4 numerical solution of nonlinear ordinary differential equations
báo cáo hóa học:" Research Article Minimal Nonnegative Solution of Nonlinear Impulsive Differential Equations on Infinite Interval" pot
... 1 x k 10k −9 /4 1/2 , x k y 2/3 , 1, 2, , 4. 2 1/2 , k 1, 2, , −t e g t Evidently, x t ≡ is not the solution of 4. 1 Conclusion Problem 4. 1 has minimal positive solution ∞ Proof It is clear ... positive solutions of multi-point boundary value problem for secondorder impulsive differential equations, ” Journal of Computational and Applied Mathematics, vol 223, no 1, pp 43 8 44 8, 2009 24 J Yan, ... and A Ouahab, “Extremal solutions of second order impulsive dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol 3 24, no 1, pp 42 5 43 4, 2006 26 M Benchohra,...
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Báo cáo hóa học: "ON DISCRETE ANALOGUES OF NONLINEAR IMPLICIT DIFFERENTIAL EQUATIONS" docx
... discrete analogues of nonlinear implicit differential equations ×10− 14 Errn 0 0.2 0 .4 0.6 0.8 tn Figure 4. 2 (Example 4. 2) h = 0.05, τ = 0.0005 Figures 4. 1 and 4. 2 show the maximal values of the local ... analogues of nonlinear implicit differential equations convergence of the explicit Euler method for nonlinear index-1 DAEs is established The results of this section are a nonlinear version” of the ... problem (3 .4) –(3.5) has a unique solution, provided h is sufficiently small and τ = αh2 , α = const Proof From the above mentioned argument we see that the problem of finding solution of system (3 .4) –(3.5)...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
... powers of h, 7 24 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations ... compare equation (4. 2 .4) with equation (16.3 .4) above You will see that the transition in Chapter to the idea of Richardson extrapolation, as embodied in Romberg integration of 4. 3, is exactly ... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521 -43 108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software...
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Tài liệu Numerical Solution of Stochastic Differential Equations with Jumps in Finance pdf
... Existence and Uniqueness of Solutions of SDEs 1.10 Exercises 1 16 23 26 34 38 45 53 57 59 Exact Simulation of Solutions of SDEs ... 10.6 Exercises 41 9 41 9 42 4 43 3 44 7 45 6 47 5 11 Monte Carlo Simulation of SDEs 11.1 Introduction to Monte Carlo Simulation ... 523 529 530 533 5 34 543 548 569 14 Numerical Stability 14. 1 Asymptotic p-Stability 14. 2 Stability of Predictor-Corrector...
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Báo cáo hóa học: "TERMINAL VALUE PROBLEM FOR SINGULAR ORDINARY DIFFERENTIAL EQUATIONS: THEORETICAL ANALYSIS AND NUMERICAL SIMULATIONS OF GROUND STATES" pdf
... the rest of this paper as “the nature of the vector field.” Therefore, a combination of properties of the associated vector field with the Kneser’s property of the cross sections of the solutions’ ... (2. 24) A terminal BVP with f continuous and let E0 be a continuum (i.e., compact and connected) subset of Rn and let ᐄ(E0 ) be the family of all solutions of 2. 24 emanating from E0 If any solution ... (3.3) r →0+ In view of Theorem 2.2 and Remark 2.10, this singular IVP has a local solution By the nature of the vector field (sign of the nonlinearity), any solution ρ = ρ(r) of (3.3) as well as...
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Numerical Solution of Stochastic Differential Equations with Jumps in Finance docx
... Existence and Uniqueness of Solutions of SDEs 1.10 Exercises 1 16 23 26 34 38 45 53 57 59 Exact Simulation of Solutions of SDEs ... 10.6 Exercises 41 9 41 9 42 4 43 3 44 7 45 6 47 5 11 Monte Carlo Simulation of SDEs 11.1 Introduction to Monte Carlo Simulation ... 523 529 530 533 5 34 543 548 569 14 Numerical Stability 14. 1 Asymptotic p-Stability 14. 2 Stability of Predictor-Corrector...
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The numerical solution of delay-differential equations P1
... Delay -differential equations In this thesis we are concerned with the numerical solution of delay -differential equations (DDE's) Delay -differential equations may best be regarded as extensions of ordinary ... we consider the development of numerical software for the solution of such problems Our discussion opens with a brief introduction to the theory of delay -differential equations Attention is paid ... for the solution of systems of delay -differential equations A novel representation for the differential equation is used to acknowledge structural differences between delay- and ordinary differential...
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The numerical solution of delay-differential equations P2
... C X •-• n4 I •-• - .4 E Z 41 4-I 4. 1 Ct E a 0 • 44 E OZ 1 -4 000 -4 ru -4 41 >4 > 4) 14 0 41 X 00 C 41 -1 -I 41 0 ILI • -I ti CI U W 11 al - .4 ) V0 41 4) 4 14 04- I '0 I II 44 -1'5C 4) 44 41 4- I 110 ... .0 -0 C 14 C rtl Z 0100 00 UM V r 4e H1-1C.>" 17 1300 40 14 CC-10 44 -4 _C00 17 a U U 40 -.-101 44 444 0 CM n I 14 041 44 1 O 00 CO M 44 14 -4 AZ V V 0. 14 14 14 1 '41 4 14 Al 0. 14 la B>,>, 144 -I 3" ... 0.-111 40 -4 V 04 O 0.1 04 • 1 .4 C 1.1 M 14 V 4n00001 MV 0'0O) / C O XI al I4 144 4 -44 04 woe 4C 00000 044 .1 044 -40 -1 c to M V 0.- 00) 00 44 14 01 "0 VO 0.>" V 0-1 0'0 10 0 44 11 c 4) 04 v 110 00 41 4...
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NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
... more accurate solutions The size of the integration interval also influences the accuracy of the solutions (should be less than 10% of the time constant) • • st Solutions of sets of order ODEs ... interval × function of gradient • This is the basis for a family of algorithms used to provide numerical solutions of ODEs • A particular class is the Runge-Kutta algorithms School of Chemical Engineering ... Engineering and Advanced Materials Newcastle University 20 30 Time 40 50 31 60 Numerical solution of ODEs: summary • • • • • All numerical ODE solution methods are based on the Taylor Series Euler’s Method...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
... subject of stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19) Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 1973, Computational Methods in Ordinary Differential Equations (New York: Wiley) Lapidus, L., and Seinfeld, J 1971, Numerical Solution of Ordinary Differential Equations (New York: Academic Press) ... 708 Chapter 16 Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521 -43 108-5) Copyright (C) 1988-1992 by...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
... free_vector(dym,1,n); 7 14 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A 19 64, Handbook of Mathematical Functions, ... http://www.nr.com or call 1-800-872- 742 3 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) x2 x1 712 Chapter 16 Integration of Ordinary Differential Equations yn yn + Figure ... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521 -43 108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
... b61=1631.0/55296.0,b62=175.0/512.0,b63=575.0/138 24. 0, b 64= 442 75.0/110592.0,b65=253.0 /40 96.0,c1=37.0/378.0, c3=250.0/621.0,c4=125.0/5 94. 0,c6=512.0/1771.0, dc5 = -277.00/ 143 36.0; float dc1=c1-2825.0/27 648 .0,dc3=c3-18575.0 /48 3 84. 0, dc4=c4-13525.0/55296.0,dc6=c6-0.25; ... Method i ci bij 1 5 3 10 40 40 10 − 10 5 − 11 54 − 70 27 35 27 1631 55296 175 512 575 138 24 442 75 110592 253 40 96 2825 27 648 j= 0 250 621 18575 48 3 84 125 5 94 13525 55296 277 143 36 512 1771 Now that ... America) static float a2=0.2,a3=0.3,a4=0.6,a5=1.0,a6=0.875,b21=0.2, b31=3.0 /40 .0,b32=9.0 /40 .0,b41=0.3,b42 = -0.9,b43=1.2, b51 = -11.0/ 54. 0, b52=2.5,b53 = -70.0/27.0,b 54= 35.0/27.0, b61=1631.0/55296.0,b62=175.0/512.0,b63=575.0/138 24. 0,...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
... 1-800-872- 742 3 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) y x ⊗ steps 726 Chapter 16 Integration of Ordinary Differential Equations n = 2, 4, 6, 8, 12, 16, 24, ... 8, 12, 16, 24, 32, 48 , 64, 96, , [nj = 2nj−2 ], (16 .4. 1) More recent work by Deuflhard [2,3] suggests that the sequence n = 2, 4, 6, 8, 10, 12, 14, , [nj = 2j], (16 .4. 2) is usually more ... sequence (16 .4. 2) the order of the method is 2k + 1: k+1,k ∼ H 2k+1 (16 .4. 4) Thus a simple estimate of a new stepsize Hk to obtain convergence in a fixed column k would be 1/(2k+1) Hk = H (16 .4. 5) k+1,k...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
... 1-800-872- 742 3 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) Then for k = 1, , m − 1, set 7 34 Chapter 16 Integration of Ordinary Differential Equations Note ... vol 27, pp 505–535 16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, the possibility of a stiff set of equations arises Stiffness occurs in ... compatibility with bsstep the arrays y and d2y are of length 2n for a system of n second-order equations The values of y are stored in the first n elements of y, while the first derivatives are stored...
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
... A31 4. 5 247 0820736 A32 4. 16352878860 C21 -5.07167533877 C31 6.02015272865 C32 0.1597506 846 73 C41 -1.856 343 618677 C42 -8.50538085819 C43 -2.0 840 7513602 B1 3.957503 746 63 B2 4. 6 248 9238836 B3 0.61 747 7263873 ... 0.61 747 7263873 B4 1.282612 945 268 E1 -2.30215 540 292 E2 -3.0736 344 8539 E3 0.873280801802 E4 1.282612 945 268 C1X GAM Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521 -43 108-5) ... g2 ) + (c31 g1 + c32 g2 )/h (1/γh − f ) · g4 = f(y0 + a41 g1 + a42 g2 + a43 g3 ) + (c41 g1 + c42 g2 + c43 g3 )/h (16.6. 24) In our implementation stiff of the Kaps-Rentrop algorithm, we have carried...
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Tài liệu Integration of Ordinary Differential Equations part 8 pdf
... 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), Chapter [1] Shampine, L.F., and Gordon, M.K 1975, Computer Solution of Ordinary Differential ... 16 Integration of Ordinary Differential Equations For multivalue methods the basic data available to the integrator are the first few terms of the Taylor series expansion of the solution at the ... 1-800-872- 742 3 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) be satisfied The second of the equations in (16.7.9) is 752 Chapter 16 Integration of Ordinary Differential...
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báo cáo hóa học: " Some new results for BVPs of first-order nonlinear integro-differential equations of volterra type" doc
... } 360 ≥ 2α(2x2 + 3y2 − = 16x2 + 24y2 − e−ts y(s)ds + xy 240 R4 320 R4 40 − − R3 240 − R3 30 − R 45 , R 360 ) for α = Clearly, ; 64 min{24y2 − 2|y|} = − x∈R 24 min{16x2 − |x|} = − x∈R Thus, ||F(t, ... Advances in Difference Equations 2011, 2011: 14 http://www.advancesindifferenceequations.com/content/2011/1/ 14 Page of 17 It completes the proof Now consider the existence of solutions of PBVP (1.1) It ... potential solutions of x = λTx, λ ∈ [0, 1], (3: 14) are bounded a priori, with the bound being independent of l With this in mind, let x be a solution of (3. 14) Note that x is also a solution of (3.6)...
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Báo cáo hóa học: " Research Article A Generalized Wirtinger’s Inequality with Applications to a Class of Ordinary Differential Equations" docx
... periodic solutions of differential delay equations, ” Nonlinear Analysis: Theory, Methods & Applications, vol 35, no 4, pp 45 7 47 4, 1999 J Li and X.-Z He, “Multiple periodic solutions of differential ... delay equations, ” Journal of Mathematical Analysis and Applications, vol 48 , no 2, pp 317– 3 24, 19 74 −g x t − ,” R D Nussbaum, “Uniqueness and nonuniqueness for periodic solutions of x t Journal of ... Science and Technology 20070 049 References M Han, “Bifurcations of periodic solutions of delay differential equations, ” Journal of Differential Equations, vol 189, no 2, pp 396 41 1, 2003 R D Nussbaum,...
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Báo cáo hóa học: " Research Article Necessary Conditions of Optimality for Second-Order Nonlinear Impulsive Differential Equations" docx
... for t ∈ tn ,T (4. 41) 14 Advances in Difference Equations Using (4. 8), (4. 33), and (4. 41), we have ϕk tn − − ϕ tn − ≤ hk ≡ bk + ck + λ(2δ + 1)ak e2αTθ , X∗ ϕk tn − − ϕ tn − X∗ (4. 42) ≤ hk , where ... (T) ˙ ˙ Obviously, if ϕ is the classical solution of (4. 9), then it must be the PCr -mild solution of (4. 6) Now we show that (4. 9) has a unique classical solution ϕ ∈ PC (I,X ∗ ) PC(I, D(A∗ )) ... estimate of mild solution in space C (I,X) which can be proved by Gronwall lemma Step by step, the existence of PCl -mild solution of (3.1) can be derived Let xu denote the PCl -mild solution of system...
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Báo cáo hóa học: " Research Article Existence of Solutions for Second-Order Nonlinear Impulsive Differential Equations with Periodic Boundary Value Conditions" potx
... ordinary differential equations, ” Nonlinear Analysis, vol 51, no 7, pp 1223–1232, 2002 [ 24] C Bai, “Existence of solutions for second order nonlinear functional differential equations with periodic ... r1 := p + p2 + 4q > 0, r2 := p − p2 + 4q < (2.7) Lemma 2.1 u ∈ PC1 ([0,T], Rn ) ∩ C ([0,T] \ {t1 }, Rn ) is a solution of (2.6) if and only if u ∈ PC1 ([0,T], Rn ) is a solution of the following ... attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems,” Mathematical and Computer Modelling, vol 40 , no 5-6, pp 509–518, 20 04 [ 14] W Zhang and...
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