Linear and nonlinear fractional order diffusion equations with initial and bowndary conditions
Keywords:
sub-diffusion equation, maximum principle, fractional differential equation, nonlinear problem, Riemann-Liouville derivative.Abstract
The usual differentiation and integration are expanded to any non-integer order in fractional calculus. The topic predates the development of differential calculus by Leibnitz and Newton and is therefore as old as classical theory. The concept of fractional calculus has generated interest not only among mathematicians but also among physicists and engineers.
This concept is calculated in the same way as the classical methods of differential and integral calculus, and also dates back to the time when Leibniz and Newton invented differential calculus. The idea of calculating fractional order is of interest not only among individual mathematicians, but also among physicists and engineers. The method of upper and lower solutions has been extended to FDEs using these minimum-maximum principles, and various existence results have been established.
In this paper, a one-dimensional subdiffusion equation is investigated using the principle of the maximum of the Riemann-Liouville derivative of fractional order. It is proved that for fractional order diffusion equations in linear and nonlinear time there is a unique classical solution to the initial boundary value problem and that the solution continuously depends on the initial and boundary conditions.
References
REFERENCES
Luchko Y. Maximum principle for the generalized time-fractional diffusion equation // Journal of Mathematical Analysis and Applications. – 2009. – Vol. 251. – P. – 218–222.
Luchko Y. Some uniqueness and existence results for the initial boundary-value problems for the generalized time-fractional diffusion equation // Computers and Mathematics with Applications. – 2010. – Vol. 59. – P. – 1766–1772.
Luchko Y. Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation // Journal of Mathematical Analysis and Applications. – 2011. – Vol. 274. – P. – 528–548.
Luchko Y. Boundary value problems for the generalized time-fractional diffusion equation of distributed order // Fractional Calculus and Applied Analysis. – 2009. – Vol. 12, No. 4. – P. –409–422.
Al-Refai M., Luchko Y. Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives // Applied Mathematics and Computation. – 2015. – Vol. 257. – P. – 40–51.
Chan C.Y., Liu H.T. A maximum principle for fractional diffusion equations // Quarterly of Applied Mathematics. – 2016. – Vol. 74, No. 2. – P. – 421–427.
Kirane M., Torebek B. T. Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations // Fractional Calculus and Applied Analysis. -2019. – Vol. 22, No. 2. – P. – 258–278.
Luchko Y., Yamamoto M. On the maximum principle for a time-fractional diffusion equation // Fractional Calculus and Applied Analysis. – 2017. – Vol. 20, No. 5. – P. – 1121–1145.
Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. – Amsterdam, London and New York: North-Holland Mathematical Studies, Elsevier (North-Holland) Science Publishers, 2006. – 522 p.
Al-Refai M., Luchko Y. Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications // Fractional Calculus and Applied Analysis. – 2014. – Vol. 17, No.2. – P. – 483–498.
