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runge-kutta method and Bolck by Block method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel

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Journal of Applied Mathematics and Physics 2020年第9期8卷 2043-2054页

作者： A. M. Al-Bugami J. G. Al-Juaid Department of Mathematics Faculty of Sciences Taif University Taif KSA

In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. runge-kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.

关键词： Nonlinear Fredholm-Volterra Integral Equation System of Nonlinear Volterra Integral Equations runge-kutta method Bolck by Block method

The Research on Calculation of Water Surface Profile in Channel by runge–kutta method

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The Research on Calculation of Water Surface Profile in Chan...

An artificial neural network based deep collocation method for the solution of transient linear and nonlinear partial differential equations

2011 International Symposium on Water Resource and Environmental Protection(ISWREP 2011)

作者： Huo Qian Li Zhong Xi Bo Guan Zhenzhen Shijiazhuang Railway University Shijiazhuang Mechanized Infantry Academy Ordnance Engineering College

ISBN: (纸本)9781612843377

The basic equation of steady gradually varied flow describes the variation of water depth with flow *** it’s difficult to be solved directly,the standard step method becomes the general way to calculate the water surface profile, which will entail many problems when the iteration method must be *** the paper does a research on runge-kutta method to solve the differential equation directly avoiding these *** calculation models for water surface profile are derived,both for prismatic channel,and also for non-prismatic *** can calculate the water surface profile from upstream or *** function ode45 in MATLAB, taking runge-kutta method as the algorithm kernel,is selected to implement the calculation of two *** two examples prove the correctness of the models,and high-precision feature of ode45 compared with the standard step *** new way of calculating water surface profile is a reliable,simple and flexible method,easy to be popularized in engineering.

关键词： water surface profile The basic equation of steady gradually varied flow channel runge-kutta method

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Frontiers of Structural and Civil Engineering 2024年第8期18卷 1296-1310页

作者： Abhishek MISHRA Cosmin ANITESCU Pattabhi Ramaiah BUDARAPU Sundararajan NATARAJAN Pandu Rang VUNDAVILLI Timon RABCZUK School of Mechanical Sciences Indian Institute of TechnologyBhubaneswar 752050India Institute of Structural Mechanics Bauhaus University of WeimarWeimar 99423Germany Department of Mechanical Engineering Indian Institute of Technology MadrasChennai 600036India

A combined deep machine learning(DML)and collocation based approach to solve the partial differential equations using artificial neural networks is *** developed method is applied to solve problems governed by the Sine–Gordon equation(SGE),the scalar wave equation and *** methods are studied:one is a space-time formulation and the other is a semi-discrete method based on an implicit runge–kutta(RK)time *** methodology is implemented using the Tensorflow framework and it is tested on several numerical *** on the results,the relative normalized error was observed to be less than 5%in all cases.

关键词： collocation method artificial neural networks deep machine learning Sine-Gordon equation transient wave equation dynamic scalar and elasto-dynamic equation runge-kutta method

runge-kutta Finite Element method Based on the Characteristic for the Incompressible Navier-Stokes Equations

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Advances in Applied Mathematics and Mechanics 2019年第6期11卷 1415-1435页

作者： Shaokai Liao Yan Zhang Da Chen College of Mechanics and Materials Hohai UniversityNanjing 210089JiangsuChina College of Harbour Coastal and Offshore EngineeringHohai UniversityNanjing 210089JiangsuChina College of Engineering&Architecture Jiaxing UniversityJiaxing 314001ZhejiangChina

In this paper,a finite element method based on the characteristic for the incompressible Navier-Stokes equations is proposed by introducing runge-kutta *** first,coordinate transformation operation is performed to obtain the alternative Navier-Stokes equations without convection ***,instead of the classical characteristic-based split(CBS)method,we use the third-order runge-kutta method along the characteristic to carry out time discretization in order to improve calculation accuracy,and segregate the calculation of the pressure from that of the velocity based on the momentum-pressure Poisson equation ***,some classical benchmark problems are used to validate the effectiveness of the present *** with the classical method,the present method has lower dissipation,larger permissible time step,and higher time *** code can be downloaded at DOI:10.13140/RG.2.2.36336.56329.

关键词： Finite element method characteristic Navier-Stokes equations runge-kutta method accuracy

SINGLY DIAGONALLY IMPLICIT runge-kutta methodS COMBINING LINE SEARCH TECHNIQUES FOR UNCONSTRAINED OPTIMIZATION

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Journal of Computational Mathematics 2005年第2期23卷 153-164页

作者： Xin-longLuo SchoolofInformationEngineering BeijingUniversityofPostsandTelecommunicationsP.O.Box101Beijing100876China

There exists a strong connection between numerical methods for the integration of ordinary differential equations and optimization problems. In this paper, we try to discover further their links. And we transform unconstrained problems to the equivalent ordinary differential equations and construct the LRKOPT method to solve them by combining the second order singly diagonally implicit runge-kutta formulas and line search *** we analyze the global convergence and the local convergence of the LRKOPT method. Promising numerical results are also reported.

关键词： Global convergence runge-kutta method Superlinear convergence Unconstrained optimization

H-STABILITY OF runge-kutta methodS WITH VARIABLE STEPSIZE FOR SYSTEM OF PANTOGRAPH EQUATIONS

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Journal of Computational Mathematics 2004年第5期22卷 727-734页

作者： YangXu Jing-junZhao Ming-zhuLiu DepartmentofMathematics HarbinInstituteofTechnologyHarbin150001China

This paper deals with H-stability of runge-kutta methods with variable stepsize for the system of pantograph equations. It is shown that both runge-kutta methods with nonsingular matrix coefficient A and stiffly accur... 详细信息

关键词： Delay differential equations runge-kutta method Stability

CONVERGENCE OF PARALLEL DIAGONAL ITERATION OF runge-kutta methodS FOR DELAY DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2004年第3期22卷 361-370页

作者： Xiao-huaDing MingzhuLiu DepartmentofMathematics HarbinInstituteofTechnology(Weihai)Weihai264209China

Implicit runge-kutta method is highly accurate and stable for stiff initial value problem. But the iteration technique used to solve implicit runge-kutta method requires lots of computational efforts. In this paper, we extend the Parallel Diagonal Iterated rungekutta(PDIRK) methods to delay differential equations(DDEs). We give the convergence region of PDIRK methods, and analyze the speed of convergence in three parts for the P-stability region of the runge-kutta corrector method. Finally, we analysis the speed-up factor through a numerical experiment. The results show that the PDIRK methods toDDEs are efficient.

关键词： runge-kutta method Parallel iteration Delay differential equation

A Class of Explicit Parallel Multistep runge-kutta methods

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Journal of Systems Engineering and Electronics 1993年第4期4卷 64-72页

作者： Xie Yajun and Liu DeguiBeijing Institute of Computer Application and Simulation Technology P.O.Box. 3929, Beijing 100854, China Beijing Institute of Computer Application and Simulation Technology P.O. Box. 3929 Beijing 100854 China

In this paper, a rather general class of explicit parallel multistep runge-kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and stability are analysed. Several parallel computational formulae are given. The numerical experiments, including accuracy, speedup, and efficiency tests show that the methods are efficient.

关键词： runge-kutta method Initial value problem.

Symmetric-Adjoint and Symplectic-Adjoint runge-kutta methods and Their Applications

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Numerical Mathematics(Theory,methods and Applications) 2022年第2期15卷 304-335页

作者： Geng Sun Siqing Gan Hongyu Liu Zaijiu Shang Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China School of Mathematics and Statistics HNP-LAMACentral South UniversityChangshaHunan 410083China Department of Mathematics City University of Hong KongHong KongChina

Symmetric and symplectic methods are classical notions in the theory of numerical methods for solving ordinary differential *** can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase *** article is mainly concerned with the symmetric-adjoint and symplectic-adjoint runge-kutta methods as well as their *** is a continuation and an extension of the study in[14],where the authors introduced the notion of symplectic-adjoint method of a runge-kutta method and provided a simple way to construct symplectic partitioned runge-kutta methods via the symplectic-adjoint *** this paper,we provide a more comprehensive and systematic study on the properties of the symmetric-adjoint and symplecticadjoint runge-kutta *** properties reveal some intrinsic connections among some classical runge-kutta ***,those properties can be used to significantly simplify the order conditions and hence can be applied to the construction of high-order runge-kutta *** a specific and illustrating application,we construct a novel class of explicit runge-kutta methods of stage 6 and order ***,with the help of symplectic-adjoint method,we thereby obtain a new simple proof of the nonexistence of explicit runge-kutta method with stage 5 and order 5.

关键词： runge-kutta method symmetric symplectic adjoint high-order explicit method