numerical methods for first order differential equations

... appeared upon acceptance Fully formatted PDF and full text (HTML) versions will be made available soon Minimal and maximal solutions to first- order differential equations with state-dependent ... problems for ordinary differential equations with deviated arguments J Optim Theory Appl 135(2), 257–269 (2007) Jankowski, T: Existence of solutions of boundary value problems for differential equations ... (greatest) solution This seems to be a peculiar feature of equations with deviated arguments, see [1] for an example with a second -order equation Therefore, we are obliged to distinguish between the concepts...

Ngày tải lên: 20/06/2014, 20:20

... Difference Equations 2011, 2011:2 http://www.advancesindifferenceequations.com/content/2011/1/2 RESEARCH Open Access Positive solutions of the three-point boundary value problem for fractional -order differential ... s)α−1 ds (α) Wang et al Advances in Difference Equations 2011, 2011:2 http://www.advancesindifferenceequations.com/content/2011/1/2 Page of 11 Hence, for each u Ỵ Ω, let t1, t2 Ỵ [0, 1], t1 < t2, ... 1] | ||u|| 0 such that f (u) ≤ τ1u for < u < r1, where τ1 >0 satisfies Wang et al Advances in Difference Equations 2011, 2011:2 http://www.advancesindifferenceequations.com/content/2011/1/2 τ1...

Ngày tải lên: 21/06/2014, 03:20

... − x1 dy = which is exact The solution is Linear Equations dx x y x = x2 = C 4.1 Definition A first- order linear differential equation has the form y + p(x)y = q(x) Example Looking back: Nguyen ... − x1 dy = which is exact The solution is Linear Equations dx x y x = x2 = C 4.1 Definition A first- order linear differential equation has the form y + p(x)y = q(x) v dI R E Example Looking back: ... − x1 dy = which is exact The solution is Linear Equations dx x y x = x2 = C 4.1 Definition A first- order linear differential equation has the form y + p(x)y = q(x) v dI R E Example Looking back:...

Ngày tải lên: 23/04/2020, 08:00

... by differential equations or systems of differential equations with different bound-ary conditions It is possible to classify the fourth -order differential equations into two forms: fully order ... Luong Tai, etc also studied methods for solving the boundary value problems for differential equations Among the differential equations, the nonlinear fourth -order differential equation has been ... approximations by approximation methods For nonlinear equations, the use of approximation methods is almost inevitable In solving differential equations for differential equations, one can find their...

Ngày tải lên: 01/04/2021, 19:17

... doi:10.1155/2009/141959 Research Article Maximum Principles and Boundary Value Problems for First- Order Neutral Functional Differential Equations Alexander Domoshnitsky, Abraham Maghakyan, and Roman Shklyar Department ... > 0for0 ≤ t<8,Ct, 0 < 0fort>8,Ct, s > 0for0 <s≤ 2, 0 ≤ t<4,Ct, s < 0for0 <s≤ 2,t>4. We see that each interval 0,ω, where ω<8, is a nonoscillation one for ... equation can be formulated in the following form of positivity of the Cauchy function Ct, s and the solution Xt. Results about comparison of solutions for delay differential equations solved...

Ngày tải lên: 22/06/2014, 03:20

... convergence rate NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTIONAL DERIVATIVES LI YONGFENG NATIONAL UNIVERSITY OF SINGAPORE 2002 NUMERICAL METHOD FOR DIFFERENTIAL EQUATIONS WITH ... filters for the simulation of flows, J Comput Phys in press 57 Name: Li Yongfeng Degree: Master of Science Department: Computational Science Thesis Title: Numerical Methods for Differential Equations ... NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTIONAL DERIVATIVES LI YONGFENG (B.Sc., Jilin University) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE...

Ngày tải lên: 27/11/2015, 11:12

... inequality for certain higher -order differential equations Appl Math Comput 134(2–3), 307–317 (2003) [17] Cakmak, D: Lyapunov-type integral inequalities for certain higher order differential equations ... appeared upon acceptance Fully formatted PDF and full text (HTML) versions will be made available soon Lyapunov-type inequalities for a class of even -order differential equations Journal of Inequalities ... Lyapunov-type inequalities for certain higher order differential equations J Math Anal Appl 195, 527–536 (1995) [9] Tang, XH, Zhang, M: Lyapunov inequalities and stability for linear Hamiltonian systems...

Ngày tải lên: 20/06/2014, 21:20

... CRITERIA FOR FIRST- ORDER FORCED NONLINEAR DIFFERENCE EQUATIONS RAVI P. AGARWAL, SAID R. GRACE, AND TIM SMITH Received 11 August 2005; Revised 20 March 2006; Accepted 25 April 2006 Some new criteria for ... (4.27) and also for forced neutral second -order nonlinear difference equations of the form Δ2 x(n) − c(n)x[n − τ] + q1 (n)xμ [n − τ + 1] = q2 (n)xλ [n − τ + 1] + e(n) (4.28) The formulations ... oscillation criteria for the neutral forced equations of type (2.26),... Discrete first -order forced oscillation The following example is illustrative Example 4.3 The forced nonlinear ...

Ngày tải lên: 22/06/2014, 22:20

... TERMINAL VALUE PROBLEM FOR SINGULAR ORDINARY DIFFERENTIAL EQUATIONS: THEORETICAL ANALYSIS AND NUMERICAL SIMULATIONS OF GROUND STATES ALEX P. PALAMIDES AND ... (0,+ ∞)andM ≤∞, combining shooting argument with vari- ational methods. For strongly singular higher -order linear d ifferential equations together with two- point conjugate and right-focal boundary ... then the uniqueness of the point... Two-point boundary value problems for higher -order linear differential equations with strong singularities, Boundary Value Problems 2006 (2006),...

Ngày tải lên: 22/06/2014, 22:20

... 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) Euler ... 1+x C + x 2 , for C an arbitrary constant, and the exact and approximate solutions are shown in Figure 200(i). 54 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 201 Some numerical experiments ... −9  . 62 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 0123 0.05 0.10 0.15 x h Figure 203(iii) Stepsize h against x for the ‘mildly stiff’ problem (203a) with variable stepsize for T =0.02...

Ngày tải lên: 13/08/2014, 05:21

... (232b) NUMERICAL DIFFERENTIAL EQUATION METHODS 95 233 Third order methods It is possible to construct methods with three stages, which have order 3 numerical behaviour. These have the form 0 ... persisted for a larger range of stepsizes for PEC methods than was the case for PECE methods NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 10−6 10 4 1 14 10−8 E 2 10−10 ... completely characterized the set of Runge–Kutta methods of order 4, and proposed the first methods of order 5. Special methods for second order differential equations were proposed by Nystr¨om (1925),...

Ngày tải lên: 13/08/2014, 05:21

... 1 6 0 . 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 ... Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-723 35- 7 138 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... − 1 2 h). 128 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 26.3 Show how to represent the PEC method based on the second order Adams–Bashforth predictor and the third order Adams–Moulton...

Ngày tải lên: 13/08/2014, 05:21

... expansions for Y 1 ,forhf (Y 1 ), for Y 2 ,forhf (Y 2 ),andsoonuntilwereachY s , hf(Y s )and finally y 1 . 160 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Theorem 313B The Taylor expansions for ... ; for. .. 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 = 1 and for ... for which there is no confusion. Hence, for methods satisfying D(1), scalar and vector order 5 conditions are equivalent. For orders 6 and higher, the confusion between the order conditions for...

Ngày tải lên: 13/08/2014, 05:21

... quadrature formula, respectively Then: I For the Radau I formula, c1 = 0 This formula is exact for polynomials of degree up to 2s − 2 II For the Radau II formula, cs = 1 This formula is exact for. ... differentialalgebraic equations, for which implicit Runge–Kutta methods also have a vital role 214 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 341 Solvability of implicit equations As ... (336a)...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)...

Ngày tải lên: 13/08/2014, 05:21

... 54 has order 5. 34.3 Find the tableau for the Gauss method with s =4andp =8. 34.4 Show that Gauss methods are invariant under reflection. 230 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ... presented in Table 352(V). 236 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Table 352(III) First sub-diagonal members of the Pad´etableN m−1,m /D m−1,m for m =1, 2, ,7 m N m−1,m D m−1,m ... evaluate XG in full 256 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Lemma 359A Let XG = W BAW, where A and B = diag(b) are as for the Gauss method of order 2s Also let 1 ξk...

Ngày tải lên: 13/08/2014, 05:21

... 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... ... Ei ), for i = 1, 2, , n, where V is the... bi bj (370a) Now consider a problem for which y Qf (y) = 0, (370b) 276 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS for all ... RUNGE–KUTTA METHODS 273 365 Effective order and DESIRE methods An alternative way of forcing singly implicit methods to be more appropriate for practical computation is to generalize the order conditions....

Ngày tải lên: 13/08/2014, 05:21

... over one step, and therefore over several steps 322 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 405 Necessity of conditions for convergence We formally prove that stability ... isamemberofeachofthem and methods can be constructed which also lie in them. We first prove the following result: 302 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Theorem 388H For any real θ and ... 1 120 . For explicit Runge–Kutta methods with fourth (effective) order, four stages are still necessary, but there is much more freedom than for methods with the same classical order. For fifth...

Ngày tải lên: 13/08/2014, 05:21

... Subsection 433 346 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Algorithm 432α Boundary locus method for low order Adams–Bashforth methods % Second order % -w = exp(i*linspace(0,2*pi));... ... vector space RkN 362 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS For U ∈ RkN , write Ui , i = 1, 2, , k, for subvectors... higher for the predictor than for the corrector ... approximation of the form k y(xn ) ≈ β0 hy (xn ) + y(xn − hθ1 ) + βi hy (xn − hθi ) i=1 (461b) 368 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ∗ To obtain order p = k for (461a),...

Ngày tải lên: 13/08/2014, 05:21

... Runge–Kutta methods, they have some desirable properties For (505a) the stage order is 2, and for (505b) the stage order is 3 384 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ... denoted by Numerical Methods for Ordinary Differential Equations, Second Edition. J. C. Butcher © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-72335-7 374 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... one of these formulations, can be transformed into the sequence that would have been generated using the alternative formulation. 376 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS It...

Ngày tải lên: 13/08/2014, 05:21

... matrix, for example a matrix with rank 1 422 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS If p = q,... some 418 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ... by Exercise 53.1 420 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 54 Methods with Runge–Kutta stability 540 Design criteria for general linear methods We consider some ... inconsistent with high order for linear multistep methods but is available for Runge–Kutta methods of any order The stability matrix for a general linear method has the form M (z) = V + zB(I...

Ngày tải lên: 13/08/2014, 05:21