A novel integration scheme for partial differential equations: An application to the complex Ginzburg-Landau equation

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

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CHAOTIC BEHAVIOR AND CHAOS CONTROL FOR A CLASS OF COMPLEX PARTIAL DIFFERENTIAL EQUATIONS

International Journal of Modern Physics C ◽

10.1142/s0129183101002073 ◽

2001 ◽

Vol 12 (06) ◽

pp. 889-899 ◽

Cited By ~ 14

Author(s):

G. M. MAHMOUD ◽

H. A. ABDUSALAM ◽

A. A. M. FARGHALY

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fixed Points ◽

Chaos Control ◽

Chaotic Behavior ◽

Point Of View ◽

Partial Differential ◽

Ginzburg Landau ◽

Practical Point ◽

Nagumo Equations

Systems of complex partial differential equations, which include the famous nonlinear Schrödinger, complex Ginzburg–Landau and Nagumo equations, as examples, are important from a practical point of view. These equations appear in many important fields of physics. The goal of this paper is to concentrate on this class of complex partial differential equations and study the fixed points and their stability analytically, the chaotic behavior and chaos control of their unstable periodic solutions. The presence of chaotic behavior in this class is verified by the existence of positive maximal Lyapunov exponent.The problem of chaos control is treated by applying the method of Pyragas. Some conditions on the parameters of the systems are obtained analytically under which the fixed points are stable (or unstable).

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On the stochastic partial differential equations of Ginzburg-Landau type

Stochastic Partial Differential Equations and Their Applications - Lecture Notes in Control and Information Sciences ◽

10.1007/bfb0007326 ◽

2005 ◽

pp. 113-122

Author(s):

T. Funaki

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Stochastic Partial Differential Equations ◽

Partial Differential ◽

Ginzburg Landau

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Discrete Temimi-Ansari method for solving a class of stochastic nonlinear differential equations

AIMS Mathematics ◽

10.3934/math.2022283 ◽

2021 ◽

Vol 7 (4) ◽

pp. 5093-5105

Author(s):

Mourad S. Semary ◽

◽

M. T. M. Elbarawy ◽

Aisha F. Fareed ◽

Keyword(s):

Differential Equations ◽

Numerical Method ◽

Landau Equation ◽

Nonlinear Differential Equations ◽

New Method ◽

Ginzburg Landau ◽

Ginzburg Landau Equation

<abstract> <p>In this paper, a numerical method to solve a class of stochastic nonlinear differential equations is introduced. The proposed method is based on the Temimi-Ansari method. The special states of the four systems are studied to show that the proposed method is efficient and applicable. These systems are stochastic Langevin's equation, Ginzburg-Landau equation, Davis-Skodje, and Brusselator systems. The results clarify the accuracy and efficacy of the presented new method with no need for any restrictive assumptions for nonlinear terms.</p> </abstract>

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Trilinear Finite Element Solution of Three Dimensional Heat Conduction Partial Differential Equations

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.36.28146 ◽

2018 ◽

Vol 7 (4.36) ◽

pp. 379

Author(s):

Erwin Sulaeman ◽

S. M. Afzal Hoq ◽

Abdurahim Okhunov ◽

Marwan Badran

Keyword(s):

Finite Element ◽

Heat Conduction ◽

Partial Differential Equations ◽

Differential Equations ◽

Error Analysis ◽

Heat Conduction Problem ◽

Promising Result ◽

Three Dimensional ◽

Integration Scheme ◽

Partial Differential

Solution of partial differential equations (PDEs) of three dimensional steady state heat conduction and its error analysis are elaborated in the present paper by using a Trilinear Galerkin Finite Element method (TGFEM). An eight-node hexahedron element model is developed for the TGFEM based on a trilinear basis function where physical domain is meshed by structured grid. The stiffness matrix of the hexahedron element is formulated by using a direct integration scheme without the necessity to use the Jacobian matrix. To check the accuracy of the established scheme, comparisons of the results using error analysis between the present TGFEM and exact solution is conducted for various number of the elements. For this purpose, analytical solution is derived in detailed for a particular heat conduction problem. The comparison shows promising result where its convergence is approximately O(h²) for matrix norms L1, L2 and L¥.

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