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## Initial-value Problem (initial-value + problem)
## Selected Abstracts## Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous walls INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010M. BabaelahiAbstract Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two-point boundary-value problem for the similarity function into an initial-value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## On the global existence and small dispersion limit for a class of complex Ginzburg,Landau equations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2009Hongjun GaoAbstract In this paper we consider a class of complex Ginzburg,Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial-value problem in d -dimensional torus ,,d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Complex-distance potential theory, wave equations, and physical wavelets MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002Gerald KaiserPotential theory in ,n is extended to ,n by analytically continuing the Euclidean distance function. The extended Newtonian potential ,(z) is generated by a (non-holomorphic) source distribution ,,(z) extending the usual point source ,(x). With Minkowski space ,n, 1 embedded in ,n+1, the Laplacian ,n+1 restricts to the wave operator ,n,1 in ,n, 1. We show that ,,(z) acts as a propagator generating solutions of the wave equation from their initial values, where the Cauchy data need not be assumed analytic. This generalizes an old result by Garabedian, who established a connection between solutions of the boundary-value problem for ,n+1 and the initial-value problem for ,n,1 provided the boundary data extends holomorphically to the initial data. We relate these results to the physical avelets introduced previously. In the context of Clifford analysis, our methods can be used to extend the Borel,Pompeiu formula from ,n+1 to ,n+1, where its riction to Minkowski space ,n, 1 provides solutions for time-dependent Maxwell and Dirac equations. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Optimal control of non-linear chemical reactors via an initial-value Hamiltonian problem OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 1 2006V. CostanzaAbstract The problem of designing strategies for optimal feedback control of non-linear processes, specially for regulation and set-point changing, is attacked in this paper. A novel procedure based on the Hamiltonian equations associated to a bilinear approximation of the dynamics and a quadratic cost is presented. The usual boundary-value situation for the coupled state,costate system is transformed into an initial-value problem through the solution of a generalized algebraic Riccati equation. This allows to integrate the Hamiltonian equations on-line, and to construct the feedback law by using the costate solution trajectory. Results are shown applied to a classical non-linear chemical reactor model, and compared against suboptimal bilinear-quadratic strategies based on power series expansions. Since state variables calculated from Hamiltonian equations may differ from the values of physical states, the proposed control strategy is suboptimal with respect to the original plant. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Accounting for an imperfect model in 4D-Var THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 621 2006Yannick Tr'emoletAbstract In most operational implementations of four-dimensional variational data assimilation (4D-Var), it is assumed that the model used in the data assimilation process is perfect or, at least, that errors in the model can be neglected when compared to other errors in the system. In this paper, we study how model error could be accounted for in 4D-Var. We present three approaches for the formulation of weak-constraint 4D-Var: estimating explicitly a model-error forcing term, estimating a representation of model bias or, estimating a four-dimensional model state as the control variable. The consequences of these approaches with respect to the implementation and the properties of 4D-Var are discussed. We show that 4D-Var with an additional model-error representation as part of the control variable is essentially an initial-value problem and that its characteristics are very similar to that of strong constraint 4D-Var. Taking the four-dimensional state as the control variable, however, leads to very different properties. In that case, weak-constraint 4D-Var can be interpreted as a coupling between successive strong-constraint assimilation cycles. A possible extension towards long-window 4D-Var and possibilities for evolutions of the data assimilation system are presented. Copyright © 2006 Royal Meteorological Society [source] ## Unified approach to KdV modulations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2001Gennady A. ElWe develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial-value problem for the zero-dispersion KdV as the steepest descent for the scalar Riemann-Hilbert problem [6] and on the method of generating differentials for the KdV-Whitham hierarchy [9]. By assuming the hyperbolicity of the zero-dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the (x, t)-plane. The resulting system effectively solves the zero-dispersion KdV with an arbitrary initial datum. © 2001 John Wiley & Sons, Inc. [source] ## A new numerical approach for solving high-order non-linear ordinary differential equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2003Songping ZhuAbstract There have been many numerical solution approaches to ordinary differential equations in the literature. However, very few are effective in solving non-linear ordinary differential equations (ODEs), particularly when they are of order higher than one. With modern symbolic calculation packages, such as Maple and Mathematica, being readily available to researchers, we shall present a new numerical method in this paper. Based on the repeated use of a symbolic calculation package and a second-order finite-difference scheme, our method is particularly suitable for solving high-order non-linear differential equations arising from initial-value problems. One important feature of our approach is that if the highest-order derivative in an ODE can be written explicitly in terms of all the other terms of lower orders, our method requires no iterations at all. On the other hand, if the highest-order derivative in an ODE cannot be written explicitly in terms of all the other lower-order terms, iterations are only required before the actual time marching begins. Copyright © 2003 John Wiley & Sons, Ltd. [source] |