Skip to content

Download E-books Structure-Preserving Algorithms for Oscillatory Differential Equations PDF

By Xiong You

Structure-Preserving Algorithms for Oscillatory Differential Equations describes a good number of powerful and effective structure-preserving algorithms for second-order oscillatory differential equations by utilizing theoretical research and numerical validation. Structure-preserving algorithms for differential equations, in particular for oscillatory differential equations, play a big function within the exact simulation of oscillatory difficulties in technologies and engineering. The booklet discusses novel advances within the ARKN, ERKN, two-step ERKN, Falkner-type and energy-preserving equipment, and so on. for oscillatory differential equations.

The paintings is meant for scientists, engineers, lecturers and scholars who're drawn to structure-preserving algorithms for differential equations. Xinyuan Wu is a professor at Nanjing college; Xiong You is an affiliate professor at Nanjing Agricultural collage; Bin Wang is a joint Ph.D pupil of Nanjing college and collage of Cambridge.

Show description

Read or Download Structure-Preserving Algorithms for Oscillatory Differential Equations PDF

Best Mathematics books

Selected Works of Giuseppe Peano

Chosen Works of Giuseppe Peano (1973). Kennedy, Hubert C. , ed. and transl. With a biographical comic strip and bibliography. London: Allen & Unwin; Toronto: college of Toronto Press.

Fourier Series and Integrals (Probability and Mathematical Statistics)

The information of Fourier have made their manner into each department of arithmetic and mathematical physics, from the idea of numbers to quantum mechanics. Fourier sequence and Integrals specializes in the intense strength and adaptability of Fourier's easy sequence and integrals and at the magnificent number of purposes during which it's the leader instrument.

Solving Mathematical Problems: A Personal Perspective

Authored through a number one identify in arithmetic, this attractive and obviously awarded textual content leads the reader during the a number of strategies all in favour of fixing mathematical difficulties on the Mathematical Olympiad point. protecting quantity thought, algebra, research, Euclidean geometry, and analytic geometry, fixing Mathematical difficulties comprises a variety of routines and version strategies all through.

Introduction to Algorithms, 3rd Edition (MIT Press)

A few books on algorithms are rigorous yet incomplete; others hide plenty of fabric yet lack rigor. creation to Algorithms uniquely combines rigor and comprehensiveness. The ebook covers a huge diversity of algorithms extensive, but makes their layout and research obtainable to all degrees of readers.

Extra info for Structure-Preserving Algorithms for Oscillatory Differential Equations

Show sample text content

P d , P d+1 , y 1 , . . . , y d , y d+1 , p J (t) = y˙ J (t), J = 1, . . . , d, p d+1 (t) = −H p1 (t), . . . , p d (t), y 1 (t), . . . , y d (t), y d+1 (t) , y d+1 (t) = t, H˜ (z) = p d+1 + H p1 , . . . , p d , y 1 , . . . , y d , y d+1 . The equations for the prolonged Hamiltonian approach (4. 34) will be written as ⎧ J ⎪ J d+1 okay ⎪ y y = − m + f J y 1 , . . . , y d , y d+1 , p ˙ ⎪ JK ⎪ ⎪ ⎪ ⎪ okay ⎪ ⎪ ⎪ ⎨ ∂U p˙ d+1 = − d+1 y 1 , . . . , y d , y d+1 , ∂y ⎪ ⎪ ⎪ ⎪ ⎪ J J ⎪ y˙ = p , J = 1, . . . , d, ⎪ ⎪ ⎪ ⎪ ⎩ d+1 y˙ = 1. J = 1, . . . , d, (4. 35) four. five. 2 Symplectic ERKN equipment for Time-Dependent Hamiltonian platforms prior to we examine the power upkeep of symplectic multidimensional ERKN equipment, we first formulate a second-order independent approach within the prolonged section area y¨ y¨ d+1 + M(y d+1 ) 0d×1 01×d zero y y d+1 = f (y, y d+1 ) , zero t ∈ [t0 , have a tendency ], (4. 36) the place y d+1 (t) = t . which will stay away from updating the significant frequency matrix M(y d+1 ) at each one step of computation, we stick to the belief in Chap. three and think about the next identical approach: y¨ y¨ d+1 + M0 01×d 0d×1 zero y y d+1 = f˜(y, y d+1 ) , zero t ∈ [t0 , have a tendency ], (4. 37) 114 four Symplectic and Symmetric Multidimensional ERKN tools the place M0 = M(t0 ), f˜ y, y d+1 = M0 − M y d+1 y + f y, y d+1 . The preliminary values are given via y(t0 ) = y0 , y(t ˙ zero ) = y˙0 , y d+1 (t0 ) = t0 , y˙ d+1 (t0 ) = 1. The strength maintenance of the tactic for the time-dependent process (4. 32) is known because the conservation of the “extended strength” H˜ , within the prolonged section house. so as to assessment H˜ , we need to receive the numerical answer of t the prolonged variable pd+1 . to do that, we denote ψ(t) = t0 (p d+1 (ξ )) dξ and ˙ g(y, t) = − 12 y T M(t)y − ∂U ∂t . Then (4. 37) is extra prolonged to the next procedure of second-order differential equations: ⎛ ⎞⎛ ⎞ ⎞ ⎛ ⎞ ⎛ y¨ y M0 0d×1 0d×1 f˜(y, y d+1 ) ⎝ y¨ d+1 ⎠ + ⎝ 01×d ⎠ , t ∈ [t0 , have a tendency ], zero zero ⎠ ⎝ y d+1 ⎠ = ⎝ zero zero zero 01×d g(y, y d+1 ) ψ ψ¨ (4. 38) and the preliminary values are y(t0 ) = y0 , y(t ˙ zero ) = y˙0 , y d+1 (t0 ) = t0 , ψ(t0 ) = zero, ˙ zero ) = −H (t0 ). ψ(t y˙ d+1 (t0 ) = 1, making use of the ERKN approach to the preliminary worth challenge of the second-order process (4. 38) yields ⎧ s ⎪ ⎪ ⎪ Yi = φ0 ci2 V yn + ci φ1 ci2 V hy˙n + h2 a¯ ij (V )f˜ Yj , Yjd+1 , ⎪ ⎪ ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎪ i = 1, 2, . . . , s, ⎪ ⎪ ⎪ ⎪ ⎪ Yid+1 = ynd+1 + ci hy˙nd+1 , i = 1, 2, . . . , s, ⎪ ⎪ ⎪ s ⎪ ⎪ (0) ⎪ d+1 + h2 ⎪ = ψ + c hp a¯ ij g Yj , Yjd+1 , i = 1, 2, . . . , s, Ψ i n i ⎪ n ⎪ ⎪ ⎪ j =1 ⎪ ⎪ s ⎪ ⎪ ⎪ 2 ⎪ ⎪ b¯i (V )f˜ Yi , Yid+1 , y = φ (V )y + hφ (V ) y ˙ + h n+1 zero n 1 n ⎨ i=1 (4. 39) d+1 d+1 + h, ⎪ y = y ⎪ n ⎪ n+1 ⎪ s ⎪ ⎪ ⎪ d+1 + h2 ⎪ b¯i(0) g Yi , Yid+1 , = ψ + hp ψ ⎪ n+1 n n ⎪ ⎪ ⎪ ⎪ i=1 ⎪ s ⎪ ⎪ ⎪ ⎪ ⎪ hy˙n+1 = −V φ1 (V )yn + hφ0 (V )y˙n + h2 bi (V )f˜ Yi , Yid+1 , ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ d+1 ⎪ = hy˙nd+1 , ⎪ hy˙n+1 ⎪ ⎪ s ⎪ ⎪ ⎪ (0) ⎪ ˙ n+1 = hψ˙ n + h2 bi g Yi , Yid+1 , h ψ ⎪ ⎩ i=1 4. 6 Concluding feedback one hundred fifteen the place V = h2 M0 . For the aim of comparing pd+1 , we really don't need to calculate ψ . conserving p d+1 = ψ˙ in brain, we will be able to simplify (4.

Rated 4.34 of 5 – based on 43 votes