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Linear Differential Equation Solution

#3327


Development of Computer Methods and Algorithms for Analytical and Numerical Solution of the Higher Order Linear Differential Equations and Systems

Tech Area / Field

  • OBS-OTH/Other/Other Basic Sciences
  • INF-SOF/Software/Information and Communications

Status
3 Approved without Funding

Registration date
22.07.2005

Leading Institute
Joint Stock Company «Konstruktorskoe Biuro Elektropribor», Russia, Saratov reg., Saratov

Collaborators

  • Indiana University Purdue University Indianapolis / Purdue School of Engineering Technology, USA, IN, Indianapolis\nGeorgetown University, USA, DC, Washington

Project summary

State of the area of study

In the mathematical theory of differential equations, the part which is considered the best explored is concerning to the ordinary linear differential equations (ODE) with the constant complex coefficients.

The various methods of analytic and numerical solution of ODE allow to solve equations of medium order (to the 30th) with various degrees of approximation. With the growth of the equation order such situations arise when computer aided tasks and algorithms become non-efficient (programs running time and accuracy, used resources, forms of the results presenting are inappropriate).

Modern Russian mathematical and practical literature does not give a complete description of possible ways of the problem statements for the ODE solution and methods of their computer aided solution. The most of the application packages (Matlab, Mathcad, Derive etc.) are aimed at the solution of specialized problems of medium complexity and are limited in their calculating abilities. The best tools of computer mathematics (Maple, Mathematica) are not available for the majority of the potential users, the most effective algorithms for solving ODE realized in them (the program code) belong to private persons, not all the relevant problems are solved, forms of presenting the results of calculation are unacceptable for analysis.

The Project Objectives:

  • Using the expertise of the group of specialists working in the sphere of development, production and usage of weapons and military technique, for the purposes of fundamental science evolution;
  • Summarizing the practical experience of the project participants in modelling linear dynamic systems in the applied theory of ordinary differential equations;
  • Developing computer methods, algorithms and programs for analytical and numerical solution of the higher order ordinary linear differential equations;
  • Publishing the results of investigation, establishing contacts with the leading scientific research institutions, working in the sphere of mathematics and practical applied modelling.

Expected Results:
  • Systematization of possible ways of problem statements for ordinary linear differential equations solution;
  • Development of methods, algorithms and programs for numerical and analytical calculation of Laplace inverse transformation of fractional complex function, not inferior and in some cases superior in calculating abilities in comparison with the best packages of computer mathematics (time of calculations, accuracy of the results, required resources and the form of presenting the results).
  • Development of methods, algorithms and programs of numerical and analytical calculation of the higher order ODEs not inferior and in some cases superior in calculating abilities in comparison with the best packages of computer mathematics (time of calculations, accuracy of the results, required resources and the form of presenting the results).
  • Development of methods, algorithms and programs for reducing initial and boundary value problems to the real form of the Cauchy problem of a minimum order.

Scientific significance of the Project Results:
  • Providing access to the new knowledge for a broad community of experts;
  • Description of methods and algorithms for computer solution of the higher order ODEs.
  • Development of new methods, algorithms and software for computer solution of the higher order ODEs.

Commercial significance of the Project Results:
  • An opportunity to create a special tool for ODEs solution, superior to all known analogues tools.

Further Perspectives of the Project:
  • Development of methods, algorithms and software for computer solutions of the higher order ordinary linear differential equations with variable coefficients;
  • Development of methods, algorithms and software for computer solution of the higher order ODE systems;
  • Development of methods, algorithms and software for computer solution of the higher order ordinary linear differential equations systems with variable coefficients;
  • Increasing the speed of calculation by using algorithms of distributed (parallel) calculations.

Competence of the Project participants.

The computational methods of ODE solution developed by the project participants were used in design of complex technical systems of the “shuttle” spacecraft “Buran”, aircrafts SU-27, SU-30, YAK-141, MIG-29, of a unique 2M velocity anti-ship missile “Moskit”, hyper-sound missile X-90, first in the world serial missile X-31 with combined ramjet.


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