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APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS

Year 2014, Volume: 2 Issue: 2, 22 - 34, 01.12.2014

Abstract

In this paper, a three–point quadrature rule for the Riemann–Stieltjes integral is introduced. As application; an error estimate for the obtained quadrature rule is provided as well

References

  • M.W. Alomari, Some Gr¨uss type inequalities for Riemann-Stieltjes integral and applications, Acta Mathematica Universitatis Comenianae, 81 (2) (2012), 211–220.
  • M.W. Alomari, A companion of Dragomir’s generalization of Ostrowski’s inequality and ap- plications in numerical integration, Ukrainian Math. J., 64(4) 2012, 491–510.
  • M.W. Alomari, A companion of Ostrowski’s inequality for the Riemann-Stieltjes integral
  • bf (t) du (t), where f is of bounded variation and u is of r-H-H¨
  • af (t) du (t), where f is of bounded variation and u is of r-H-H¨
  • older type and applications,
  • Appl. Math. Comput., 219 (2013), 4792–4799.
  • M.W. Alomari, A companion of Ostrowski’s inequality for the Riemann-Stieltjes integral
  • bf (t) du (t), where f is of r-H-H¨ a
  • submitted. Avalibale at: http://ajmaa.org/RGMIA/papers/v14/v14a59.pdf. older type and u is of bounded variation and applications,
  • M.W. Alomari, A new generalization of Gr¨uss type inequalities for the Stieltjes integral and Applications, submitted. Avaliable at: http://ajmaa.org/RGMIA/papers/v15/v15a36.pdf
  • M.W. Alomari, New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann–Stieltjes integrals and applications, Ukrainian Mathematical Journal, 65 (7) 2013, 995–1018.
  • M.W. Alomari and S.S. Dragomir, Mercer–Trapezoid rule for the Riemann–Stieltjes integral with applications, Journal of Advances in Mathematics, 2 (2) (2013), 67–85.
  • M.W. Alomari and S.S. Dragomir, A three–point quadrature rule for the Riemann–Stieltjes integral, Southeast Bulletin Journal of Mathematics, accepted.
  • N.S. Barnett, S.S. Dragomir and I. Gomma, A companion for the Ostrowski and the generalised trapezoid inequalities, Math. and Comp. Mode., 50 (2009), 179–187.
  • N.S. Barnett, W.-S. Cheung, S.S. Dragomir, A. Sofo, Ostrowski and trapezoid type inequal- ities for the Stieltjes integral with Lipschitzian integrands or integrators, Comp. Math. Appl. , 57 (2009), 195–201.
  • P. Cerone and S.S. Dragomir, Midpoint-type rules from an inequalities point of view, Hand- book of Analytic-Computational Methods in Applied Mathematics, Editor: G.A. Anastassiou, CRC Press, N.Y. (2000), 135-200.
  • P. Cerone, W.S. Cheung, S.S. Dragomir, On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation, Comp. Math. Appl., 54 (2007), 183–191.
  • P. Cerone, S.S. Dragomir, New bounds for the three-point rule involving the Riemann– Stieltjes integrals, in: C. Gulati, et al. (Eds.), Advances in Statistics Combinatorics and Related Areas, World Science Publishing, 2002, pp. 53–62.
  • P. Cerone, S.S. Dragomir, Approximating the Riemann–Stieltjes integral via some moments of the integrand, Mathematical and Computer Modelling, 49 (2009), 242–248.
  • P. Cerone, S.S. Dragomir and C.E.M. Pearce, A generalized trapezoid inequality for functions of bounded variation, Turk. J. Math., 24 (2000), 147–163.
  • W.-S. Cheung and S.S. Dragomir, Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions, Bull. Austral. Math. Soc., 75 (2007), 299–311.
  • S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 59–66.
  • S.S. Dragomir, On the Ostrowski inequality for Riemann–Stieltjes integral bf (t)du(t) where af (t)du(t) where
  • f is of H¨older type and u is of bounded variation and applications, J. KSIAM, 5 (2001), 35–45.
  • S.S. Dragomir, On the Ostrowski’s inequality for Riemann–Stieltes integral and applications, Korean J. Comput. & Appl. Math., 7 (2000), 611–627.
  • S.S. Dragomir, C. Bu¸se, M.V. Boldea and L. Braescu, A generalisation of the trapezoid rule for the Riemann–Stieltjes integral and applications, Nonlinear Anal. Forum, 6 (2) (2001) 337–351.
  • S.S. Dragomir, Some inequalities of midpoint and trapezoid type for the Riemann-Stieltjes integral, Nonlinear Anal., 47 (4) (2001), 2333–2340.
  • S.S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for func- tions of bounded variation, Arch. Math., 91 (2008), 450–460
  • S.S. Dragomir, Approximating the Riemann–Stieltjes integral in terms of generalised trape- zoidal rules, Nonlinear Anal. TMA 71 (2009), e62–e72.
  • S.S. Dragomir, Approximating the Riemann–Stieltjes integral by a trapezoidal quadrature rule with applications, Mathematical and Computer Modelling 54 (2011), 243–260.
  • S.S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applica- tions, Comp. Math. Appl., 38 (1999), 33–37.
  • S.S. Dragomir and I. Fedotov, An inequality of Gruss type for Riemann—Stieltjes integral and applications for special means, Tamkang J. Math., 29(4) (1998), 287–292.
  • R.P. Mercer, Hadamard’s inequality and trapezoid rules for the Riemann–Stieltjes integral, J. Math. Anal. Appl., (344) (2008), 921-927.
  • M. Tortorella, Closed Newton–Cotes quadrature rules for Stieltjes integrals and numerical convolution of life distributions. SIAM J. Sci. Statist. Comput. 11 (1990), no. 4, 732–748. Department of Mathematics, Faculty of Science and Technology, Irbid National
  • University, Jordan. E-mail address: mwomath@gmail.com
Year 2014, Volume: 2 Issue: 2, 22 - 34, 01.12.2014

Abstract

References

  • M.W. Alomari, Some Gr¨uss type inequalities for Riemann-Stieltjes integral and applications, Acta Mathematica Universitatis Comenianae, 81 (2) (2012), 211–220.
  • M.W. Alomari, A companion of Dragomir’s generalization of Ostrowski’s inequality and ap- plications in numerical integration, Ukrainian Math. J., 64(4) 2012, 491–510.
  • M.W. Alomari, A companion of Ostrowski’s inequality for the Riemann-Stieltjes integral
  • bf (t) du (t), where f is of bounded variation and u is of r-H-H¨
  • af (t) du (t), where f is of bounded variation and u is of r-H-H¨
  • older type and applications,
  • Appl. Math. Comput., 219 (2013), 4792–4799.
  • M.W. Alomari, A companion of Ostrowski’s inequality for the Riemann-Stieltjes integral
  • bf (t) du (t), where f is of r-H-H¨ a
  • submitted. Avalibale at: http://ajmaa.org/RGMIA/papers/v14/v14a59.pdf. older type and u is of bounded variation and applications,
  • M.W. Alomari, A new generalization of Gr¨uss type inequalities for the Stieltjes integral and Applications, submitted. Avaliable at: http://ajmaa.org/RGMIA/papers/v15/v15a36.pdf
  • M.W. Alomari, New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann–Stieltjes integrals and applications, Ukrainian Mathematical Journal, 65 (7) 2013, 995–1018.
  • M.W. Alomari and S.S. Dragomir, Mercer–Trapezoid rule for the Riemann–Stieltjes integral with applications, Journal of Advances in Mathematics, 2 (2) (2013), 67–85.
  • M.W. Alomari and S.S. Dragomir, A three–point quadrature rule for the Riemann–Stieltjes integral, Southeast Bulletin Journal of Mathematics, accepted.
  • N.S. Barnett, S.S. Dragomir and I. Gomma, A companion for the Ostrowski and the generalised trapezoid inequalities, Math. and Comp. Mode., 50 (2009), 179–187.
  • N.S. Barnett, W.-S. Cheung, S.S. Dragomir, A. Sofo, Ostrowski and trapezoid type inequal- ities for the Stieltjes integral with Lipschitzian integrands or integrators, Comp. Math. Appl. , 57 (2009), 195–201.
  • P. Cerone and S.S. Dragomir, Midpoint-type rules from an inequalities point of view, Hand- book of Analytic-Computational Methods in Applied Mathematics, Editor: G.A. Anastassiou, CRC Press, N.Y. (2000), 135-200.
  • P. Cerone, W.S. Cheung, S.S. Dragomir, On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation, Comp. Math. Appl., 54 (2007), 183–191.
  • P. Cerone, S.S. Dragomir, New bounds for the three-point rule involving the Riemann– Stieltjes integrals, in: C. Gulati, et al. (Eds.), Advances in Statistics Combinatorics and Related Areas, World Science Publishing, 2002, pp. 53–62.
  • P. Cerone, S.S. Dragomir, Approximating the Riemann–Stieltjes integral via some moments of the integrand, Mathematical and Computer Modelling, 49 (2009), 242–248.
  • P. Cerone, S.S. Dragomir and C.E.M. Pearce, A generalized trapezoid inequality for functions of bounded variation, Turk. J. Math., 24 (2000), 147–163.
  • W.-S. Cheung and S.S. Dragomir, Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions, Bull. Austral. Math. Soc., 75 (2007), 299–311.
  • S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 59–66.
  • S.S. Dragomir, On the Ostrowski inequality for Riemann–Stieltjes integral bf (t)du(t) where af (t)du(t) where
  • f is of H¨older type and u is of bounded variation and applications, J. KSIAM, 5 (2001), 35–45.
  • S.S. Dragomir, On the Ostrowski’s inequality for Riemann–Stieltes integral and applications, Korean J. Comput. & Appl. Math., 7 (2000), 611–627.
  • S.S. Dragomir, C. Bu¸se, M.V. Boldea and L. Braescu, A generalisation of the trapezoid rule for the Riemann–Stieltjes integral and applications, Nonlinear Anal. Forum, 6 (2) (2001) 337–351.
  • S.S. Dragomir, Some inequalities of midpoint and trapezoid type for the Riemann-Stieltjes integral, Nonlinear Anal., 47 (4) (2001), 2333–2340.
  • S.S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for func- tions of bounded variation, Arch. Math., 91 (2008), 450–460
  • S.S. Dragomir, Approximating the Riemann–Stieltjes integral in terms of generalised trape- zoidal rules, Nonlinear Anal. TMA 71 (2009), e62–e72.
  • S.S. Dragomir, Approximating the Riemann–Stieltjes integral by a trapezoidal quadrature rule with applications, Mathematical and Computer Modelling 54 (2011), 243–260.
  • S.S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applica- tions, Comp. Math. Appl., 38 (1999), 33–37.
  • S.S. Dragomir and I. Fedotov, An inequality of Gruss type for Riemann—Stieltjes integral and applications for special means, Tamkang J. Math., 29(4) (1998), 287–292.
  • R.P. Mercer, Hadamard’s inequality and trapezoid rules for the Riemann–Stieltjes integral, J. Math. Anal. Appl., (344) (2008), 921-927.
  • M. Tortorella, Closed Newton–Cotes quadrature rules for Stieltjes integrals and numerical convolution of life distributions. SIAM J. Sci. Statist. Comput. 11 (1990), no. 4, 732–748. Department of Mathematics, Faculty of Science and Technology, Irbid National
  • University, Jordan. E-mail address: mwomath@gmail.com
There are 36 citations in total.

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Journal Section Articles
Authors

M.W. Alomarı This is me

Publication Date December 1, 2014
Submission Date April 4, 2015
Published in Issue Year 2014 Volume: 2 Issue: 2

Cite

APA Alomarı, M. (2014). APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS. Konuralp Journal of Mathematics, 2(2), 22-34.
AMA Alomarı M. APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS. Konuralp J. Math. October 2014;2(2):22-34.
Chicago Alomarı, M.W. “APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS”. Konuralp Journal of Mathematics 2, no. 2 (October 2014): 22-34.
EndNote Alomarı M (October 1, 2014) APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS. Konuralp Journal of Mathematics 2 2 22–34.
IEEE M. Alomarı, “APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS”, Konuralp J. Math., vol. 2, no. 2, pp. 22–34, 2014.
ISNAD Alomarı, M.W. “APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS”. Konuralp Journal of Mathematics 2/2 (October 2014), 22-34.
JAMA Alomarı M. APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS. Konuralp J. Math. 2014;2:22–34.
MLA Alomarı, M.W. “APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS”. Konuralp Journal of Mathematics, vol. 2, no. 2, 2014, pp. 22-34.
Vancouver Alomarı M. APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A THREE-POINT QUADRATURE RULE AND APPLICATIONS. Konuralp J. Math. 2014;2(2):22-34.
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