On an operator method for constructing solutions to integral and differential equations with a singularity
On an operator method for constructing solutions to integral and differential equations with a singularity
Anahtar Kelimeler:
fractional order integral, fractional derivative, integral equation, differential equation, normalized system, operator method, explicit solution, Mittag-Leffler type function.Özet
In this paper, we consider new classes of fractional integral and derivative. These operators are generalizations of well-known integrals and the Riemann-Liouville derivative and the Caputo derivative. The article considers an operator method for solving integral and differential equations of fractional order. This method is based on the construction of normalized systems with respect to integral and differential operators. The algorithm for constructing normalized systems is given in the form of four steps. This method is first used to construct solutions to linear integral equations with constant coefficients. A theorem on the existence and uniqueness of a solution of the considered integral equation is proved. Solutions are defined explicitly and their representability in terms of multivariant functions of the Mittag-Leffler type is shown. These solutions are defined explicitly and they are represented in terms of multivariant functions of the Mittag-Leffler type.
Referanslar
Kochubei A., Luchko Y. Handbook of Fractional Calculus with Applications. Volume 2:
Fractional Differential Equations. Edited by J. A. Tenreiro Machado. Berlin, Boston: De
Gruyter, 2019. –527 p.
Kochubei A., Luchko Y. Handbook of Fractional Calculus with Applications. Volume 4:
Applications in Physics, Part A. Edited by V. E. Tarasov. Berlin, Boston: De Gruyter,
– 314 p.
Қ.А. Ясауи атындағы Халықаралық қазақ-түрік университетінің хабарлары
(математика, физика, информатика сериясы), №3 (26), 2023
Kochubei A., Luchko Y. Handbook of Fractional Calculus with Applications. Volume 7: Applications in Engineering, Life and Social Sciences, Part A. Edited by D. Baeanu, A. M. Lopes. Berlin, Boston: De Gruyter, 2019. – 259 p. 4. Pskhu A.V. Initial-value problem for a linear ordinary differential equation of noninteger order// Sb. Math., – 2011. – – Vol.202, No. 4. – P.571–582. 5. Kilbas A. A. Новые направления в теории дробных интегральных и дифференциальных уравнений// Учён. зап. Казан. гос. ун-та. Сер. Физ.-матем. науки. – 2005. – T. 147. – C.72–106. 6. Мажгихова М.Г.Обобщенная задача Дирихле для обыкновенного дифференциального уравнения с запаздывающим аргументом с производной Джрбашяна – Нерсесяна//Доклады АМАН. – 2022. – Vol. 22, No.4. – P.11 – 17. 7. Al-Refai M., Luchko Y. The General Fractional Integrals and Derivatives on a Finite Interval. Mathematics. – 2023. – Vol.11, No.1031. – P.1 – 13. 8. Tarasov V.E. Scale-Invariant General Fractional Calculus: Mellin Convolution Operators// Fractal and Fractional. – 2023. – Vol.7, No.481. – P.1 – 25. 9. Saleh M. H., Mohamed D.Sh., Ahmed M.H., Marjan M.K. System of Linear Fractional Integro-Differential Equations by using Adomian Decomposition Method//International Journal of Computer Applications. – 2015. – Vol.121, No.24. – P.1 – 11. 10. Бондаренко Б. А. Операторные алгоритмы в дифференциальных уравнениях. - Ташкент : Фан, 1984. - 183 с. 11. Karachik V.V. Normalized system of functions with respect to the Laplace operator and its applications// Journal of Mathematical Analysis and Applications. – 2003. – Vol.287, No.2. – P. 577–592. 12. Karachik V.V. Method for constructing solutions of linear ordinary differential equations with constant coefficients // Computational Mathematics and Mathematical Physics. – 2012. – Vol.52. – P. 219–234. 13. Ashurov P., Cabada A., Turmetov B. Operator method for construction of solutions of linear fractional differential equations with constant coefficients// Fractional Calculus and Applied Analysis. – 2016. – Vol.19, No.1. – P. 229–252. 14. Shinaliyev K., Turmetov B., Umarov S.A fractional operator algorithm method for construction of solutions of fractional order differential equations//Fractional Calculus and Applied Analysis. – 2012. – Vol.15, No.2. – P. 267–281. 15. Turmetov B. Kh. On a method for constructing a solution of integro-differential equations of fractional order //Electronic Journal of Qualitative Theory of Differential Equations. – 2018. – No. 25. – P.1–14. 16. Turmetov B.K., Usmanov K.I., Nazarova K.Z. On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives// Fractal Fractional. – 2021. – Vol.5, No.109. – P.1 – 21. 17. Turmetov B. On Certain Operator Method for Solving Differential Equations// Filomat. – 2017. – Vol.31. – P.4275–4286. 18. Katugampola U.N. A new approach to generalized fractional derivatives//Bull. Math. Anal. Appl. – 2014. – Vol.6. -- P. 1-15. 19. Jarad F., Ugurlu E., Abdeljawad T., Baleanu D. On a new class of fractional operators // Advances in Difference Equations. – 2017. – Vol.2017, No.247. – P.1 –16. 20. Hadid S.B., Luchko Y. An operational method for solving fractional differential equations of an arbitrary real order// Panamerican Mathematical Journal. – 1996. – Vol.6. – P. 57-73. 21. Gorenflo R., Luchko Y. Operationl method for solving generalized Abel integral equation of second kind // Integral Transforms and Special Functions. – 1997. – Vol.5, No.1-2. – P.47 – 58.
REFERENCES
Kochubei A., Luchko Y. Handbook of Fractional Calculus with Applications. Volume 2: Fractional Differential Equations. Edited by J. A. Tenreiro Machado. Berlin, Boston: De Gruyter, 2019. –527 p. 2. Kochubei A., Luchko Y. Handbook of Fractional Calculus with Applications. Volume 4: Applications in Physics, Part A. Edited by V. E. Tarasov. Berlin, Boston: De Gruyter, 2019. – 314 p. 3. Kochubei A., Luchko Y. Handbook of Fractional Calculus with Applications. Volume 7: Applications in Engineering, Life and Social Sciences, Part A. Edited by D. Baeanu, A. M. Lopes. Berlin, Boston: De Gruyter, 2019. – 259 p. 4. Pskhu A.V. Initial-value problem for a linear ordinary differential equation of noninteger order// Sb. Math., – 2011. – – Vol.202, No. 4. – P.571–582. 5. Kilbas A. A. Новые направления в теории дробных интегральных и дифференциальных уравнений// Учён. зап. Казан. гос. ун-та. Сер. Физ.-матем. науки. – 2005. – T. 147. – C.72–106. 6. Мажгихова М.Г.Обобщенная задача Дирихле для обыкновенного дифференциального уравнения с запаздывающим аргументом с производной Джрбашяна – Нерсесяна//Доклады АМАН. – 2022. – Vol. 22, No.4. – P.11 – 17. 7. Al-Refai M., Luchko Y. The General Fractional Integrals and Derivatives on a Finite Interval. Mathematics. – 2023. – Vol.11, No.1031. – P.1 – 13. 8. Tarasov V.E. Scale-Invariant General Fractional Calculus: Mellin Convolution Operators// Fractal and Fractional. – 2023. – Vol.7, No.481. – P.1 – 25. 9. Saleh M. H., Mohamed D.Sh., Ahmed M.H., Marjan M.K. System of Linear Fractional Integro-Differential Equations by using Adomian Decomposition Method//International Journal of Computer Applications. – 2015. – Vol.121, No.24. – P.1 – 11. 10. Бондаренко Б. А. Операторные алгоритмы в дифференциальных уравнениях. - Ташкент : Фан, 1984. - 183 с. 11. Karachik V.V. Normalized system of functions with respect to the Laplace operator and its applications// Journal of Mathematical Analysis and Applications. – 2003. – Vol.287, No.2. – P. 577–592. 12. Karachik V.V. Method for constructing solutions of linear ordinary differential equations with constant coefficients // Computational Mathematics and Mathematical Physics. – 2012. – Vol.52. – P. 219–234. 13. Ashurov P., Cabada A., Turmetov B. Operator method for construction of solutions of linear fractional differential equations with constant coefficients// Fractional Calculus and Applied Analysis. – 2016. – Vol.19, No.1. – P. 229–252. 14. Shinaliyev K., Turmetov B., Umarov S.A fractional operator algorithm method for construction of solutions of fractional order differential equations//Fractional Calculus and Applied Analysis. – 2012. – Vol.15, No.2. – P. 267–281. 15. Turmetov B. Kh. On a method for constructing a solution of integro-differential equations of fractional order //Electronic Journal of Qualitative Theory of Differential Equations. – 2018. – No. 25. – P.1–14.
Turmetov B.K., Usmanov K.I., Nazarova K.Z. On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives// Fractal Fractional. – 2021. – Vol.5, No.109. – P.1 – 21.
Turmetov B. On Certain Operator Method for Solving Differential Equations// Filomat. – 2017. – Vol.31. – P.4275–4286.
Katugampola U.N. A new approach to generalized fractional derivatives//Bull. Math. Anal. Appl. – 2014. – Vol.6. -- P. 1-15. 19. Jarad F., Ugurlu E., Abdeljawad T., Baleanu D. On a new class of fractional operators // Advances in Difference Equations. – 2017. – Vol.2017, No.247. – P.1 –16. 20. Hadid S.B., Luchko Y. An operational method for solving fractional differential equations of an arbitrary real order// Panamerican Mathematical Journal. – 1996. – Vol.6. – P. 57-73. 21. Gorenflo R., Luchko Y. Operationl method for solving generalized Abel integral equation of second kind // Integral Transforms and Special Functions. – 1997. – Vol.5, No.1-2. – P.47 – 58.
