Linear and nonlinear fractional order diffusion equations with initial and bowndary conditions

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.