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A General Initial Value Problem [PDF]

, 1971
Suppose r ≥ O is a given real number, R = (−∞,∞), Rn is a real or complex n-dimensional linear vector space with norm |•|, C([a,b],Rn) is the Banach space of continuous functions mapping the interval [a,b] into Rn with the topology of uniform convergence. If [a,b] = [−r,0] we let C = C([−r,0],Rn) and designate the norm of an element ϕ in C by \(\left| {
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Initial Value and Boundary Value Problems

2011
The energy balance models by Sellers (1969) and Budyko (1969) result in a linear partial differential equation of 1st order in time and 2nd order in space, (4.9).
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Initial-value problem for a linear ordinary differential equation of noninteger order

, 2011
An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely ...
A. Pskhu
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Inhomogeneous inflation: The initial-value problem

Physical Review D, 1991
We present a spatially three-dimensional study for solving the initial-value problem in general relativity for inhomogeneous cosmologies. We use York's conformal approach to solve the constraint equations of Einstein's field equations for scalar field sources and find the initial data which will be used in the evolution. This work constitutes the first
Richard A. Matzner, Pablo Laguna, Hannu Kurki-Suonio   +2 more
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The General Initial Value Problem

2004
We now consider the Initial Value Problem or IVP for a system of nonlinear differential equations of the form: Find u : [0, 1] →ℝ d such that $$u'(x) = f(u(x),x)for0 < x1,u(0) = {u^0}$$ (40.1) where f : ℝ d × [0, 1] → ℝ d is a given bounded and Lipschitz continuous function, u 0 ∈ ℝ d is a given initial value, and d ≥ 1 is the dimension of ...
Kenneth Eriksson, Donald Estep, Claes Johnson   +2 more
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The Viscous Initial Value Problem

2001
This chapter will reexamine the equations governing the evolution of small disturbances in a viscous fluid. The emphasis will be different, though. Rather than concentrating on the eigenvalue problem, we will investigate the equations in the form of an initial value problem.
Peter J. Schmid, Dan S. Henningson
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The Initial-Boundary Value Problem

2004
In this chapter, we extend the analysis of Chapter 7 and consider the evolution of the scalar initial-boundary value problem (6.16)–(6.20), namely, $$ u_t = u_{xx} + f(u), x,t > 0, $$ (1) $$ f(u) = \left\{ {\begin{array}{*{20}c} {(1 - u)u^m - ku^n ,u > 0,} \\ {0, u \leqslant 0,} \\ \end{array} } \right. $$ (2) $$ u(x,0) = \left\{
D. J. Needham, J. A. Leach
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Differentiation and initial value problems

2018
Differential equations describe a wide variety of physical phenomena, however not all systems of differential equations can be solved analytically. Thus, numeric approximations fill an important void, providing us with methods to model and analyse physical systems when analytic tools fall short.
Jingbo Wang, Joshua Izaac
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Initial Value Problem for the Pair Transition Coupled Nonlinear Schrödinger Equations via the Riemann–Hilbert Method

, 2020
Wei-Qi Peng, Shou‐Fu Tian, Tian‐Tian Zhang   +2 more
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Collocation for initial value problems

BIT, 1976
A collocation method for initial value problems, parametrized byn + 1, the number of collocation points, and δ, the step size, is shown (using Kantorovich's methods) to produce errors which are uniformlyO[δ/n]p+1 for linear time varying systems of ordinary differential equations whose solutions arepth order continuous. Using Wright's method, the single
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