EXAMPLES OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 Copyright C 1988-1992 by Cambridge University P[r]

How is ∆0, the desired accuracy, related to some looser prescription like “get a solution good to one part in 10 6 ”? That can be a subtle question, and it depends on exactly what your application is! You may be dealing with a set of equations whose dependent variables differ enormousl[r]

The Runge-Kutta method treats every step in a sequence of steps in identical manner. Prior behavior of a solution is not used in its propagation. This is mathematically proper, since any point along the trajectory of an ordinary differential equation can serve as a[r]

Chúng tôi đã tiến hành tổ chức nhóm nghiên cứu về phương trình vi phân phi tuyến và phương trình tích phân phi tuyến để thực hiện đề tài nghiên cứu như đăng ký, các kết quả thu được đã đ[r]

722 Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Re[r]

3. Step forward in time using a Runge-Kutta ordinary differential equation solver to get ω at a small time in the future.
This new value of ω is then used in step 1 and the process can continue indef- initely. We will examine various techniques for solving the matrix equation[r]

20. Li, YK, Shu, J: Multiple positive solutions for first-order impulsive integral boundary value problems on time scales. Bound Value Probl. 2011 , 12 (2011). doi:10.1186/1687-2770-2011-12
21. Li, YK, Shu, J: Solvability of boundary value problems with Riemann-Stieltjes -integral condition[r]

As we discussed in the proof of Theorem 3.1, T is compact. Taking into account that the family of BVP (1.2) is equivalent to the family of problem x = Tx , our problem is reduced to show that T has a least one fixed point. For this purpose, we apply Schae- fer ’ s Theorem by sho[r]

y(t) uniquely from its n th derivative, we need n additional pieces of information (constraints) about
y(t) . These constraints are also called auxiliary conditions. When these conditions are given at t = 0 ,
they are called initial conditions.
We discuss here two systematic procedures[r]

3. Step forward in time using a Runge-Kutta ordinary differential equation solver to get ω at a small time in the future.
This new value of ω is then used in step 1 and the process can continue indef- initely. We will examine various techniques for solving the matrix equation[r]

732 Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Re[r]

15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painlev´e Test, Using Truncated Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
15.12. Methods of the Inverse Scattering Problem (Soliton Theory) . . . . . .[r]

WITH THE ABOVE APPROACH, IT CAN BE SHOWN THAT THE NONLINEAR WAVE EQUATION 15.3.3.11 ALSO TRANG 2 676 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TABLE 15.2 Invariant solutions that may be o[r]