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2012
In the previous chapter we derived a simple finite difference method, namely the explicit Euler method, and we indicated how this can be analysed so that we can make statements concerning its stability and order of accuracy. If Euler’s method is used with constant time step h then it is convergent with an error of order O(h) for all sufficiently smooth
Karline Soetaert +2 more
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In the previous chapter we derived a simple finite difference method, namely the explicit Euler method, and we indicated how this can be analysed so that we can make statements concerning its stability and order of accuracy. If Euler’s method is used with constant time step h then it is convergent with an error of order O(h) for all sufficiently smooth
Karline Soetaert +2 more
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2014
This chapter discusses the basic problems for solutions of initial value problems: existence and uniqueness, continuation, and dependence on parameters and initial conditions.
S.P. Venkateshan, Prasanna Swaminathan
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This chapter discusses the basic problems for solutions of initial value problems: existence and uniqueness, continuation, and dependence on parameters and initial conditions.
S.P. Venkateshan, Prasanna Swaminathan
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1997
We begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to $$ \left\{ {\begin{array}{*{20}{c}}{y' =
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We begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to $$ \left\{ {\begin{array}{*{20}{c}}{y' =
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Bounds for Initial Value Problems
Journal of Applied Mechanics, 1973A new method has been developed for finding rigorous upper and lower bounds to the solution of a wide class of initial value problems. The method is applicable to initial value problems of the following type: x(¨t)+f(t,x,x)˙=0,x(0)=X0,x(˙0)=V0, where f is continuous with continuous first derivatives, Lipschitzian, and ∂f/∂x ≥ 0.
Bell, C. A., Appl, F. C.
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Initial-Value Problems for ODE
2012The ability to reliably solve initial-value problems for ordinary differential equations is essential in order to understand the evolution of dynamical systems. In this chapter we deal with methods of advancing the given initial state of a system to later times, explaining clearly the role of stiffness, local discretization and round-off errors, and ...
Simon Širca, Martin Horvat
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Rapidly Forced Initial Value Problems
SIAM Journal on Applied Mathematics, 1993The Cauchy problem for the evolution equation of the type \(u_ t + u_{xx} = f(x,t/ \varepsilon)\), \(t>0\), \(x \in \mathbb{R}^ 1\), with smooth initial function is considered. The function \(f(x, \tau)\) is periodic in \(\tau\). An asymptotic expansion of the solution as \(\varepsilon\to 0\) is constructed in the form of the \(\varepsilon\)-power ...
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1998
Historically, the study of differential equations originated in the beginnings of calculus with Newton and Leibniz in the seventeenth century and is closely interwoven with the general development of mathematics. To a substantial degree, the central role of differential equations within mathematics is due to the fact that many important problems in ...
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Historically, the study of differential equations originated in the beginnings of calculus with Newton and Leibniz in the seventeenth century and is closely interwoven with the general development of mathematics. To a substantial degree, the central role of differential equations within mathematics is due to the fact that many important problems in ...
openaire +1 more source