Quadrature formula for singular integral computation of special type
Keywords:
Fractional integral, Riemann-Liouville integral, fractional integration, fractional differentiation, quadrature formula, fractional calculus, singular integral.
Abstract
In this paper we develop the quadrature formula for the singular integral with the Cauchy kernel from the fractional Riemann-Liouville integral. The derivation is based on the quadrature formula obtained previously to calculate the Riemann-Liouville fractional integral. During the development of the quadrature formula to calculate a singular integral with the Cauchy kernel, we use the formulas for the exact calculation of Cauchy type integral principal value. The formula for the remainder of the quadrature formula is derived. The error estimation was performed. A special case of the developed quadrature formula was considered. The calculations were performed in Wolfram Mathematica system.
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References
Agarwal, P; Nieto, JJ; Luo, MJ. (2017). Extended Riemann-Liouville type fractional derivative operator with applications// OPEN MATHEMATICS. 15, 1667-1681
Galimyanov, A. F.; Gilemzyanov, A. F.; Minnegalieva, C. B. (2017). SQUARE FORMULAS FOR WEIL FRACTIONAL INTEGRAL BASED ON TRIGONOMETRIC POLYNOM // JOURNAL OF FUNDAMENTAL AND APPLIED SCIENCES. 9, 1934 – 1944
Golovchun A., Karimova B., Zhunissova M., Ospankulova G., Mukhamadi K. (2017). Content And Language Integrated Learning In Terms Of Multilingualism: Kazakhstani Experience, Astra Salvensis, Supplement No. 10, p. 297-306.
Gorskaya T.Yu., Galimyanov A.F. (2014). Generalized Bubnov-Galerkin method for the equations with a fractional-integral operator / Bulletin from KGASU, №4 (30). Kazan, pp. 341- 345.
Gorskaya T.Yu., Galimyanov A.F. (2017). Approximation of fractional integrals by partial sums of Fourier series / KGASU Bulletin, №3 (41). Kazan, pp. 261-265.
Grinko A.P., A.A. Kilbas. (2007). Integral equation of Abel type and local fractional integrals and derivatives, BSU Bulletin. SERIES 1, Physics. Mathematics. Informatics, Minsk, 1, 71- 77.
Labora, DC; Rodriguez-Lopez, R. (2017). FROM FRACTIONAL ORDER EQUATIONS TO INTEGER ORDER EQUATIONS// FRACTIONAL CALCULUS AND APPLIED ANALYSIS. 20(6), 1405-1423
Li, Y; Xiao, W. (2017). FRACTAL DIMENSION OF RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OF CERTAIN UNBOUNDED VARIATIONAL CONTINUOUS FUNCTION // FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY. 25(5), 1750047
Micula, S. (2018). An iterative numerical method for fractional integral equations of the second kind // JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS. 339, 124-133
Parovik R.I. (2017). Fractional calculus in the theory of oscillatory systems // Modern science-intensive technologies. 1, pp. 61-68.
Pshu A.V. (2017). The solution of the multidimensional integral Abel equation of the second kind with partial fractional integrals // Differential equations. 53 (9), 1195
Villalobos Antúnez, J.V., Popper K.R. (2017). Heráclito y la invención del logos. Un contexto para la Filosofía de las Ciencias Sociales, Opción, vol. 33, núm. 84, diciembre, pp. 4-11.
Zhu, SG; Fan, ZB; Li, G. (2018). APPROXIMATE CONTROLLABILITY OF RIEMANN-LIOUVILLE FRACTIONAL EVOLUTION EQUATIONS WITH INTEGRAL CONTRACTOR ASSUMPTION // JOURNAL OF APPLIED ANALYSIS AND COMPUTATION. 8 (2), 532-548.
Galimyanov, A. F.; Gilemzyanov, A. F.; Minnegalieva, C. B. (2017). SQUARE FORMULAS FOR WEIL FRACTIONAL INTEGRAL BASED ON TRIGONOMETRIC POLYNOM // JOURNAL OF FUNDAMENTAL AND APPLIED SCIENCES. 9, 1934 – 1944
Golovchun A., Karimova B., Zhunissova M., Ospankulova G., Mukhamadi K. (2017). Content And Language Integrated Learning In Terms Of Multilingualism: Kazakhstani Experience, Astra Salvensis, Supplement No. 10, p. 297-306.
Gorskaya T.Yu., Galimyanov A.F. (2014). Generalized Bubnov-Galerkin method for the equations with a fractional-integral operator / Bulletin from KGASU, №4 (30). Kazan, pp. 341- 345.
Gorskaya T.Yu., Galimyanov A.F. (2017). Approximation of fractional integrals by partial sums of Fourier series / KGASU Bulletin, №3 (41). Kazan, pp. 261-265.
Grinko A.P., A.A. Kilbas. (2007). Integral equation of Abel type and local fractional integrals and derivatives, BSU Bulletin. SERIES 1, Physics. Mathematics. Informatics, Minsk, 1, 71- 77.
Labora, DC; Rodriguez-Lopez, R. (2017). FROM FRACTIONAL ORDER EQUATIONS TO INTEGER ORDER EQUATIONS// FRACTIONAL CALCULUS AND APPLIED ANALYSIS. 20(6), 1405-1423
Li, Y; Xiao, W. (2017). FRACTAL DIMENSION OF RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OF CERTAIN UNBOUNDED VARIATIONAL CONTINUOUS FUNCTION // FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY. 25(5), 1750047
Micula, S. (2018). An iterative numerical method for fractional integral equations of the second kind // JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS. 339, 124-133
Parovik R.I. (2017). Fractional calculus in the theory of oscillatory systems // Modern science-intensive technologies. 1, pp. 61-68.
Pshu A.V. (2017). The solution of the multidimensional integral Abel equation of the second kind with partial fractional integrals // Differential equations. 53 (9), 1195
Villalobos Antúnez, J.V., Popper K.R. (2017). Heráclito y la invención del logos. Un contexto para la Filosofía de las Ciencias Sociales, Opción, vol. 33, núm. 84, diciembre, pp. 4-11.
Zhu, SG; Fan, ZB; Li, G. (2018). APPROXIMATE CONTROLLABILITY OF RIEMANN-LIOUVILLE FRACTIONAL EVOLUTION EQUATIONS WITH INTEGRAL CONTRACTOR ASSUMPTION // JOURNAL OF APPLIED ANALYSIS AND COMPUTATION. 8 (2), 532-548.
Published
2018-08-30
How to Cite
Galimyanov, A., Gilemzyanov, A., & Minnegalieva, C. (2018). Quadrature formula for singular integral computation of special type. Amazonia Investiga, 7(15), 69-73. Retrieved from https://amazoniainvestiga.info/index.php/amazonia/article/view/393
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