Functional differential equation

L. H. Erbe et ai. Oscillation Theory for Functional Differential Equations (1995)... 

By functional differentiation, Equation 4.22 leads us to the Euler-Lagrange deterministic equation for the electron density, viz.,... 

Curtis, W. K., R. O. Fox, and K. Halasi (1992). Numerical stability analysis of a class of functional differential equations. SIAM Journal of Applied Mathematics 52,... 

What is of interest are the time rate of change of carbon in the landscape and its forcing functions. Differentiating equation... 

In the case of a maximum mixedness reactor, one works best with the life expectancy b. The life expectancy distribution in the feed stream,/(f>), is exactly the same as the residence time distribution in the product stream,/(/). One can generalize the Zwietering (1959) equation for a maximum mixedness reactor to the case of continuous mixtures (Astarita and Ocone, 1990) to obtain the following functional differential equation for cix,b) ... 

Three types of new directions are discussed. In two of these, ordinary differential equations are not an adequate model to describe the phenomenon of interest functional differential equations and partial differential equations provide the appropriate setting. In the remaining case ordinary differential equations are appropriate but the modeling is not complete. Improving the model would result in a larger system for which the techniques of monotone dynamical systems are inappropriate. The problems will be described and results indicated, but no proofs are given. In all cases, much more work needs to be done before the problem is appropriately modeled and analyzed. 

Delay models were discussed in Chapter 10. We repeat here that the most interesting problem is a modeling one. Since the problem is sensitive to how the delay is introduced, care must be taken in the modeling. A physical delay is caused by the physiology of the cell, so model equations must be modified to consider or approximate the cell physiology. Once a model is known, analysis of the corresponding system of equations (either functional differential equations or hyperbolic partial differential equations of a structured model) would be an important contribution. It is likely, however, that the delay will be state-dependent, and the theory for such equations is not well developed. A model with delays due to both cell physiology and diffusion in an unstirred chemostat would also be of interest. 

HI] J. K. Hale (1977), Theory of Functional Differential Equations. New York Springer. 

Chapters 10 and 11 are intended to steer the interested reader into open problems in the subject. Chapter 10 deals with three specific types of problems where preliminary analysis of the model has been carried out but many open questions remain. It is felt that these new directions are worthy of serious study. Some of the problems are modeling problems and some are mathematical ones. In two cases the models are not ordinary differential equations, but rather functional differential equations and partial differential equations. In Chapter 11 specific technical open questions are... 

V.A.Rvachev Compactly supported solutions of functional differential equations and their applications. Russian Math Surveys 45,1 pp87-120... 

Simplifying Eq. (29) yields a functional differential equation for the ground-state density,... 

A rather large class of functional differential equations and systems... 

Because the Laplace transform of a time derivative (of any order) is an algebraic function, differential equations involving time are converted to algebraic expressions by a Laplace transformation. Thus, differential equations involving time can often be solved in the transform domain by using usual algebraic techniques provided that the result can be inverted back to the time domain. 

The design of data flow diagrams has been discussed in chapter two. They consist of functions represented by a circle, connected by arrows representing the data that is transported. The sign next to the arrow indicates whether the function has a positive or negative contribution to another function. Differential equations can be recognized by a square, indicating a status buffer since such an equation has a memoiy. 

Another excursion to chain dynamics provides this final model. The dynamics of a single chain is described by the Rouse equation, and we use the Rouse theory in its continuous version where the chain variables can be brought in indirectly. The Rouse equation for the probability is given by the functional differential equation... 

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