Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method
Abstract
:1. Introduction
2. Definition of the Problem
3. Preliminaries
3.1. Fractional Operators
3.2. Review Some of Integrated Formulas: [23,24,25,28]
4. Proposed Method
4.1. Two-Block Method
Algorithm 1: Two-Block Method (BMP2) |
Step 1:
Step 2: for each :
For all doing steps 3–8: Step 3: For each :
Step 4: For each . For each , If then else evaluate: Step 5: For each :
Step 6: For each . For each , If then put else evaluate: Step 7: for each . Compute and for all ; , from the system: and Step 8: for each : set |
4.2. Three-Block Method
Algorithm 2: Three-Block Method (BMP3) |
Step 1:
Step 5: For each :
Step 7: For each :
Step 9: For each . Compute and for all ; , from the system: and Step 10: For each :
Step 12: For each :
Step 14: For each :
Step 16: for each . Compute and for all ; , from the system: And Step 17: for each : set ; . |
5. Implementation of the Method
6. Discussion
- The numerical experiments have shown that the three-block method (BMP3) gives better accuracy than the two-block method (BMP2);
- Where are sufficiently large number, the error to be minimizes and the results are approached to the exact solution;
- The two-block method (BMP2) is faster, and lower in accuracy, than the three-block method (BMP3).
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Ahmed, S.S.; Hamasalih, S.A. Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method. Symmetry 2023, 15, 607. https://doi.org/10.3390/sym15030607
Ahmed SS, Hamasalih SA. Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method. Symmetry. 2023; 15(3):607. https://doi.org/10.3390/sym15030607
Chicago/Turabian StyleAhmed, Shazad Shawki, and Shokhan Ahmed Hamasalih. 2023. "Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method" Symmetry 15, no. 3: 607. https://doi.org/10.3390/sym15030607
APA StyleAhmed, S. S., & Hamasalih, S. A. (2023). Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method. Symmetry, 15(3), 607. https://doi.org/10.3390/sym15030607