ALA 20210 On the operational solution of the system of fractional differential equations

1. ALA 20210On the operational solution of the system of fractional differential equations Đurđica Takači Department of Mathematics and Informatics Faculty of Science, University of Novi Sad Novi Sad, Serbia djtak@dmi.uns.ac.rs

2. The Mikusinski operator field The set of continuous functions with supports in with the usual addition and the multiplication given by the convolution is a commutative ring without unit element. By the Titchmarsh theorem, it has no divisors of zero; its quotient field is called the Mikusinski operator field

3. The Mikusinski operator field The elements of the Mikusinski operator field are convolution quotients of continuous functions

4. The Mikusinski operator field The Wright function The character of the operational function

5. The matrices with operators , square matrix, is a given vector, is the unknown vector

6. Example

7. The matrices with operators , square matrix, is a given vector, is the unknown vector

8. The matrices with operators • The exact solution of • The approximate solution

9. Fractional calculus The origins of the fractional calculus go back to the end of the 17th century, when L'Hospital asked in a letter to Leibniz about the sense of the notation the derivative of order Leibniz replied: “An apparent paradox, from which one day useful consequences will be drawn"

10. Fractional calculus The Riemann-Liouville fractional integral operator of order Fractional derivative in Caputo sense

11. Fractional calculus Basic properties of integral operators

12. Fractional calculus Relations between fractional integral and differential operators

13. Relations between the Mikusiński and the fractional calculus

14. On the character of solutions of the time-fractional diffusion equation to appear in Nonlinear Analysis Series A: Theory, Methods & Applications Djurdjica Takači, Arpad Takači,Mirjana Štrboja

15. The time-fractional diffusion equation

16. The time-fractional diffusion equation with the conditions

17. The time-fractional diffusion equation

18. The time-fractional diffusion equation In the field of Mikusinski operators the time-fractional diffusion equation has the form

19. The time-fractional diffusion equation • The solution is • The character of operational functions • The Wright function

20. The time-fractional diffusion equation • The exact solution

21. A numerical example • The exact solution • In the Mikusinski field

22. The solution has the form A numerical example

23. A numerical example The exact solution

24. A numerical example

25. A numerical example

26. The system of fractional differential equationsInitial value problem (IVP) Caputo fractional derivative, order

29. The initial value problem (IVP) has a unique continuous solution

30. References • Caputo, M., Linear models of dissipation whose Q is almost frequency independent- II, Geophys. J. Royal Astronom. Soc., 13, No 5 (1967), 529-539 (Reprinted in: Fract. Calc. Appl. Anal.,11, No 1 (2008), 3-14.) • Mainardi, F., Pagnini, G., The Wright functions as the solutions of time-fractional diffusion equation, Applied Math. and Comp., Vol.141, Iss.1, 20 August 2003, 51-62. • Podlubny, I., Fractional Differential Equations, Acad. Press, San Diego (1999). • Ross, B., A brief history and exposition of fundamental theory of fractional calculus, In: "Fractional Calculus and Its Applications" (Proc. 1st Internat. Conf. held in New Haven, 1974; Ed. B. Ross), Lecture Notes in Math. 457, Springer Verlag, N. York (1975), pp. 1-37.

31. Podlubny, I., Fractional Differential Equations, Acad. Press, San Diego (1999). • Ross, B., A brief history and exposition of fundamental theory of fractional calculus, In: "Fractional Calculus and Its Applications" (Proc. 1st Internat. Conf. held in New Haven, 1974; Ed. B. Ross), Lecture Notes in Math. 457, Springer-Verlag, N. York (1975), pp. 1-37.

32. Thank you for your attention!