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Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, USA. Weiming Yu & Diego A. Murio. IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa.
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Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, USA Weiming Yu & Diego A. Murio IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
1. Lotka (1924) and Volterra (1926) established their original and famous predator-prey/competing species model 2.Theaim of the model is the quantification of the interaction within and between species and their environment IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
3. The range of applications is pretty wide: ecological systems, conduction in nerves, epidemics, carbon monoxide poisoning, diffusion systems, chemical reactions, etc. 4. Modification of the equation IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
Inverse Problem: Given values of uat times T1 and T2, identify the coefficients a(x) and b(x) from the PDE problem IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
Solution: IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
Estimates: IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
Example: IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
Graph of the solution function u(x,t) IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
a(x), epsilon = 0.010, 0.005 and 0.001 IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
b(x), epsilon = 0.010, 0.005 and 0.001 IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
Future Work: • Extension to higher dimensions • Extension to systems • Relation to IHCP IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
Brief References • Anthony W. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, Kluwer Academic Publishers, 1989. • D.A. Murio, C.E. Mejía and S. Zhan, ‘Discrete Mollification and Automatic Numerical Differentiation’, Computers Math. Applic. 35(5), 1-16, (1998). IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa
THANK YOU! IPES 2003Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa