1. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, USA IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa Weiming Yu & Diego A. Murio

2. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS IPES 2003 Inverse 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. The aim of the model is the quantification of the interaction within and between species and their environment

3. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS 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 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa

4. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS Inverse Problem: Given values of u at times T1 and T2, identify the coefficients a(x) and b(x) from the PDE problem IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa

5. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS Solution: IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa

6. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS Estimates: IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa

7. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS Example: IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa

8. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS Graph of the solution function u(x,t) IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa

9. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa Relative L2 Error Norms in [0.3, 0.7]  x =  t = 1/128 Measurement Times: T1 = 0.2 and T2 = 0.8 epsilona(x)b(x) 0.0017 x 10(- 5)6 x 10(- 6) 0.0053 x 10(- 4)2 x 10(- 5) 0.0102 x 10(- 2)3 x 10(- 5)

10. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa a(x), epsilon = 0.010, 0.005 and 0.001

11. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa b(x), epsilon = 0.010, 0.005 and 0.001

12. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa Extension to higher dimensions Extension to systems Relation to IHCP Future Work:

13. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa 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). Brief References

14. COEFFICIENT IDENTIFICATION IN REACTION- DIFFUSION EQUATIONS REACTION- DIFFUSION EQUATIONS IPES 2003 Inverse Problems in Engineering Symposium The University of Alabama, Tuscaloosa THANK YOU !