


Paper's Title:
Numerical Solution of Certain Types of FredholmVolterra IntegroFractional Differential Equations via Bernstein Polynomials
Author(s):
Alias B. Khalaf^{1}, Azhaar H. Sallo^{2} and Shazad S. Ahmed^{3}
^{1}Department
of Mathematics, College of Science,
University of Duhok,
Kurdistan Region,
Iraq.
Email: aliasbkhalaf@uod.ac
^{2}Department
of Mathematics, College of Science,
University of Duhok,
Kurdistan Region,
Iraq.
Email: azhaarsallo@uod.ac
^{3}Department
of Mathematics, College of Science,
University of Sulaimani,
Kurdistan Region,
Iraq.
Email: shazad.ahmed@univsul.edu
Abstract:
In this article we obtain a numerical solution for a certain fractional order integrodifferential equations of FredholmVolterra type, where the fractional derivative is defined in Caputo sense. The properties of Bernstein polynomials are applied in order to convert the fractional order integrodifferential equations to the solution of algebraic equations. Some numerical examples are investigated to illustrate the method. Moreover, the results obtained by this method are compared with the exact solution and with the results of some existing methods as well.
Paper's Title:
Viability
Theory And Differential Lanchester Type Models For Combat.
Differential Systems.
Author(s):
G. Isac and A. Gosselin
Department Of
Mathematics, Royal Military College Of Canada,
P.O. Box 17000, Stn Forces, Kingston,
Ontario, Canada K7k 7b4
isacg@rmc.ca
gosselina@rmc.ca
URL:
http://www.rmc.ca/academic/math_cs/isac/index_e.html
URL:
http://www.rmc.ca/academic/math_cs/gosselin/index_e.html
Abstract:
In 1914, F.W. Lanchester proposed several mathematical models based on differential equations to describe combat situations [34]. Since then, his work has been extensively modified to represent a variety of competitions including entire wars. Differential Lanchester type models have been studied from many angles by many authors in hundreds of papers and reports. Lanchester type models are used in the planning of optimal strategies, supply and tactics. In this paper, we will show how these models can be studied from a viability theory stand point. We will introduce the notion of winning cone and show that it is a viable cone for these models. In the last part of our paper we will use the viability theory of differential equations to study Lanchester type models from the optimal theory point of view.
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