Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

Yokuş, Asıf; Baskonus, Hacı Mehmet; Sulaıman, Tukur Abdulkadir; Bulut, Hasan

doi:10.1002/num.22192

Cited by 126 publications

(55 citation statements)

References 37 publications

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“…The first example presents that the convergence order of difference Scheme (2.23)–(2.27) is O( τ 2 + h 2 ).Example In (1.1)–(1.5), let L = 1, T = 1. In order to test the convergence rate of the proposed method, we consider the exact solution of the problem (1.1)–(1.5) as follows .

u (x, t) = x^{2} {((), x - 1)}^{2} e^{t}, v (x, t) = t^{2} \sin (π x),

which determine

\begin{array}{l} f ((), x, t) = x^{2} {(x - 1)}^{2} e^{t} - 3 π^{2} t^{2} normalsin ((), π italicx), \\ g ((), x, t) = 2 t normalsin ((), π italicx) - ([], 2 {(x - 1)}^{2} e^{t} + 8 x ((), x - 1) e^{t} + 2 x^{2} e^{t}) - 4 {[x (x - 1)]}^{4} e^{2 t}, \\ ϕ ((), x) = x^{2} {(x - 1)}^{2}, ψ ((), x) = 0 . \end{array}

…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Some numerical examples are provided. In , the forward finite difference method was used in analyzing the numerical behavior of the two‐component second‐order KdV evolutionary system, the stability of this scheme was analyzed by using the Fourier‐Von Neumann analysis. The accuracy of this difference scheme was checked in L 2 and L ∞ norm.…”

Section: Introductionmentioning

confidence: 99%

“…Our motivation of this work is to fill this gap. For generality, we consider the following nonlinear KdV equation with nonhomogeneous source ,

u_{t} + 3 v_{italicxx} = f (x, t), x \in [0, L], t \in [0, T],

v_{t} - u_{italicxx} - 4 u^{2} = g (x, t), x \in [0, L], t \in [0, T],

u (x, 0) = ϕ (x), v (x, 0) = ψ (x), x \in [0, L],

u (0, t) = 0, u (L, t) = 0, t \in [0, T],

v (0, t) = 0, v (L, t) = 0, t \in [0, T],

where f ( x , t ), g ( x , t ), ϕ ( x ), and ψ ( x ) are known smooth functions.…”

Section: Introductionmentioning

confidence: 99%

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Two‐level linearized and local uncoupled difference schemes for the two‐component evolutionary Korteweg‐de Vries system

Shen

Sun

2019

Numerical Methods Partial

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The two‐level linearized and local uncoupled spatial second order and compact difference schemes are derived for the two‐component evolutionary system of nonhomogeneous Korteweg‐de Vries equations. It is shown by the mathematical induction that these two schemes are uniquely solvable and convergent in a discrete L∞ norm with the convergence order of O(τ2 + h2) and O(τ2 + h4), respectively, where τ and h are the step sizes in time and space. Three numerical examples are given to confirm the theoretical results.

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u (x, t) = x^{2} {((), x - 1)}^{2} e^{t}, v (x, t) = t^{2} \sin (π x),

which determine

\begin{array}{l} f ((), x, t) = x^{2} {(x - 1)}^{2} e^{t} - 3 π^{2} t^{2} normalsin ((), π italicx), \\ g ((), x, t) = 2 t normalsin ((), π italicx) - ([], 2 {(x - 1)}^{2} e^{t} + 8 x ((), x - 1) e^{t} + 2 x^{2} e^{t}) - 4 {[x (x - 1)]}^{4} e^{2 t}, \\ ϕ ((), x) = x^{2} {(x - 1)}^{2}, ψ ((), x) = 0 . \end{array}

…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Our motivation of this work is to fill this gap. For generality, we consider the following nonlinear KdV equation with nonhomogeneous source ,

u_{t} + 3 v_{italicxx} = f (x, t), x \in [0, L], t \in [0, T],

v_{t} - u_{italicxx} - 4 u^{2} = g (x, t), x \in [0, L], t \in [0, T],

u (x, 0) = ϕ (x), v (x, 0) = ψ (x), x \in [0, L],

u (0, t) = 0, u (L, t) = 0, t \in [0, T],

v (0, t) = 0, v (L, t) = 0, t \in [0, T],

where f ( x , t ), g ( x , t ), ϕ ( x ), and ψ ( x ) are known smooth functions.…”

Section: Introductionmentioning

confidence: 99%

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Two‐level linearized and local uncoupled difference schemes for the two‐component evolutionary Korteweg‐de Vries system

Shen

Sun

2019

Numerical Methods Partial

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“…The mathematical groundwork for derivatives of fractional order was laid through combined efforts of pioneers such are Liouville, Riemann, Caputo, Miller and Ross, Podlubny, and others. The theory of fractional‐order calculus has been related to practical projects, and it has been applied to chaos theory, signal processing, electrodynamics, fluid dynamics, finance, and other areas …”

Section: Introductionmentioning

confidence: 99%

“…The theory of fractional-order calculus has been related to practical projects, and it has been applied to chaos theory, 8 signal processing, 9 electrodynamics, 10 fluid dynamics, 11 finance, 12 and other areas. [13][14][15][16][17][18][19][20][21][22][23] The Burgers equations are the fundamental nonlinear partial differential equations that exist in distinct connected branches of science and technology. These equations can be seen as a simplified model of fluid dynamics, and it is used as a computational tool in order to deal with more complex problems.…”

mentioning

confidence: 99%

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

Prakasha

Veeresha

Rawashdeh

2019

Math Methods in App Sciences

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The aim of the present work is to find the numerical solutions for time‐fractional coupled Burgers equations using a new novel technique, called fractional natural decomposition method (FNDM). Two examples are considered in order to illustrate and validate the efficiency of the proposed algorithm. The numerical simulation has been conducted to ensure the exactness of the present method, and the obtained solutions are offered graphically to reveal the applicability and reliability of the FNDM. The outcomes of the study reveal that the FNDM is computationally very effective and accurate to study the (2 + 1)‐dimensional coupled Burger equations of arbitrary order.

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Dynamics of exact soliton solutions in the double‐chain model of deoxyribonucleic acid

Bilal

Younas

Ren

2021

Math Methods in App Sciences

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In this research, we study analytically the double‐chain model. The model consists of two long elastic homogeneous strands (or rods), which represent two polynucleotide chains of the deoxyribonucleic acid molecule, connected with each other by an elastic membrane (or some linear springs) representing the hydrogen bonds between the base pairs of the two chains. The new extended direct algebraic method and the generalized Kudryashov method are successfully utilized to discuss the exact soliton solutions to the double‐chain model of deoxyribonucleic acid that plays an important role in biology. The solutions obtained by these mechanisms can be divided into solitary, singular, kink, single wave, combine behavior as well as hyperbolic, plane wave, and trigonometric solutions with arbitrary parameters. Some solutions have been exemplified by graphics to understand the physical meaning of the DNA model. The accomplished solutions seem with all essential constraint conditions, which are obligatory for them to subsist. Hence, our techniques via fortification of symbolic computations provide an active and potent mathematical implement for solving diverse benevolent nonlinear wave problems. The results show that the system theoretically has extremely rich exact wave structures of biological relevance.

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Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

Cited by 126 publications

References 37 publications

Two‐level linearized and local uncoupled difference schemes for the two‐component evolutionary Korteweg‐de Vries system

Two‐level linearized and local uncoupled difference schemes for the two‐component evolutionary Korteweg‐de Vries system

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

Dynamics of exact soliton solutions in the double‐chain model of deoxyribonucleic acid

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