On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs

2017 ◽  
Vol 8 (1-2) ◽  
pp. 17
Author(s):  
Truong Nguyen-Ba ◽  
Abdulrahman Alzahrani ◽  
Thierry Giordano ◽  
Remi Vaillancourt
Keyword(s):  
Relative Error ◽  
Upper Bounds ◽  
Initial Value ◽  
Maximum Order ◽  
First Order ◽  
Cpu Time ◽  
Good Shape ◽  
Nonnegative Coefficients ◽  
Contractivity Preserving ◽  
Predictor Corrector

New optimal, contractivity-preserving (CP), \(d\)-derivative, 2- and 3-step, predictor-corrector,  Hermite-Birkhoff-Obrechkoff series methods, denoted by \(HBO(d,k,p)\), \(k=2,3\), with  nonnegative coefficients are constructed  for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\).  The upper bounds \(p_u\) of order \(p\) of \(HBO(d,k,p)\), \(k=2,3\) methods are approximately 1.4 and 1.6 times the number  of derivatives \(d\), respectively.  Their stability regions have generally  a good shape and grow with decreasing \(p-d\).  Two selected CP HBO methods: 9-derivative 2-step HBO of order 13, denoted by HBO(9,2,13),  which has maximum order 13 based on the CP conditions, and  8-derivative 3-step HBO of order 14, denoted by HBO(8,3,14), compare well  with Adams-Cowell of order 13 in PECE mode, denoted by AC(13),  in solving standard N-body problems over an interval of 1000 periods  on the basis of the relative error of energy as a function of the CPU time.  They also compare well with AC(13) in solving standard N-body problems on the basis of the growth of relative error of energy and 10000 periods of integration.  The coefficients of selected HBO methods are listed in the  appendix.

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

2017 ◽  
Vol 8 (1-2) ◽  
pp. 139
Author(s):  
Truong Nguyen-Ba ◽  
Thierry Giordano ◽  
Huong Nguyen-Thu ◽  
Remi Vaillancourt
Keyword(s):  
Relative Energy ◽  
Positional Error ◽  
Maximum Order ◽  
Energy Error ◽  
Good Shape ◽  
Regions Of Stability ◽  
Nonnegative Coefficients ◽  
Contractivity Preserving ◽  
Predictor Corrector ◽  
Numer Math

The contractivity-preserving 2- and 3-step predictor-corrector series methods for ODEs  (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt,  On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs,  J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17--39. doi:10.20454/jmmnm.2017.1130)  are expanded into  new optimal, contractivity-preserving (CP), d-derivative, k-step, predictor-corrector,  Hermite- Birkhoff--Obrechkoff series methods, denoted by HBO(d,k,p), k=4,5,6,7, with nonnegative coefficients  for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\).  The main reason for considering this class of formulae is to obtain a set of methods  which have larger regions of stability and generally higher upper bound \(p_u\) of  order \(p\) of HBO(d,k,p) for a given d. Their stability regions have generally  a good shape and grow generally with decreasing \(p-d\).  A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14)  which has maximum order 14 based on the CP conditions compares satisfactorily  with Adams--Cowell of order 13 in PECE mode, denoted by AC(13),  in solving standard N-body problems over an interval of 1000 periods  on the basis of the relative error of energy as a function of the CPU time.  HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems  on the basis of the growth of relative positional error, relative energy error  and 10000 periods of integration.  The coefficients of HBO(6,4,14) are listed in the appendix.

Continuous linear multistep method for the general solution of first order initial value problems for Volterra integro-differential equations

2018 ◽  
Vol 14 (5) ◽  
pp. 960-969
Author(s):  
Nathaniel Mahwash Kamoh ◽  
Terhemen Aboiyar
Keyword(s):  
General Solution ◽  
Approximation Method ◽  
Legendre Polynomial ◽  
Basis Function ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
Content Type ◽  
First Order ◽  
Predictor Corrector

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

Parallel iteration of two-step Runge-Kutta methods

2021 ◽  
Vol 66 (1) ◽  
pp. 12-24
Author(s):  
Thuy Nguyen Thu
Keyword(s):  
Iteration Process ◽  
Parallel Computers ◽  
Initial Value ◽  
Parallel Methods ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Parallel Iteration ◽  
Predictor Corrector ◽  
First Order Differential Equations

In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

Mathematical Models of Papermaking

2001 ◽  
Vol 6 (1) ◽  
pp. 9-19 ◽  
Author(s):  
A. Buikis ◽  
J. Cepitis ◽  
H. Kalis ◽  
A. Reinfelds ◽  
A. Ancitis ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Initial Value Problems ◽  
Moisture Transfer ◽  
Bound Water ◽  
Drying Process ◽  
Initial Value ◽  
First Order ◽  
Heat And Moisture ◽  
The Mathematical Model ◽  
The Web

The mathematical model of wood drying based on detailed transport phenomena considering both heat and moisture transfer have been offered in article. The adjustment of this model to the drying process of papermaking is carried out for the range of moisture content corresponding to the period of drying in which vapour movement and bound water diffusion in the web are possible. By averaging as the desired models are obtained sequence of the initial value problems for systems of two nonlinear first order ordinary differential equations. 

Improved Predictor Corrector Method for solving fuzzy initial value problem

2009 ◽  
Author(s):  
T. Allahviranloo ◽  
N. Ahmady ◽  
E. Ahmady ◽  
Alberto Cabada ◽  
Eduardo Liz ◽  
...  
Keyword(s):  
Initial Value Problem ◽  
Initial Value ◽  
Predictor Corrector

scholarly journals A graphical solution of a differential equation with application to hill’s treatment of nerve excitation

1937 ◽  
Vol 123 (832) ◽  
pp. 382-395 ◽  
Keyword(s):  
Viscous Medium ◽  
Graphical Solution ◽  
Initial Value ◽  
First Order ◽  
Rate Dependent ◽  
Monomolecular Reaction ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Physical And Chemical ◽  
Nerve Excitation

Linear differential equations with constant coefficients are very common in physical and chemical science, and of these, the simplest and most frequently met is the first-order equation a dy / dt + y = f(t) , (1) where a is a constant, and f(t) a single-valued function of t . The equation signifies that the quantity y is removed at a rate proportional to the amount present at each instant, and is simultaneously restored at a rate dependent only upon the instant in question. Familiar examples of this equation are the charging of a condenser, the course of a monomolecular reaction, the movement of a light body in a viscous medium, etc. The solution of this equation is easily shown to be y = e - t / a { y 0 = 1 / a ∫ t 0 e t /a f(t) dt , (2) where y 0 is the initial value of y . In the case where f(t) = 0, this reduces to the well-known exponential decay of y .

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