Volume 13, no. 4Pages 60 - 67
Numerical Analysis of Fractional Order Integral Dynamical Models with Piecewise Continuous KernelsA. Tynda, D. Sidorov, I. Muftahov
Volterra integral equations find their application in many areas, including mathematical physics, control theory, mechanics, electrical engineering, and in various industries. In particular, dynamic Volterra models with discontinuous kernels are effectively used in power engineering to determine the operating modes of energy storage devices, as well as to solve the problem of load balancing. This article proposes the numerical scheme for solution of the fractional order linear Volterra integral equations of the first kind with piecewise continuous kernels. The developed approach is based on a polynomial collocation method and effectively approximate such a weakly singular integrals. The efficiency of proposed numerical scheme is illustrated by two examples.Full text
- Volterra integral equations; numerical method; convergence; discontinuous kernel; singularity; fractional integral.
- 1. Linz P. Analytical and Numerical Methods for Volterra Equations. Philadelphia, SIAM, 1985. DOI: 10.1137/1.9781611970852
2. Sidorov D. Volterra Equations of the First Kind with Discontinuous Kernels in the Theory of Evolving Systems Control. Studia Informatica Universalis, 2011, vol. 9, no. 3, pp. 135-146, arXiv: arxiv.org/abs/1111.5903v1.
3. Sidorov D. Integral Dynamical Models: Singularities, Signals and Control. Singapore, London, World Scientific, 2015. DOI: 10.1142/9278
4. Atangana A., Bildik N. The Numerical Solution of the Volterra Fractional Integral Equations of the Second Kind. Mathematical Problems in Engineering, 2013, article ID: 981526, 11 p. DOI: 10.1155/2013/981526
5. Baleanu D., Diethelm K., Scalas E., Trujillo J.J. Fractional Calculus Models and Numerical Methods. Singapore, World Scientific Publishing, 2016. DOI: 10.1142/10044
6. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications. Yveron, Gordon and Breach, 1993.
7. Diethelm K., Ford N.J. Analysis of Fractional Differential Equations. Journal of Mathematical Analysis and Applications, 2002, vol. 265, pp. 229-248. DOI: 10.1006/jmaa.2000.7194
8. Diethelm K., Freed A.D. The Frac PECE Subroutine for the Numerical Solution of Differential Equations of Fractional Order. Forschung und wissenschaftliches rechnen, 1998, pp. 57-71.
9. Diethelm K. An Algorithm for the Numerical Solution of Differential Equations of Fractional Order. Electronic Transactions on Numerical Analysis, 1997, vol. 5, pp. 1-6.
10. Sidorov D.N. On Parametric Families of Solutions of Volterra Integral Equations of the First Kind with Piecewise Smooth Kernel. Differential Equations, 2013, vol. 49, no 2, pp. 210-216. DOI: 10.1134/S0012266113020079
11. Sidorov D.N. Solvability of Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels.Russian Mathematics, 2013, vol. 57, pp. 54-63. DOI: 10.3103/S1066369X13010064
12. Sidorov N.A., Sidorov D.N. On the Solvability of a Class of Volterra Operator Equations of the First Kind with Piecewise Continuous Kernels. Mathematical Notes, 2014, vol. 96, pp. 811-826. DOI: 10.4213/mzm10220
13. Muftahov I., Tynda A., Sidorov D. Numeric Solution of Volterra Integral Equations of the First Kind with Discontinuous Kernels. Journal of Computational and Applied Mathematics, 2017, vol. 313, pp. 119-128. DOI: 10.1016/j.cam.2016.09.003
14. Sidorov D., Tynda A., Muftahov I., Dreglea A., Liu F. Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation. Mathematics, 2020, vol. 8, no. 8, pp. 1257. DOI: 10.3390/math8081257
15. Sidorov D., Muftahov I., Tomin, N., Karamov D., Panasetsky D., Dreglea A., Liu F. A Dynamic Analysis of Energy Storage with Renewable and Diesel Generation Using Volterra Equations. IEEE Transactions on Industrial Informatics, 2020, vol. 16, no. 5, article ID: 8784402, pp. 3451-3459. DOI: 10.1109/TII.2019.2932453
16. Sidorov D., Panasetsky D., Tomin N., Karamov D., Zhukov A., Muftahov I., Dreglea A., Liu F., Li Y. Toward Zero-Emission Hybrid AC/DC Power Systems with Renewable Energy Sources and Storages: A Case Study from Lake Baikal Region. Energies, 2020, vol. 13, no. 5, pp. 1226. DOI: 10.3390/en13051226
17. Sidorov N., Sidorov D., Sinitsyn A. Toward General Theory of Differential-Operator and Kinetic Models. Singapore, London, World Scientific, 2020. DOI: 10.1142/11651
18. Sizikov V., Sidorov D. Generalized Quadrature for Solving Singular Integral Equations of Abel Type in Application to Infrared Tomography. Applied Numerical Mathematics, 2016, vol. 106, pp. 69-78. DOI: 10.1016/j.apnum.2016.03.004
19. Brunner H., Pedas A., Vainikko G. The Piecewise Polynomial Collocation Method for Nonlinear Weakly Singular Volterra Equation. Mathematics of Computation, 1999, vol. 68, no. 227, pp. 1079-1095. DOI: 10.1090/S0025-5718-99-01073-X