Abstract
The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.
Highlights
Fractional differential operators are applied successfully to model various processes with anomalous dynamics in science and engineering [1,2]
Note the monotone iterative techniques combined with lower and upper solutions are applied in the literature to solve various problems in ordinary differential equations [3], differential equations with maxima [4], difference equations with maxima [5], Caputo fractional differential equations [6], Riemann-Liouville fractional differential equations [7,8,9,10]
The main aim of the paper is to suggest a scheme for the approximate solving of the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval
Summary
Fractional differential operators are applied successfully to model various processes with anomalous dynamics in science and engineering [1,2]. Only a small number of fractional differential equations could be solved explicitly It requires the application of different approximate methods for solving nonlinear factional equations. Note the monotone iterative techniques combined with lower and upper solutions are applied in the literature to solve various problems in ordinary differential equations [3], differential equations with maxima [4], difference equations with maxima [5], Caputo fractional differential equations [6], Riemann-Liouville fractional differential equations [7,8,9,10]. We consider an initial value problem for a scalar nonlinear Riemann-Liouville fractional differential equation with a constant delay on a finite interval. Mathematics 2020, 8, 477 lower and upper solutions and monotone-iterative technique to suggest an algorithm for approximate solving of the studied problem. The suggested and well-grounded algorithm is used in an appropriate computer environment and it is applied to a particular problem to illustrate the practical usefulness
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