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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 ...
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2003
It is not easy to see how a uniform or nearly uniform wave train can realistically emerge from some general initial condition or from a realistic forcing unless the initial condition or the forcing is periodic. That turns out not to be the case, and the ideas we have so far developed about group velocity and energy propagation turn out to be invaluable
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It is not easy to see how a uniform or nearly uniform wave train can realistically emerge from some general initial condition or from a realistic forcing unless the initial condition or the forcing is periodic. That turns out not to be the case, and the ideas we have so far developed about group velocity and energy propagation turn out to be invaluable
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Function and therapeutic value of astrocytes in neurological diseases
Nature Reviews Drug Discovery, 2022Hong-Gyun Lee +2 more
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