Results 221 to 230 of about 12,954,008 (270)
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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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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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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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2017
This chapter is devoted to initial values problems for ordinary differential equations. It discusses theory for existence, uniqueness and continuous dependence on the data of the problem. Special techniques for linear ordinary differential equations with constant coefficients are discussed in terms of matrix exponentials and their approximations. Next,
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This chapter is devoted to initial values problems for ordinary differential equations. It discusses theory for existence, uniqueness and continuous dependence on the data of the problem. Special techniques for linear ordinary differential equations with constant coefficients are discussed in terms of matrix exponentials and their approximations. Next,
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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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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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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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