The implicit midpoint rule applied to discontinuous differential equations

Computing ◽  
1992 ◽  
Vol 49 (1) ◽  
pp. 45-62 ◽  
Author(s):  
Alois E. Kastner-Maresch
Keyword(s):  
Differential Equations ◽  
Discontinuous Differential Equations ◽  
Implicit Midpoint Rule ◽  
Midpoint Rule

Lyapunov Instability for Discontinuous Differential Equations

2021 ◽  
Vol 2 (2) ◽  
pp. 49-58
Author(s):  
Iguer Luis Domini dos Santos
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Discontinuous Differential Equations ◽  
Lyapunov Instability

The present work studies the Lyapunov instability for discontinuous differential equations through the use of the notion of Carathéodory solution to differential equations. From Lyapunov's first instability theorem and Chetaev's instability theorem, which deal with instability to ordinary differential equations, two Lyapunov instability results for discontinuous differential equations are obtained.

On the Number of Limit Cycles of Discontinuous Liénard Polynomial Differential Systems

2018 ◽  
Vol 28 (14) ◽  
pp. 1850175
Author(s):  
Fangfang Jiang ◽  
Zhicheng Ji ◽  
Yan Wang
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Second Order ◽  
Discontinuous Differential Equations ◽  
Differential Systems ◽  
Lower Upper ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Averaging Theorem

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

scholarly journals On New Picard-Mann Iterative Approximations with Mixed Errors for Implicit Midpoint Rule and Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Teng-fei Li ◽  
Heng-you Lan
Keyword(s):  
Optimal Control Problems ◽  
Nonexpansive Mappings ◽  
Elliptic Boundary ◽  
Iteration Process ◽  
Implicit Midpoint Rule ◽  
Mann Iteration ◽  
Midpoint Rule ◽  
Elliptic Boundary Value ◽  
Mann Iteration Process ◽  
Iteration Processes

In order to solve (partial) differential equations, implicit midpoint rules are often employed as a powerful numerical method. The purpose of this paper is to introduce and study a class of new Picard-Mann iteration processes with mixed errors for the implicit midpoint rules, which is different from existing methods in the literature, and to analyze the convergence and stability of the proposed method. Further, some numerical examples and applications to optimal control problems with elliptic boundary value constraints are considered via the new Picard-Mann iterative approximations, which shows that the new Picard-Mann iteration process with mixed errors for the implicit midpoint rule of nonexpansive mappings is brand new and more effective than other related iterative processes.

scholarly journals Petrov-Galerkin Method for the Coupled Schrödinger-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
M. S. Ismail ◽  
Farida M. Mosally ◽  
Khadeejah M. Alamoudi
Keyword(s):  
Galerkin Method ◽  
Fourth Order ◽  
Kdv Equation ◽  
Approximation Technique ◽  
Implicit Midpoint Rule ◽  
Von Neumann ◽  
Midpoint Rule ◽  
B Splines ◽  
Runge Kutta Method ◽  
Second Order In Time

Petrov-Galerkin method is used to derive a numerical scheme for the coupled Schrödinger-KdV (SKdV) equations, where we have used the cubic B-splines as a test functions and a linear B-splines as a trial functions. Product approximation technique is used to deal with the nonlinear terms. An implicit midpoint rule and the Runge-Kutta method of fourth-order (RK4) are used to discretize in time. A block nonlinear pentadiagonal system is obtained. We solve this system by the fixed point method. The resulting scheme has a fourth-order accuracy in space direction and second-order in time direction in case of the implicit midpoint rule and it is unconditionally stable by von Neumann method. Using the RK4 method the scheme will be linear and fourth-order in time and space directions, and it is also conditionally stable. The exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness and the efficiency of the proposed schemes.

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