Results 111 to 120 of about 665,985 (155)
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Journal of Partial Differential Equations, 2002
Summary: The Boussinesq approximation, where the viscosity depends polynomially on the shear rate, finds frequent use in geological practice. In this paper, we consider the periodic initial value problem and initial value problem for this modified Boussinesq approximation with the viscous part of the stress tensor \(\tau^v=\tau ({\mathbf e})-2 \mu_1 ...
Guo, Boling, Shang, Yadong
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Summary: The Boussinesq approximation, where the viscosity depends polynomially on the shear rate, finds frequent use in geological practice. In this paper, we consider the periodic initial value problem and initial value problem for this modified Boussinesq approximation with the viscous part of the stress tensor \(\tau^v=\tau ({\mathbf e})-2 \mu_1 ...
Guo, Boling, Shang, Yadong
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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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2016
In this chapter we derive numerical methods to solve the first-order differential equation $$\displaystyle{ \frac{dy} {dt} = f(t,y),\;\;\text{ for }\;0
Richard Khoury, Douglas Wilhelm Harder
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In this chapter we derive numerical methods to solve the first-order differential equation $$\displaystyle{ \frac{dy} {dt} = f(t,y),\;\;\text{ for }\;0
Richard Khoury, Douglas Wilhelm Harder
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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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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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2014
This chapter discusses the basic problems for solutions of initial value problems: existence and uniqueness, continuation, and dependence on parameters and initial conditions.
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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.
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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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2013
Abstract In Chapter 22, we turn to the geometric formulation of the initial value problem for the Einstein–Vlasov equations. In particular, we prove, given initial data, that there is a globally hyperbolic development. Moreover, we prove that two globally hyperbolic developments of the same data are extensions of a common globally ...
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Abstract In Chapter 22, we turn to the geometric formulation of the initial value problem for the Einstein–Vlasov equations. In particular, we prove, given initial data, that there is a globally hyperbolic development. Moreover, we prove that two globally hyperbolic developments of the same data are extensions of a common globally ...
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