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Page 3712 of Mathematical Reviews Vol. , Issue 96f [page]

1996 Mathematical Reviews  
961:65090 96f:65090. 65L06 34A09 34A45 70-08 70E15 von Schwerin, Reinhold (D-HDBG-SC; Heidelberg) ; Bock, Hans Georg (D-HDBG-SC; Heidelberg) A Runge-Kutta starter for a multistep method for differential  ...  In this paper we present a Runge-Kutta starter for an Adams-Bashforth-Moulton predictor-corrector method by giving a detailed description of its construction as well as the ideas behind it with a special  ... 

Page 1399 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
[Evans, David John'] (4-NTTR; Nottingham) New Runge-Kutta starters for multistep methods. (English summary) Int. J. Comput. Math. 71 (1999), no. 1, 99-104.  ...  Summary: “In this paper the use of new improved fourth- and fifth-order linear and nonlinear Runge-Kutta methods in starting procedures for the well-known k-step multistep methods is shown to give greater  ... 

Page 5188 of Mathematical Reviews Vol. , Issue 81M [page]

1981 Mathematical Reviews  
The author discusses Runge-Kutta-like formulas which enable multistep methods to start or restart after one Runge-Kutta step.  ...  {For the entire collection see MR 81j:65008.} W. H. Enright (Toronto, Ont) Gear, C. W. Runge-Kutta starters for methods. ACM Trans. Math. Software 6 (1980), no. 3, 263-279.  ... 

Page 1575 of Mathematical Reviews Vol. , Issue 84d [page]

1984 Mathematical Reviews  
Software 6 (1980), 263-279; MR 81m:65119] for the design of Runge-Kutta starters for linear multistep methods. The first part concerns the nonstiff case and the use of block-explicit formulas.  ...  R. 84d:65044b Block Runge- Kutta methods for the numerical integration of initial value problems in ordinary differential equations. II. The stiff case. Math. Comp. 40 (1983), no. 161, 193-206.  ... 

Block Runge-Kutta Methods for the Numerical Integration of Initial Value Problems in Ordinary Differential Equations Part I. The Nonstiff Case

J. R. Cash
1983 Mathematics of Computation  
Gear in developing Runge-Kutta starters for linear multistep methods.  ...  Some numerical results are given to illustrate the algorithms developed for both the stiff and nonstiff cases and comparisons with standard Runge-Kutta methods are made.  ...  In [11] Runge-Kutta methods are proposed that yield Oihp+x) accurate approximations to hsy(s\xn) for s = 1,2,...,p.  ... 
doi:10.2307/2007368 fatcat:aarblfusobgfdl7kbkxw57de2u

Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. I. The nonstiff case

J. R. Cash
1983 Mathematics of Computation  
Gear in developing Runge-Kutta starters for linear multistep methods.  ...  Some numerical results are given to illustrate the algorithms developed for both the stiff and nonstiff cases and comparisons with standard Runge-Kutta methods are made.  ...  In [11] Runge-Kutta methods are proposed that yield Oihp+x) accurate approximations to hsy(s\xn) for s = 1,2,...,p.  ... 
doi:10.1090/s0025-5718-1983-0679439-3 fatcat:s4mgvca4uncf3gthsx7sg4m5v4

A special class of continuous general linear methods

D.G. Yakubu, A.M. Kwami, M.L. Ahmed
2012 Computational and Applied Mathemathics  
Their stability properties are similar to those for Runge-Kutta methods. These methods provide a unifying scope for many families of traditional methods.  ...  methods that are acceptable for solving stiff initial value problems.  ...  The first author wishes to record his thanks to the referee for his/her constructive suggestions and comments that have led to a number of improvements to this paper.  ... 
doi:10.1590/s1807-03022012000200003 fatcat:zrd52ykdkzet7m6doec5udvoii

Page 1064 of American Society of Civil Engineers. Collected Journals Vol. 114, Issue 5 [page]

1989 American Society of Civil Engineers. Collected Journals  
The system of equations (34) was integrated using a multistep self-adapt- ing Adams Bashforth predictor-corrector method with a fourth order RungeKutta method as a starter.  ...  Here, the calculated results for a constant load, with p held constant at the aforementioned value, are presented.  ... 

Numerical Solutions of Second Order Initial Value Problems of Bratu-Type Equations using Predictor-Corrector Method

Mengistu Tola MEKONNEN, Genanaw GOFE, Habtamu Garoma
2020 International Journal of Engineering and Applied Sciences  
The stability and convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation.  ...  The numerical results are tabulated in terms of maximum absolute errors and it is observed that the present method is more accurate and convergent and it also improves the results of the methods existing  ...  calculate at 1 n x + from the solution at a number of previous solutions using Runge-Kutta method as self-starter.  ... 
doaj:d8ed3fce018c47fd8590c7798c231131 fatcat:27m7sxzu7jbwpi25q42dq7wwzq

Gear and Runge–Kutta Numerical Discretization Methods in Differential Equations of Adsorption in Adsorption Heat Pump

Katarzyna Zwarycz-Makles, Dorota Majorkowska-Mech
2018 Applied Sciences  
We focused on the comparison of results obtained when the set of heat and mass balance ODEs for an adsorber was solved using: (1) the RungeKutta fixed step size fourth-order method (RKfixed), (2) the  ...  RungeKutta–Fehlberg 4.5th-order method with a variable step size (RK45), and (3) the Gear Backward Differentiation Formulae numerical (Gear BDF) methods.  ...  Runge-Kutta-Fehlberg 4.5th-Order Method A popular method for integrating equations in time is the Runge-Kutta-Fehlberg (RK45 order) method.  ... 
doi:10.3390/app8122437 fatcat:hbwaqeaoorci7itamtar56tuqu

Solving Delay Differential Equations of Small and Vanishing Lag Using Multistep Block Method

Nurul Huda Abdul Aziz, Zanariah Abdul Majid, Fudziah Ismail
2014 Journal of Applied Mathematics  
This paper considers the numerical solution of delay differential equations for solving the problem of small and vanishing lag using multistep block method.  ...  The proposed approach that is based on interpolation of Newton divided difference has been implemented by adapting this problem to the multistep block method.  ...  Karoui and Vaillancourt [2] developed a SYSDEL code which is based on the numerical method of Runge-Kutta for the formula pair of order (5, 6) for solving the case of vanishing lag and asymptotically  ... 
doi:10.1155/2014/348912 fatcat:gdwcwk6fsvhwbl5tnexexgcgte

Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations

Maziar Raissi, Paris Perdikaris, George Em Karniadakis
2018 SIAM Journal on Scientific Computing  
Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors.  ...  Runge-Kutta methods.  ...  In the following, we will generalize the framework outlined above to arbitrary linear multistep methods, originally proposed by Bashforth and Adams [5] , as well as Runge-Kutta methods, generally attributed  ... 
doi:10.1137/17m1120762 fatcat:jwjiodpcmveanccvh6rzclrrry

Special Issue on the Method of Lines: Dedicated to Keith Miller

Alain Vande Wouwer, W.E. Schiesser
2005 Journal of Computational and Applied Mathematics  
The method of lines (MOL) is generally recognized as a comprehensive and powerful approach to the numerical solution of time-dependent partial differential equations (PDEs).  ...  This method proceeds in two separate steps: • Spatial derivatives are first replaced with finite difference, finite volume, finite element or other algebraic approximations. • The resulting system of,  ...  In particular, "Runge-Kutta starters" are derived for use after grid adaptation, which allow a considerable saving in computational effort.  ... 
doi:10.1016/j.cam.2004.12.029 fatcat:ggkmj2d2qzf7riwot7yawwwxka

A computational procedure for interaction of high-speed vehicles on flexible structures without assuming known vehicle nominal motion

L. Vu-Quoc, M. Olsson
1989 Computer Methods in Applied Mechanics and Engineering  
Predictor/corrector algorithms, which combine Runge-Kutta methods and linear multistep methods with an unconditionally stable algorithm for structural dynamics, are proposed to solve the partitioned DAEs  ...  Moreover, stiff ODES can be efficiently solved using existing specialized software, which is mostly based on linear multistep methods, see [17] .  ...  ) Runge-Kutta method.  ... 
doi:10.1016/0045-7825(89)90058-3 fatcat:pbexwl2qszdwdoedsd73ces5dy

Convolutional Neural Networks combined with Runge-Kutta Methods [article]

Mai Zhu, Bo Chang, Chong Fu
2022 arXiv   pre-print
We consider that the iterations of implicit Runge-Kutta methods are fused into the training of these models.  ...  Moreover, we propose a novel approach to constructing network models based on high-order Runge-Kutta methods in order to achieve higher efficiency.  ...  The Runge-Kutta Neural Network (RKNN) is proposed for the identification of unknown time-invariant dynamical systems by Wang and Lin (1998) .  ... 
arXiv:1802.08831v7 fatcat:m35fjhaw65gsdbhnqzvqv4q6e4
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