Optimal Control of a Class of Parabolic Partial Fractional Differential Equations
American Journal of Theoretical and Applied Statistics
Volume 6, Issue 5-1, September 2017, Pages: 66-70
Received: Jul. 28, 2017;
Accepted: Jul. 31, 2017;
Published: Aug. 9, 2017
Views 1805 Downloads 161
Mahmoud M. El-borai, Faculty of Science, Alexandria University, Alexandria, Egypt
Mohamed A. Abdou, Faculty of Education, Alexandria University, Alexandria, Egypt
Mai Taha Elsayed, Faculty of Education, Alexandria University, Alexandria, Egypt
In this paper, the existence of the solution of the parabolic partial fractional differential equation is studied and the solution bound estimate which are used to prove the existence of the solution of the optimal control problem in a Banach space is also studied, then apply the classical control theory to parabolic partial differential equation in a bounded domain with boundary problem. An expansion formula for fractional derivative, optimal conditions and a new solution scheme is proposed.
Mahmoud M. El-borai,
Mohamed A. Abdou,
Mai Taha Elsayed,
Optimal Control of a Class of Parabolic Partial Fractional Differential Equations, American Journal of Theoretical and Applied Statistics. Special Issue: Statistical Distributions and Modeling in Applied Mathematics.
Vol. 6, No. 5-1,
2017, pp. 66-70.
A. Debbouche and M. M. El-Borai, "Weak almost periodic and optimal mild solutions of fractional evolution equations". Electronic Journal of Differential Equations, vol. 2009, pp. 1-8, 2009.
M. M. El-Borai, "Some probability densities and fundamental solutions of fractional evolution equations". Chaos, Solitons & Fractals, vol. 14, pp. 433-440, 2002.
M. M. El-Borai, K. E. S. El-Nadi, and E. G. El-Akabawy, "On some fractional evolution equations". Computers and mathematics with applications, vol. 59, pp. 1352-1355, 2010.
O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynam. 38 (2004), no. 1-4, 323-337.
O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems. J. Vib. Control 14 (2008), no. 9-10, 1291-1299.
O. P. Agrawal, O. Defterli and D. Baleanu, Dumitru Fractional optimal control problems with several state and control variables. J. Vib. Control 16 (2010), no. 13, 1967-1976.
G. S. F. Frederico and D. F. M. Torres, Fractional optimal control in the sense of Coputo and the fractional Noether’s theorem. Int. Math. Forum 3 (2008), no. 9-12, 479-493.
G. S. F. Frederico and D. F. M. Torres. Fractional conservation laws in optimal control theory, Nonlinear Dynam. 53 (2008), no. 3, 215-222.
C. Tricaud and Y. Chen. An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. 59 (2010), no. 5, 1644-1655.
C. Tricaud and Y. Chen. Time Optimal Control of Systems with Fractional Dynamics, Int. J. Differ. Equ. Appl., Volume 2010 (2010). Article ID 461048, 16 pages.
M. M. El-Borai, W. G. Elsayed and R. M. Al-Masroub, Exact Solutions for Some Nonlinear Fractional Parabolic Equations, Inter. J. Adv. Eng. Res. (IJAER), vol. 10, No. III, Sep. 2015, 106-122.
M. M. El-Borai, W. G. Elsayed and F. N. Ghaffoori, On the Cauchy Problem for Some Parabolic Fractional Partial Differential Equations with Time Delays, J. Math. & System Scie. 6(2016), 194-199.
M. M. El-Borai, W. G. Elsayed and R. M. Al-Masroub, Exact Solutions for Some Nonlinear Partial Differential Equations via Extended (G'/G) – Expansion Method, Inter. J. Math. Trends and Tech. (IJMTT) – Vol. 36, No. 1-Aug 2016, 60-71.
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, North-Holland Mathematics Studies, 2006, 204.
M. M. El-Borai, W. G. Elsayed and M. Taha, On The Fractional Optimal Control Problem wit free End Point, American Journal of Theoretical and Applied Statistics, ISSN: 2326-8999; (2017).
Zoran D, Nebosa Petrovacki, Optimality condition and a solution scheme for fractional optimal control problems, Struct. Multidisc Optim. 10.1007/s00158-008-0307-7, (2009)
Atanackovic TM, Stankovic B (2004) An expansion formula for fractional derivatives and its applications. Fractional Calculus and Applied Analysis 7(3): 365–378.
Atanackovic TM, Stankovic B (2007a) On a class of differential equations with left and right fractional derivatives. ZAMM, Z Angew Math Mech 87: 537–546.
Atanackovic TM, Stankovic B (2007b) On a differential equation with left and right fractional derivatives. Fractional Calculus Applied Analysis 10: 138–150.
Mahmoud M. El-borai, M. A. Abdou, E. M. Youssef, On some Approximate analytical solution for mathematical model of carcinogenesis using Adomian decomposition method.
Dummit, D. S., Statistics and probability, Prentice Hall, John Wiley & Sons, Hoboken, NJ.
Knapp, W.,' Advanced Real Analysis, Birkhauser', Boston, 2005.
Yuan L, Agrawal OP (2002) A numerical scheme for dynamic systems containing fractional derivatives. J Vib Acoust 124: 321–324.
M. El-Borai, K. El-Nadi, O. Mostafa, and H. Ahmed, "Numerical methods for some nonlinear stochastic differential equations," JOURNAL-KOREA SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS, vol. 9, p. 79, 2005.
M. M. El-Borai, "The fundamental solutions for fractional evolution equations of parabolic type," International Journal of Stochastic Analysis, vol. 2004, pp. 197-211, 2004.
M. M. El-Borai, "On some fractional differential equations in the Hilbert space," Discrete and Continuous Dynamical Systems. Series A, pp. 233-240, 2005.
M. M. El-Borai, A. El-Banna, and W. H. Ahmed, "Optimal Control of a Class of Parabolic Partial Differential Equations," International Journal of Advanced Computing, vol. 36, 2013.
M. M. El-Borai, A.-Z. H. El-Banna, and W. H. Ahmed, "On Some Fractional-Integro Partial Differential Equations," International Journal of Basic & Applied Sciences, vol. 13, 2013.
Gelfand, I. M.; Fomin, S. V. Silverman, Richard A., Calculus of variations. Mineola, New York: p. 3. ISBN 978-0486414485. (2000).
Knapp, A. W., Advanced Real Analysis, Birkhauser, Boston, 2005.