On the numerical solution of nonlinear system of coupled sine-Gordon equations

Yıldırım, Özgür; Uzun, Meltem; Altun, Oğuz

doi:10.1063/1.5049056

Cited by 4 publications

(6 citation statements)

References 21 publications

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“…which describes the open states in DNA double helices, is studied by many researchers (see, [18,34] and the references given therein). Note that some applications and numerical results of the present study, without proof, are presented in [32,33]. Unique solvability of problem (10) is presented as the limit of first order of accuracy unconditionally stable difference scheme…”

Section: Introductionmentioning

confidence: 94%

Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

Yıldırım

Uzun

2020

NAMC

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In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

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Section: Introductionmentioning

confidence: 94%

Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

Yıldırım

Uzun

2020

NAMC

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“…Sadighi et al [13] employed the homotopy perturbation method (HPM) for solving both sine-Gordon and coupled sine-Gordon equations. In paper [14], the authors obtained numerical solution of inhomogeneous systems of sine-Gordon equation ( 1) by finite difference method with fixed-point iteration.…”

Section: Introductionmentioning

confidence: 99%

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

Deresse

2022

Advances in Mathematical Physics

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In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

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“…It is known that classical numerical techniques like finite difference method preserve their importance due to their well‐established theory and useful properties like stability 21–27 . So finite difference method will be implemented and analyzed on a multidimensional time fractional nonlinear Schrödinger integro‐differential equation for the first time 28–31 …”

Section: Introductionmentioning

confidence: 99%

“…[21][22][23][24][25][26][27] So finite difference method will be implemented and analyzed on a multidimensional time fractional nonlinear Schrödinger integro-differential equation for the first time. [28][29][30][31] In the present study, initial value problem for the FSDE is considered, which is stated as i du dt þ Au ¼ ∫ t 0 f s; D α s uðsÞ À Á ds; 0 < t < T; 0 ≤ α < 1;…”

Section: Introductionmentioning

confidence: 99%

“…Assume that function f : ½0; T × C α ð Þ ½0; T; L 2 ðΩÞ À Á →L 2 ðΩÞ is continuous, that is,‖ f ðt; D α t uðtÞÞ‖ L 2 ðΩÞ ≤ M (32) in ½0; T × C α ð Þ ½0; T; L 2 ðΩÞ À Áand Lipschitz condition is satisfied with respect to t uniformly‖ f ðt; D α t uÞ− f ðt; D α t vÞ‖ L 2 ðΩÞ ≤ L‖D α t u − D α t v‖ L 2 ð‾Ω Þ;The proof of Theorem 2.3 is obtained using proof of Theorem 2.1, and the symmetry properties of the operator A x specified by formula(30) and the following coercivity theorem 45 : Theorem 2.4. The solutions of the elliptic problem A x uðxÞ ¼ wðxÞ; x ∈ Ω;…”

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confidence: 99%

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Multidimensional problems for nonlinear fractional Schrödinger differential and difference equations

Ashyralyev

Hicdurmaz

2019

Math Methods in App Sciences

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In the present paper, a nonlinear fractional Schrödinger integro‐differential equation is considered in a Hilbert space. Operator approach is applied on multidimensional problems with nonlinearity that deserve a studious treatment. In this paper, theorems on existence and uniqueness of a bounded solution for the abstract problem are achieved. Additionally, existence theorems are obtained for first and second orders of accuracy difference schemes of the abstract problem. Furthermore, theorems are applied on a one‐dimensional problem with nonlocal condition and a multidimensional problem with Dirichlet boundary condition. Numerical results and illustrations are presented to show the effectiveness of the theoretical results.

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On the numerical solution of nonlinear system of coupled sine-Gordon equations

Cited by 4 publications

References 21 publications

Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

Multidimensional problems for nonlinear fractional Schrödinger differential and difference equations

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