Description
Obtaining conservative laws of partial differential equations based on the concepts of adjoint equation and nonlinearly self-adjoint equation as discussed in the theoretical framework is a very recent task, since it arises from a new theorem given by Ibragimov in 2007. In the literature there are not many papers that address the problem of finding nonlinearly self-adjoint equations for nonlinear evolution equations of higher order. The project needed to treat the problem of finding nonlinearly self-adjoint equations for nonlinear evolution equations of higher order and to extend it for difference equations. The objective of the project is to derive conservation laws to equations of Physical interest and Financial Mathematics The project expected to extend the adjoint equation method and to search for new applications. The project addresses these challenges by using: theory of difference equations, partial differential equations, Lie groups, conservation laws. The donor partner, Norwegian School of Economics, is skilled in the study of difference equations and the lambda-symmetries. The project promoter, University of Cádiz is expert in the study of conservative laws. Partner’s universities will obtain benefit because starting from the obtained results; they intend to enlarge the research teams in order to broaden the study of conservative laws and the application to new mathematical models.
Summary of project results
Partners have derived first integrals of ODEs (Ordinary Differential Equations) via exponential symmetries. They have considered a system which admits a nonlocal exponential symmetry. They have constructed a formal Lagrangian and the adjoint system. From the adjoint system they have derived a first integral depending on the nonlocal variable. After obtaining two first integrals of the nonlocal variable they were able to derive proper first integrals eliminating the nonlocal variable. They have applied this approach to the second order and third order Riccati equations. In order to do this study, three different programmes have been designed with free software MAXIMA. These programmes allow to determine the adjoint equation, the conditions of self-adjointness and the first integrals of the Riccati equations. The grantee prof. Gandarias from University of Cadiz made a stay at the Derpartment of Finance and Management Science of the Norwegian School of Economics, During the short stay, the grantee was working with Prof. Kozlov and had talks and seminars with other professors of the Department. Two different seminars were organized, on which both parts presented their previous works. Partners have developed a scientific paper with the theoretical results obtained and the application of such results to Riccati II and Riccati II equations. The results are not completed yet nor finished, so partners plan to perform some future exchanges in order to continue this research. They plan to apply this results to different equations as well as to go on with the research of conservation laws for partial differential equations. They also plan to perform an Erasmus mobility allowing them to further exchange results.
Summary of bilateral results
The grantee from University of Cadiz started a research cooperation with professor Kozlov at the Norwegian School of Economics. This collaboration will lead to new research lines and joint research with professors and students of his research group. Prof. Kozlov is an expert on searching for first integrals by using classical symmetries, lambda symmetries and the condition of adjoint equation proposed by Nail Ibragimov. The grantee had several meetings in which professors at University of Cadiz and professors of the Norwegian School of Economics have exchanged both research and academic issues. Other meetings were focused on sharing knowledge concerning first integrals by using nonlocal symmetries. There was also a meeting between Prof. Gandarias (University of Cadiz) with Prof. Ibragimov concerning Ibragimov’s algorithm for conservation laws of ordinary differential equations. Prof. Ibragimov is one of the most important researchers in this field, he is director of the GAMMET Group Analysis of Mathematical Models in Naturaland Engineering Sciences of the State Technical University of Aeronautics of Russia.