV and time‐fractional Boussenisq‐Burger's equation are carried. Semi‐self‐similar SS wave solutions are obtained. That is, there exists a soliton wave propagates along a specific characteristic curve in the xt‐ plane. This may be due to the effects associated with the distributed time delay. The effects of space fractional derivative on a system are attributed to the transition states. Thus, anomalous wave transport is produced mainly near the origin.

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“…We present some approach to study the stability of solutions of Equation Global stability: To study the global stability of Equation , we consider the integral

E false(τ_{1} false) = true \int_{0}^{x} u^{2} false(x, t, τ_{1}, x_{1}, x_{2} false) d x,

subjected to the boundary conditions BC

u false(x = X, x^{γ}, x^{β}, . . . false) = 0, u^{'} false(0, τ_{1}, t false) = 0

and

u^{''} false(0, τ_{1}, t false)

. …”

Section: Stability Approachmentioning

confidence: 99%

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

2020

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“…Here, the key ideas of the unified method [27][28][29][30][31][32][33][34][35][36] for extracting some analytical wave solutions for Equation 1 are described.…”

Section: Analytical Solutionsmentioning

confidence: 99%

Analytical and numerical solutions of mathematical biology models: The Newell‐Whitehead‐Segel and Allen‐Cahn equations

İnan

Osman

et al. 2019

Math Methods in App Sciences

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In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell‐Whitehead‐Segel–type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.

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“…Here, we are concerning with studying the singular Sturm–Liouville eigenvalue problem to second order equations in Eq. .Again, by topological invariance, we write the eigenfunction solutions as follow :

\begin{matrix} u_{1} (ξ) & = p_{0} ξ {(1 - ξ)}^{m} {(1 + ξ)}^{n}, v_{1} (ξ) = q_{0} ξ {(1 - ξ)}^{m} {(1 + ξ)}^{n}, \\ ξ & = \tanh (\frac{c_{2} ζ \sqrt{λ_{1} μ (λ_{1} μ - 4 γ^{2})}}{2 \sqrt{2} a_{1} γ μ}), \end{matrix}

where m > 0, n > 0, in order to satisfy the boundary condition and p 0 , q 0 are arbitrary constants. In this case, u 0 ( ζ ) and v 0 ( ζ ) are given by Eqs.…”

Section: The Stability Analysis Of Traveling Wave Solutionsmentioning

confidence: 99%

“…Here, u 0 ( ζ ) and v 0 ( ζ ) are given by Eqs. and , respectively.By bearing in mind the eigenfunction solution when λ = 0, then by topological invariance, we have

u_{1} (ξ) = \frac{p_{0} {(1 - ξ)}^{m} {(1 + ξ)}^{n}}{{((), d_{0} c + \sqrt{c_{0}} d_{1} ξ)}^{r}}, v_{1} (ξ) = \frac{q_{0} {(1 - ξ)}^{m} {(1 + ξ)}^{n}}{{((), d_{0} c + \sqrt{c_{0}} d_{1} ξ)}^{r}}, ξ = \sin (c ζ),

where m > 0, n > 0, and r > 0, in order to satisfy the boundary condition.By substituting form Eq. into Eq.…”

Section: The Stability Analysis Of Traveling Wave Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

Dynamic of DNA's possible impact on its damage

Abdel‐Gawad

Tantawy

Osman

2015

Math Methods in App Sciences

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In this paper, we investigate the dynamic of DNA described via DNA double-stranded model with transverse and longitudinal motions. This model admits solitary, soliton, periodic, or chirped wave solution. It is justified that the most admissible physical solution is the soliton or chirped wave solution. The stability analysis of all these solutions is performed by using the Sturm-Liouville problem and the topological invariance. We found that soliton and chirped waves are unstable so that the unbounded amplitude may occur. In the view of these models, damage of DNA membrane or bases may occur under small disturbance. Also, the suggested models will be indispensable when inhomogeneity or medium dissipation is taken into account.

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On the Variational Approach for Analyzing the Stability of Solutions of Evolution Equations

Cited by 71 publications

References 20 publications

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

Analytical and numerical solutions of mathematical biology models: The Newell‐Whitehead‐Segel and Allen‐Cahn equations

Dynamic of DNA's possible impact on its damage

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