scholarly journals A Class of New Pouzet-Runge-Kutta-Type Methods for Nonlinear Functional Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Chengjian Zhang
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Numerical Stability ◽  
Numerical Experiments ◽  
Integrodifferential Equations ◽  
Stability Criteria ◽  
Nonlinear Functional ◽  
Runge Kutta ◽  
Theoretical Results

This paper presents a class of new numerical methods for nonlinear functional-integrodifferential equations, which are derived by an adaptation of Pouzet-Runge-Kutta methods originally introduced for standard Volterra integrodifferential equations. Based on the nonclassical Lipschitz condition, analytical and numerical stability is studied and some novel stability criteria are obtained. Numerical experiments further illustrate the theoretical results and the effectiveness of the methods. In the end, a comparison between the presented methods and the existed related methods is given.

scholarly journals Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

2017 ◽  
Vol 17 (3) ◽  
pp. 479-498 ◽  
Author(s):  
Raphael Kruse ◽  
Yue Wu
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Quadrature Rule ◽  
Time Variable ◽  
Important Ingredient ◽  
Riemann Sum ◽  
Runge Kutta ◽  
Theoretical Results

AbstractThis paper contains an error analysis of two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the {L^{p}(\Omega;{\mathbb{R}}^{d})}-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

scholarly journals Stability of Runge-Kutta Methods for Neutral Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Liping Wen ◽  
Xiong Liu ◽  
Yuexin Yu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Stability ◽  
Neutral Delay ◽  
Numerical Examples ◽  
Neutral Delay Differential Equations ◽  
Runge Kutta ◽  
Delay Differential ◽  
Stability Results ◽  
Theoretical Results

This paper is concerned with the numerical stability of a class of nonlinear neutral delay differential equations. The numerical stability results are obtained for(k,l)-algebraically stable Runge-Kutta methods when they are applied to this type of problem. Numerical examples are given to confirm our theoretical results.

scholarly journals Continuous stage stochastic Runge–Kutta methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xuan Xin ◽  
Wendi Qin ◽  
Xiaohua Ding
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Additive Noise ◽  
Rooted Tree ◽  
Order Theory ◽  
Order Conditions ◽  
Quadrature Formulas ◽  
General Order ◽  
Runge Kutta ◽  
Theoretical Results

AbstractIn this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

scholarly journals Stability Analysis of Runge-Kutta Methods for Nonlinear Functional Differential and Functional Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yuexin Yu ◽  
Zhongyan Liu ◽  
Liping Wen
Keyword(s):  
Stability Analysis ◽  
Numerical Stability ◽  
Functional Equations ◽  
Numerical Test ◽  
Sufficient Conditions ◽  
Nonlinear Functional ◽  
Runge Kutta ◽  
Functional Differential ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the numerical stability of Runge-Kutta methods for a class of nonlinear functional differential and functional equations. The sufficient conditions for the stability and asymptotic stability of(k,l)-algebraically stable Runge-Kutta methods are derived. A numerical test is given to confirm the theoretical results.

scholarly journals Collocation for High-Order Differential Equations with Lidstone Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Francesco Costabile ◽  
Anna Napoli
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Experiments ◽  
Recurrence Formula ◽  
High Order ◽  
Theoretical Results

A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.

Mechanical Systems With Impacts: Comparison of Several Numerical Methods

1999 ◽  
Author(s):  
C. H. Lamarque ◽  
O. Janin
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Infinite Number ◽  
Mechanical Systems ◽  
Numerical Results ◽  
Degree Of Freedom ◽  
Periodic Behavior ◽  
Numerical Tests ◽  
Runge Kutta ◽  
Theoretical Results

Abstract We study the performances of several numerical methods (Paoli-Schatzman, Newmark, Runge-Kutta) in order to compute periodic behavior of a simple one-degree-of-freedom impacting oscillator. Some theoretical results are given and numerical tests are performed. We compare mathematical and numerical results using our simple example exhibiting either finite or infinite number of impacts per period. Comparison of exact and numerical solution provides a practical order for each scheme. We conclude about the use of the different numerical methods.

scholarly journals Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 142 ◽  
Author(s):  
Junaid Ahmad ◽  
Yousaf Habib ◽  
Shafiq Rehman ◽  
Azqa Arif ◽  
Saba Shafiq ◽  
...  
Keyword(s):  
Hamiltonian System ◽  
Numerical Methods ◽  
Energy Conservation ◽  
Hamiltonian Systems ◽  
Numerical Experiments ◽  
Effective Order ◽  
Good Energy ◽  
Runge Kutta ◽  
Separable Hamiltonian System ◽  
Selection Of

A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system confirming the efficiency of the approach is also provided with good energy conservation.

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