power series solution of ordinary differential equations pdf

... Delay-differential equations In this thesis we are concerned with the numerical solution of delay-differential equations (ĐÉs) Delay-differential equations may best be regarded as extensions of ordinary ... the propagation of derivative discontinuities through systems of equations is presented Following a short resumd of standard techniques for the solution of ordinary differential equations (ODÉs), ... for the solution of systems of delay-differentialequations A novel representation for the differential equation isused to acknowledge structural differences between delay- and ordinarydifferential

Ngày tải lên: 11/10/2014, 04:02

... The Numerical Solution of Delay-differential Equations by David Richard Wile Department of Mathematics University of Manchester volume II of II A thesis submitted to the University of Manchester ... appendix we give a formal proof of the result (1.1.3:14). For any given time let the order of continuity of the component y i be Z i . Our task is thus to obtain a set of lower bounds {c i : > ... closure of E. Heuristically, by an extension of the above arguments, we may think of Ei as defining the set of points whose continuity bounds can affect the continuity bound, c i , of y i .

Ngày tải lên: 11/10/2014, 04:03

... Taylor Series • Note: any function can be expanded as a Taylor Series • The series can be used to solve Taylor Series expansion of x(t) about t0 Trang 4Example: solution via Taylor SeriesTrang 5Solution ... 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) • Solutions of sets of 1st order ODEs ... SeriesTrang 5Solution using Taylor Series• Accuracy of this solution is dependent on the ‘order’ of the Taylor Series expansion • Need to have the higher order differentials of x(t) to maintain accuracy

Ngày tải lên: 19/03/2015, 08:49

... instability of an orbit A more generalmethod is the Lyapunov function In fact, any one of the numbers ofdifferent stabilization standards is used Therefore, we will study thestability of ordinary differential ... r(0) 6= 0 solutions tend to infinity in finite time We caninfer to this idea of stability the system is unstable, not because solutiondiverge, but because the solution of the original equations ... Definition 1.3.3 Suppose that ˙x = f (x) The solution of this differ-ential equation define a flow, ϕ(x, t), such that ϕ(x, t) is solution of the differential equation at time t with initial

Ngày tải lên: 23/12/2019, 16:17

... instability of an orbit A more generalmethod is the Lyapunov function In fact, any one of the numbers ofdifferent stabilization standards is used Therefore, we will study thestability of ordinary differential ... r(0) 6= 0 solutions tend to infinity in finite time We caninfer to this idea of stability the system is unstable, not because solutiondiverge, but because the solution of the original equations ... Definition 1.3.3 Suppose that ˙x = f (x) The solution of this differ-ential equation define a flow, ϕ(x, t), such that ϕ(x, t) is solution of the differential equation at time t with initial

Ngày tải lên: 07/04/2021, 08:05

... 2.6 ) admits a global solution. Proof Let ρ be a solution of (2.6) Thenρ ∈ ᐄ(P), the family of all solutions emanating u 1=max where u denotes the usual sup-norm ofu on [0, T] On the other ... paper as “the nature of the vec-tor field.” Therefore, a combination of properties of the associated vector field with theKneser’s property of the cross sections of the solutions’ funnel is the ... proposition, there is a solution ρ ∈ ᐄ(P) that solves the Taking into account the classical theorem of the extendability of solutions, we imposeone more condition on the desired solution lim Trang

Ngày tải lên: 22/06/2014, 22:20

... and 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 ... pagesdoi:10.1155/2011/192156 Research Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations Jian Zu College of Mathematics, Jilin University, Changchun 130012, ... about the existence of periodic solutions for second-order differential equations have been obtained by combining the classical method of lower and upper solutions and the method of alternative problemsThe

Ngày tải lên: 14/07/2023, 09:44

... eration to eventually positive solutions of (A), since if y(t) is a solution of (A), then so consid-is−y(t) Let P denote the totally of eventually positive solutions of (A) It will be shownthat ... y(t) is a solution of (A) belonging toP(I j) This completes the Unlike the solutions of classP(I) it seems to be very difficult (or impossible) to acterize the existence of solutions of classP(II), ... satisfies the integral equations (4.13), (4.16), and (4.19) on [t ∗,∞), respectively,that is, a positive solution of (4.4) This completes the proof ofLemma 4.4  Proof of Theorem 4.2 It is sufficient

Ngày tải lên: 22/06/2014, 21:20

... are a number of definitions for the stability of stochastic systems Acommon choice of definition amongst numerical analysts investigating stochastic differ-ential equations is that of mean square ... trivial solution of the initial functional differential equation is asymptoticallymean square stable then there exists a method and a step of discretization of this equa-tion so that the trivial solution ... stability of the trivial solution of ( 1.1 ). 3 Stability of difference analogues to the integro-differential equation Let{ Ω,Ᏺ,P }be a basic probability space, f i ∈ Ᏺ, i ∈ Z = {0, 1, }be a sequence of

Ngày tải lên: 22/06/2014, 22:20

... DIFFERENTIAL EQUATIONS Lecturer : Dr Nguyễn Hữu Thọ Department of Mathematics - WRU Trang 2Slide 11 - 2 Dr Nguyen Huu Tho - Department of Mathematics Unit 1 Introduction to Differential Equations ... Nguyen Huu Tho - Department of Mathematics Trang 26Slide 11 - 26 Dr Nguyen Huu Tho - Department of Mathematics Unit 2 Differential Equations and solutions Separable Equations Trang 28Slide 11 ... Dr Nguyen Huu Tho - Department of Mathematics Trang 46Slide 11 - 46 Dr Nguyen Huu Tho - Department of Mathematics Unit 3 Linear Equations Exact Differential Equations Trang 47Slide 11 - 47

Ngày tải lên: 16/12/2020, 00:10

... DIFFERENTIAL EQUATIONS Lecturer : Dr Nguyễn Hữu Thọ Department of Mathematics - WRU Trang 2Slide 11 - 2 Dr Nguyen Huu Tho - Department of Mathematics Unit 1 Introduction to Differential Equations ... Nguyen Huu Tho - Department of Mathematics Trang 26Slide 11 - 26 Dr Nguyen Huu Tho - Department of Mathematics Unit 2 Differential Equations and solutions Separable Equations Trang 28Slide 11 ... Dr Nguyen Huu Tho - Department of Mathematics Trang 46Slide 11 - 46 Dr Nguyen Huu Tho - Department of Mathematics Unit 3 Linear Equations Exact Differential Equations Trang 47Slide 11 - 47

Ngày tải lên: 21/03/2021, 18:30

... Bernstein Series Solution of a Class of Lane-Emden Type Equations 1 Elementary Mathematics Education Program, Faculty of Education, Mugla Sitki Kocman University, 48000 Mugla, Turkey 2 Department of ... classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain A matrix method was given by Yuzbasi for solving nonlinear Lane-Emden type equations ... Comparison of𝑦(𝑥), between present method and HFC method forExample 10, digits: 50.𝑡 Bernstein series solutions Corrected Bernstein series solution HFC method [18] Table 6: The∞-norms of the absolute

Ngày tải lên: 01/11/2022, 08:57

... discussion of SVD thus far Trang 5Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)A ⋅ x = b SVD “solution” of A ⋅ x = c solutions of A ⋅ x = c′ solutions of ... combination of the set of equations that we are trying to solve The resolution of the paradox is that we are throwing away precisely a combination of equations that is so corrupted by roundoff error ... particular solution closest to zero, as shown The point c lies outside of the range of A, so A· x = c has no solution SVD finds the least-squares best compromise solution, namely a solution of A·

Ngày tải lên: 24/12/2013, 12:16

... also be aware of some of the special sparse forms that occur in the solution of partial differential equations in two or more dimensions See Chapter 19 If your particular pattern of sparsity is ... applications.) • Each of the first N locations of ija stores the index of the array sa that contains the first off-diagonal element of the corresponding row of the matrix (If there are no off-diagonal ... THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)2.7 Sparse Linear Systems A system of linear equations is called sparse if only a relatively small number devoted to solving the set of equations

Ngày tải lên: 24/12/2013, 12:16

... Trang 1Periodic solutions of some linear evolution systems of natural differential equations on 2-dimensional tore Dang Khanh Hoi* Department of General Education, Hoa Binh University, ... and close Let € = OF gf? = OP gAM(T*X) @C is the complexified cotangent bundle of manifolds X, C°(€) is the space of smooth differential forms and H*(€) is the Sobolev space of differential forms ... phrases: Natural differential operators, small denominators, spectrum of compact operators 1 Introduction Beside authors, as from A.A Dezin (see, [1]), considered the linear differential equations

Ngày tải lên: 14/03/2014, 13:20

... of solutions In fact, equations in three (ormore) dimensions will often exhibit chaotic behavior which makes a simpledescription of solutions impossible! Problem 1.21 Generalize the concept of ... 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 ... 2004 Copyright c Trang 4§1.2 Classification of differential equations 5§1.3 First order autonomous equations 8 §1.5 Qualitative analysis of first order equations 16 §2.2 The basic existence and

Ngày tải lên: 31/03/2014, 15:56

... Trang 2of entire solutions of some classes of algebraic differential equations Qi Jianming1,3, Li Yezhou2 and Yuan Wenjun∗3 University, Shanghai 200240, People’s Republic of China 2School of Science, ... algebraic differential equations of higher order In 2009, Yuan et al [9], improved their results and gave a general estimate of order of w(z), which depends on the degrees of coefficients of differential ... this article, by means of the normal family theory, we estimate the growth order of entire solutions of some algebraic differential equations and improve the related results of Bergweiler, Barsegian,

Ngày tải lên: 20/06/2014, 20:20

... t ≥ s, and the zero solution of (8) is stable Thus, the zero solution of (8) is asymptotically stable, and hence the zero solution of (1) is asymptotically stable The proof is complete.□ Remark ... asymptotic stability of the non-linear impulsive stochastic differential equations and the impulsive stochastic partial differential equations with infinite delays by means of the fixed point theory ... results on stability of the zero solution of Equation 1 Then, we generalized the results of the stability to Equation 2 Consider the first-order delay neutral differential equation of the form x(t)

Ngày tải lên: 20/06/2014, 22:20

... Recently, there are a large number of papers dealing with the existence of solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis (fixed point ... works above, in this article we investigate the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders ⎧ ⎪ ⎪ −D α 0 +u(t) = f (t, v(t)), ... Then (u, v) Î C[0, 1] × C[0, 1] is a solution of (1.1) if and only if (u, v) Î C[0, 1] × C[0, 1] is a solution of system(3.1) This proof is similar to that of Lemma 3.3 in [25], so is omitted

Ngày tải lên: 21/06/2014, 03:20

... corresponding solutions of the integer-order equations are found to follow as special cases of those of fractional-order equations We consider the system of fractional-order equations of the form ... coefficients of like powers of p, we get the following set of equations: coefficient of p0: Y0 L−1 ∞  n 0 , coefficient of p1 : Y1 L−1 i   L−1 i Y i,1 − L−1 i Ni Y i,0 . 3.11 Now, we solve these equations ... introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations In this approach, the solution is considered as a Taylor series expansion

Ngày tải lên: 21/06/2014, 05:20