There is a large class of scientific and technological problems whose exact solution is very difficult or even impossible. In some cases the difficulty could be connected with certain classes of functions whose integrals cannot be expressed by the elementary functions. Similar mathematical objects can also complicate the process of solving systems of differential equations of various type, describing the initial scientific or technological problem. Our application, Functional Analyzer , may help in finding the approximate solution of the problem by replacing of some class of functions by another class, whose behavior is less complicated. The method is based on well-known results of linear algebra and functional analysis. Certainly, our simple program cannot replace the very sophisticated, complicated (and usually commercial) software for numerical mathematics and its applications. Its value is in its simplicity and flexibility, since it can help the user during the decision process, which class of functions is the most suitable for the particular problem. The user may choose and experiment with almost any finite collection of functions (the limitation is given by their integrability over the unit interval) that will be used for generating the considered function space. Although the polynomials are the typical candidates, our application is not limited on them. The user also need not care about the process of the orthogonalization of the generators of the function space, or about the question which orthogonal system is the best.Such questions are usually subsidiary at the very initial stage. Our application automatically calculates the orthogonal projection of the investigated function to the given function space and the function is then approximated by its orthogonal projection. Finally, the projection is graphically compared with the original function and its Taylor polynomial, calculated at the point x=0.
The orthogonal projection of the studied function:
The orthogonal projection of the studied function with numerical values of the coefficients:
The Taylor polynomial at x=0:
Color key: Studied function,Ortogonal projection,Taylor polynomial.
The application Functional Analyzer is written in Java powered by Wolfram webMathematica 3.0. The application is hosted at the server of the Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology. For research and scientific activities the software is available free of charge. In all other cases, please contact RNDr. M. Novák, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, phone: +420 5 4114 3135. Acknowledgement: FEKT-S-11-2/921 "Vlastnosti řešení funkcionálních diferenciálních a diferenčních rovnic"