numerical methods for ordinary differential equations

... reliable numerical solution is greatly desired.High-order numerical methods for solving ordinary differential equations (ODEs) are very common For instance, there are many variations of multistep methods ... financial market?Stochastic Differential Equations 0.004 10Stochastic differential equations (SDEs) are defined by a vector field that includes a random (Brownian) forcing Such equations have become ... Differential Equations 0.00000, 57We are primarily interested in numerical methods of order 1 to determine discrete- time, strong (path-wise) approximate solutions to autonomous stochastic differential

Ngày tải lên: 02/10/2024, 01:54

... interval, the equations are solved one at a time, using the same methods as for single ODEs, Trang 26Solving higher order ODEs• All numerical ODE solution methods are designed for applications ... Trang 1NUMERICAL SOLUTIONSOF ORDINARY DIFFERENTIAL EQUATIONS Trang 2ODEs in Chemical Engineering• The ODEs that occur in Chemical Engineering tend to be of the following form: • Following ... algorithms for the numerical solution of ODEs – Improved Euler’s Method – Modified Euler’s Method – Runge-Kutta algorithms Trang 14Improved Euler’s Method• The basic Euler’s Method formula is

Ngày tải lên: 19/03/2015, 08:49

... about numerical methods for both fractionaldifferential equations and integral equations To state only a few, we start from men-tioning two classic monographs concerning numerical methods for Volterra ... general methods for ordinaryfractional differential equations while [29] gives a analysis of a non-uniform gridapproximation Lastly, we would like to mention several papers discussing numericalapproaches ... particular structure of these operators A review of numericalmethods (as well as analytical results) for ordinary fractional differential equationshas been given in [6], where a modern overview

Ngày tải lên: 04/12/2022, 15:46

... Trang 1DIFFERENTIAL EQUATIONS: THEORETICAL ANALYSISAND NUMERICAL SIMULATIONS OF GROUND STATES ALEX P PALAMIDES AND THEODOROS G ... obvious therefore, that the technique presented here is dif-ferent from those employed in the previous papers [6,12], but closely related, at the sametime, to the methods of [9,11] or [10] For the ... we imposeone more condition on the desired solution lim Trang 6(pρ ) < 0 forρ ∈(−∞,−1) ∪(0,ξ),(pρ ) > 0 forρ ∈(−1, 0)∪(ξ, + ∞) (2.22) Thus, it is obvious that any solution of (2.6)

Ngày tải lên: 22/06/2014, 22:20

... Trang 1NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTIONAL DERIVATIVESLI YONGFENG (B.Sc., Jilin University) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE ... why we will consider numerical meth-ods for solving this kind of differential equations However, the exact solution inhand can be used as benchmark to evaluate numerical methods In the followingpart, ... Please refer to [12] for moredetails Other than finite element method, in this paper, we will discuss an alternativenumerical method to handle the equations with distributions For the purpose of

Ngày tải lên: 27/11/2015, 11:12

... solutions Many numerical meth-ods have been developed for DAEs Most numerical methods for differentialalgebraic equations based on standard methods from the theory of ordinarydifferential equations ... Most numerical methods for differential-algebraic equations (DAEs) arebased on standard methods from the theory of ordinary differential equation(ODEs) It is well known that the robust and numerically ... thesis Therefore, this thesis concentrates on numerical methods for semi-explicitDAEs of index 2 Here, we are concerned with one-step methods for index 2 DAEs in senberg form These methods combine

Ngày tải lên: 03/01/2018, 08:38

... Existence, blow-up and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions, Mathematical Methods in the Applied Sciences, 37 (4) (2014) 464-487 ... Thanh Long, On the existence of asymptotically stable solutions for a mixed functional integral equation in N variables, Differential Equations & Applications, 6 (2) (2014) 187-208. ISSN: 1847-120X ... Long, Asymptotically stable solutions for a nonlinear functional integral equation, Acta Mathematica Vietnamica [Ref.: Ms No AMVI-D-13-00050R2] (accepted for publication) [Feb 10, 2014] ISSN:

Ngày tải lên: 23/01/2021, 11:09

... This modification allows for the extension of well-established high-order explicit Runge-Kutta methods for ordinary differential equations (ODEs) to half-explicit forms for differential-algebraic ... descriptor form.Implicit Runge-Kutta methodsFormula of implicit Runge-Kutta methodsThe s-stage Runge-Kutta method for semi-explicit index 2 differential-algebraic equations (DAEs) is defined for a ... x˙ ) = 0 can be transformed into an ordinary differ- ential equation (ODE) of the form x˙ = f (t, x) Numerical methods forODE models have been already well discussed Therefore, the most interesting

Ngày tải lên: 23/12/2021, 21:29

... Trang 1Numerical Methods for Maxwell EquationsJoachim Sch¨ oberl SS05 AbstractThe Maxwell equations describe the interaction of electric and magnetic ... required In this lecture we formulate the Maxwell equations, and discuss the finite elementmethod to solve them Involved topics are partial differential equations, variationalformulations, edge elements, ... chanical force (Lorentz force) arising from electric current in the magnetic field is used totransform electromagnetic energy into motion, and vice versa This requires the coupling me-of Maxwell equations

Ngày tải lên: 26/01/2022, 15:15

... uniqueness of a periodic solution for second-order ordinary differential equations with some assumption 1 Introduction The study on periodic solutions for ordinary differential equations is a very important ... ordinary differential equation,” Nonlinear Analysis Theory, Methods & Applications, vol 32, no 7, pp 881–890, 1998. 11 Y Li, “Boundary value problems for nonlinear ordinary differential equations,” ... Differential Equations, vol 243, no 2, pp 536–557, 2007. 2 J Ehme, P W Eloe, and J Henderson, “Upper and lower solution methods for fully nonlinear boundary value problems,” Journal of Di fferential Equations,

Ngày tải lên: 14/07/2023, 09:44

... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Note that for compatibility with bsstep the arrays y and d2y are of length 2n for a system of n second-order equations ... This nice feature of implicit methods holds only for linear systems, but even in the general case implicit methods give better stability. Trang 3Sample page from NUMERICAL RECIPES IN C: THE ART ... all equations are linear with constant coefficients, unfortunately! For the system implicit differencing gives yn+1= yn+ hf(yn+1) (16.6.15) In general this is some nasty set of nonlinear equations

Ngày tải lên: 24/12/2013, 12:16

... b. Example 2.2 Ordinary differential equations: a two-point boundary-value problem Large sparse sets of linear equations arise in the numerical solution of differential Trang 31Formal problems ... work are numerical linear algebra and functionminimisation Why not differential equations? Quite simply because I have hadvery little experience with the numerical solution of differential equations ... indirections relevant to compact numerical methods to allow for a suitable algorithm to be included For example, over the last 15 years I have been interested in methodsfor the mathematical programming

Ngày tải lên: 31/03/2014, 15:02

... already for second order equations explicitly solvable ones are rare.Alternatively, we can also ask a symbolic computer program like Math-ematica to solve differential equations for us For example, ... the differential equation for y = y(x) you have to solve.) 1.5 Qualitative analysis of first order equations As already noted in the previous section, only very few ordinary differentialequations ... = 0, for all t ∈ I (1.16)This implicitly implies (t, φ(t), φ(1)(t), , φ(k)(t)) ∈ U for all t ∈ I Unfortunately there is not too much one can say about differential tions in the above form

Ngày tải lên: 31/03/2014, 15:56

... which of various alternative numerical methods should be used for a specific problem, or even for a large class of problems 56 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Table 201(II) h ... 12 12 −9 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 0.05 h 0.10 0.15 62 Figure 203(iii) x Stepsize h against x for the ‘mildly stiff’ problem (203a) with variable stepsize for T = 0.02 ... function, y, on [x0 , x] by the formula y(x) = y(xk−1 ) + (x − xk−1 )f (xk−1 , y(xk−1 )), x ∈ (xk−1 , xk ], (210b) 66 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS for k = 1, 2, , n If we...

Ngày tải lên: 13/08/2014, 05:21

... converge for large stepsizes (not shown in the diagrams) This effect persisted for a larger range of stepsizes for PEC methods than was the case for PECE methods NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... than for corresponding explicit NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 10−6 10−4 104 −8 E 10 10 −10 10−4 Figure 239(ii) 10−3 h 10−2 Runge–Kutta methods with cost corrections methods ... with Figure 252(i) shows the new methods to be slightly more accurate for the same stepsizes 122 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS The final numerical result in this subsection...

Ngày tải lên: 13/08/2014, 05:21

... by Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-72335-7 138 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ... 124 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 262 Generalized linear multistep methods These methods, known also as hybrid methods or modified linear multistep methods, generalize ... interpolation formula in the form y(xn−1 + ht) ≈ (1 + 2t)(1 − t)2 y(xn−1) + (3 − 2t)t2 y(xn ) + t(1 − t)2 hy (xn−1 ) − t2 (1 − t)hy (xn ) 132 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS...

Ngày tải lên: 13/08/2014, 05:21

... of the matrix A For i corresponding to a member of row k for k = 1, 2, , m, the only non-zero 190 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS aij are for j = and for j corresponding ... ) For explicit methods, D(2) cannot hold, for similar reasons to the impossibility of C(2) For implicit methods D(s) is possible, as we shall see in Section 342 174 NUMERICAL METHODS FOR ORDINARY ... 31.3 For an arbitrary Runge–Kutta method, find the order condition corresponding to the tree 170 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 32 Low Order Explicit Methods 320 Methods...

Ngày tải lên: 13/08/2014, 05:21

... p + (333g) 204 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Proof For a given tree t, let Φ(t) denote the elementary weight for (333a) and Φ(t) the elementary weight for (333b) Because ... I formula, c1 = This formula is exact for polynomials of degree up to 2s − II For the Radau II formula, cs = This formula is exact for polynomials of degree up to 2s − III For the Lobatto formula, ... of degree 216 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ∗ less than n − A simple calculation shows that Q is orthogonal to Pk for ∗ k < n − Hence, (342f) follows except for the value...

Ngày tải lên: 13/08/2014, 05:21

... 230 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 35 Stability of Implicit Runge–Kutta Methods 350 A-stability, A(α)-stability and L-stability We recall that the stability function for ... choose Z = −t diag(ej ), for t positive The value of R(Z) becomes R(Z) = − tbj + O(t2 ), 248 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS which is greater than for t sufficiently small Now ... non-linear equations We consider how to solve these equations using a full Newton method This requires going through the following steps: 260 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS...

Ngày tải lên: 13/08/2014, 05:21

... Runge–Kutta methods exist for which A is lower triangular? 280 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 38 Algebraic Properties of Runge–Kutta Methods 380 Motivation For any specific ... then the sub-forest induced by V is the forest (V , E), where E is the intersection of V × V and E A special 288 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS case is when a sub-forest (V ... acceptable for many problems In each of the values of ξ for which there is a single underline, the method is A(α)-stable with α ≥ 1.55 ≈ 89◦ 268 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS...

Ngày tải lên: 13/08/2014, 05:21