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RESEARCH PROGRAM

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Research Units
Similar research programs:
Scientific and education field classification
International Patent Classification
  • HUMAN NECESSITIES
  • PHYSICS
    • COMPUTING; CALCULATING; COUNTING (score computers for games A63; combinations of writing applicances with computing devices B43K29/08)
      • ANALOGUE COMPUTERS (analogue optical computing devices G06E3/00)
    • MEASURING (counting G06M); TESTING
      • RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES [N: (for special applications, see the relevant subclasses, e.g. A61B, G01F, G01N, G02B; measuring dimensions or angles of objects G01B; navigation in general G01C; measuring infrasonic, sonic or ultrasonic vibrations in general G01H; measuring infra-red, visible, or ultra-violet radiation in general G01J; transducers per se, see the relevant subclasses, e.g. G01L, H01L, H04R; measuring direction or velocity of flowing fluids by reception or emission of radiowaves or other waves and based on propagation effects caused in the fluid itself G01P; measuring electric or magnetic variables in general G01R]; (detecting masses or objects by methods not involving reflection or radiation of radio, acoustic or other waves G01V; [N: time-interval measuring G04F]; aerials H01Q) [C9504]
Geographical classification
Keywords
DIFFERENTIAL EQUATIONS; PARTIAL FUNCTIONAL EQUATIONS; DELAY EQUATIONS; NUMERICAL METHODS; NUMERICAL CODES; CHARACTERISTIC ROOTS; DIFFERENCE EQUATIONS; STABILITY

Numerical methods for functional differential equations

Università degli Studi di Trieste
Abstract
The research will concern the following issues.
1) The study of efficient numerical methods for approximating the solution of evolutionary functional differential problems deriving from the mathematical modelling of real-life phenomena. Particular relevance will be given to problems with delayed terms (delay equations). Problems governed by more general functional differential equations will be considered as well.
2) The study of the discrete problems generated by the application of the methods used in the various cases, as well as of the corresponding solution techniques.
3) The development and the validation of corresponding computational codes, as well as the improvement of the software previously developed by researches in this project.

Principal Investigator
Marino ZENNARO Università degli Studi di TRIESTE
Research Objectives
The aim of the research is the derivation of efficient numerical methods for the treatment of functional differential equations of various kinds and the development of corresponding computational codes.
Concerning the differential problems which will be investigated, we mention the following list:
a) problems with discrete delayed terms (delay equations);
b) problems governed by more general functional equations.
The treatment of such problems requires the development of the following items:
1) additional insight (if necessary) about the continuous problem;
2) further analysis about existing methods and/or derivation of new ones;
3) study of the properties of the discrete problems generated by the application of the methods, under various points of view;
4) study of suitable implementation techniques for the methods themselves, in view of their use in the conscruction of computational codes.
Concerning the software development, the activity will be substantially divided in three sections:
I) the revision of existing computational codes;
II) the development of new computational codes;
III) the experimentation with the newly produced codes and their comparison with existing codes.

First Results
The expected results for this research consist in the improvement of the knowledge of numerical methods for functional differential equations, with particular attention to delay differential equations and to the corresponding implementation techniques. At the same time, we expect to upgrade, produce and test some computational codes.

Timescale
24 months
National and international background
More and more frequently, modern applied sciences use mathematical models of the phenomena to be studied. The reasons for this are various, among which, for example, the need to simulate quantitatively the evolution of the phenomenon itself. To this purpose, it must be stressed that the mathematical simulation of the phenomenon often is much cheaper than constructing corresponding experimental trials, that are sometimes even impossible. When one is interested in the time-space evolution of a phenomenon, the corresponding mathematical model is made up of evolutionary equations. Since such equations are often very entangled, their solution in closed form is, in practice, never available. Therefore, it is customary to resort to suitable numerical methods of approximation. These methods are available to scientists as computational codes and/or software packages. In this respect, it must be stressed that a computational code must not be considered as a static product: as matter of facts, the problems to be solved, and/or their size, change with the time, so that computational codes that were once adequate, may be no more appropriate at the present or in the near future. Thus it is of paramount importance that the "mathematics - computer science technology" at the root of a modern computational code evolves in time, in order to meet the needs of the applications. This evolution consists in both the improvement of the existing codes, making them adequate to the new demands, and the >>>