An Operator-Based Scheme for the Numerical Integration of FDEs
Abstract
:1. Introduction
2. Preliminaries and Motivation
2.1. The Generalized Differential Operator Scheme for ODEs
2.1.1. The Construction of Analytic Solutions to ODEs in the Series Form
2.1.2. The Construction of Closed-Form Solutions to ODEs
2.1.3. Truncated Series and Shifted Centers of the Expansion
2.2. The Ordinary Riccati Equation and Its Solution
2.3. The Fractional Power Series and Caputo Differentiation
2.4. Motivation: The Fractional Riccati Equation
2.5. Transformation of the FDE into the Characteristic ODE
3. The Development of the Numerical FDE Integration Scheme
3.1. Adaptive Step-Size Selection Strategy for the FDE Integration Scheme
- Step 1. Let . The absolute differences are computed for and , where L is the upper bound of x. The contour plot depicting various levels of is presented in Figure 1a. It can be observed that for a fixed value of N the value of increases as x increases. New initial values , for the next approximation are computed as follows:
- Step . Analogous computations are performed for steps . Firstly, differences are evaluated for and and then new initial values , are computed:
3.2. The Implementation of the Numerical FDE Integration Scheme
- Transform the FDE (40)–(41) into the characteristic ODE using the procedure described in Section 2.5:
- Obtain analytic expressions of coefficients in the series solution (35) to the ODE (42)–(43) (see Section 2.1.1).
- Fix the values of the following parameters: the order of the approximation N, the upper bound of the independent variable L, the upper bound for the step-size , the upper bound for the change in the numerical solution . Note that the recommended values for the parameters and are derived from the study presented in the previous section (Figure 5). The value corresponds to the highest value of h on the regression line, while the value corresponds to the highest value of on the regression line.
- Repeat the following steps until the upper bound L is reached ():
- Evaluate coefficients .
- Find the lowest value of x at which at least one of the following conditions is violated:
- Assign new initial values:
4. The Application of the Proposed Numerical FDE Integration Scheme
5. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Appendix A. Analytical Expressions of the Coefficients pj (c,s) for the ODE (30)–(31)
Appendix B. Transformation of FDE (49) into the Characteristic ODE (51)
Appendix C. Analytical Expressions of the Coefficients pj(c,s) for the ODE (51)–(52)
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Timofejeva, I.; Navickas, Z.; Telksnys, T.; Marcinkevicius, R.; Ragulskis, M. An Operator-Based Scheme for the Numerical Integration of FDEs. Mathematics 2021, 9, 1372. https://doi.org/10.3390/math9121372
Timofejeva I, Navickas Z, Telksnys T, Marcinkevicius R, Ragulskis M. An Operator-Based Scheme for the Numerical Integration of FDEs. Mathematics. 2021; 9(12):1372. https://doi.org/10.3390/math9121372
Chicago/Turabian StyleTimofejeva, Inga, Zenonas Navickas, Tadas Telksnys, Romas Marcinkevicius, and Minvydas Ragulskis. 2021. "An Operator-Based Scheme for the Numerical Integration of FDEs" Mathematics 9, no. 12: 1372. https://doi.org/10.3390/math9121372
APA StyleTimofejeva, I., Navickas, Z., Telksnys, T., Marcinkevicius, R., & Ragulskis, M. (2021). An Operator-Based Scheme for the Numerical Integration of FDEs. Mathematics, 9(12), 1372. https://doi.org/10.3390/math9121372