Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights

Affiliation(s)

^{1}
School of Mathematics, KNT University of Technology, Tehran, Iran.

^{2}
Department of Mathematics, College of Basic Sciences, Karaj branch Islamic Azad University, Alborz, Iran.

ABSTRACT

In this paper, we present a procedure for the numerical q-calculation of the q-integrals based on appropriate nodes and weights which are determined such that the error of q-integration is mini-mized; a system of linear and nonlinear set of equations respectively are prepared to obtain the nodes and weights simultaneously; the error of q-integration is considered to be minimized under this condition; finally some application and numerical examples are given for comparison with the exact solution. At the end, the related tables of approximations are presented.

In this paper, we present a procedure for the numerical q-calculation of the q-integrals based on appropriate nodes and weights which are determined such that the error of q-integration is mini-mized; a system of linear and nonlinear set of equations respectively are prepared to obtain the nodes and weights simultaneously; the error of q-integration is considered to be minimized under this condition; finally some application and numerical examples are given for comparison with the exact solution. At the end, the related tables of approximations are presented.

Cite this paper

Hashemiparast, S. , Ghondaghsaz, D. and Maghasedi, M. (2015) Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights.*Applied Mathematics*, **6**, 958-966. doi: 10.4236/am.2015.66088.

Hashemiparast, S. , Ghondaghsaz, D. and Maghasedi, M. (2015) Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights.

References

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http://dx.doi.org/10.2298/AADM0701311R

[2] Bostan, A., Salvy, B., Chowdhury, M.F.I., Schost, E., Lebreton, R. and Max, E. (2014) Power Series Solution of Singular q-Differential Equations. Journal of Combinatorial Theory, Series A, 121, 45-63.

http://dx.doi.org/10.1016/j.jcta.2013.09.005

[3] Kim, T. (2007) On the Analogs of Euler Number and Polynomials Associated with p-Adic q-Integral on Z_{p} at q = -1. Journal of Mathematical Analysis and Applications, 331, 779-792.

http://dx.doi.org/10.1016/j.jmaa.2006.09.027

[4] Lim, S.C., Eab, C.H., Mak, K.H., Li, M. and Chen, S.Y. (2012) Solving Linear Coupled Fractional Differential Equations by Direct Operational Method and Some Applications. Mathematical Problems in Engineering, 2012, Article ID: 653939. http://dx.doi.org/10.1155/2012/653939

[5] Foupouagnigni, M., Koepf, W. and Ronveaux, A. (2004) On Factorization and Solutions of q-Difference Equations Satisfied by Some Classes of Orthogonal Polynomials. Journal of Computational and Applied Mathematics, 162, 299- 326. http://dx.doi.org/10.1016/j.cam.2003.04.005

[6] Bowman, D. and Sohn, J. (1999) Partial q-Differences Equations for Basic Hypergeometric Function and Their q-Con- tinued Fractions. University of Illinois.

[7] De la Sen, M. (2014) On Nonnegative Solutions of Fractional q-Linear Time-Varying Dynamics. Hindawi Publisher Co. Abstract and Applied Analysis, 2014, Article ID: 247375.

http://dx.doi.org/10.1155/2014/247375

[8] Simsek, Y. (2006) q-Dedekind Type Sums Related to q-Zeta Function and Basic L-Series. Journal of Mathematical Analysis and Applications, 318, 333-351.

http://dx.doi.org/10.1016/j.jmaa.2005.06.007

[9] Abdeljavad, T., Benli, B. and Baleanu, D. (2012) A Generalized q-Mittag-Leffler Function by q-Captuo Fractional Linear Equations. Hindawi Publisher Co. Abstract and Applied Analysis, 2012, Article ID: 546062.

[10] Ismail, M.E.H. and Stanton, D. (2003) q-Taylor Theorems, Polynomial Expansions and Interpolation of Entire Functions. Journal of Approximation Theory, 123, 125-146.

http://dx.doi.org/10.1016/S0021-9045(03)00076-5

[11] Stankovic, M.S., Rajkovic, P.M. and Marinkovic, S.D. (2006) Inequalities Which Includes q-Integrals. Bulletin: Classe des sciences mathematiques et natturalles, 133, 137-146.

[12] Wu, G.-C. and Baleanu, D. (2013) New Application of the Variation Iteration Method from Differential Equation to q-Fractional Difference Equations. Advanced in Difference Equation, 21.

[13] Hashemiparast, S.M. (2011) Numerical Solution of the Integrals by Using Appropriate Nodes and Weights. Proceedings of the ICMS Conference, Istanbul.

[14] Ernst, T. (2003) A Method for q-Calculus. Journal of Nonlinear Mathematical Physics, 10, 487-525.

http://dx.doi.org/10.2991/jnmp.2003.10.4.5

[15] Jackson, F.H. (1910) On q-Definite Integrals. Quarterly Journal of Pure and Applied Mathematics, 41, 193-203.

[16] Koekoek, R., Alesky, P. and Swarrouw, R. (2010) Hyper Geometric Orthogonal Polynomials and Their q-Analogues. Cambridge University Press, Cambridge.

[17] Miao, Y. and Feng, Q. (2009) Several q-Integrals Inequalities. Journal of Mathematical Inequalities, 3, 115-121. http://dx.doi.org/10.7153/jmi-03-11

[18] Hashemiparast, S.M., Eslahchi, M.R. and Dehghan, M. (2007) Determination of Nodes in Numerical Integration Rules Using Difference Equations. Applied Mathematics and Computation, 176, 117-122.

[19] Rietsch, K. (2001) Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties. Journal of the American Mathematics Society, 16, 363-392.

[20] Andersen, J.E. and Berg, C. (2009) Quantum Hilbert Matrices and Orthogonal Polynomials. Journal of Computational and Applied Mathematics, 233, 723-729.

[21] Cooper, A.P. (2011) The Quantum Matrix. The Author and Copyrights ©2011.

[22] Gray, R.M. (2006) Toeplitz and Circulant Matrices: A Review. Department of Electrical Engineering, Stanford University, Stanford.

[23] Lv, X.-G. and Huang, T.-Z. (2007) A Note on Inversion of Toeplitz Matrices. Applied Mathematics Letters, 20, 1189- 1193. http://dx.doi.org/10.1016/j.aml.2006.10.008

[1] Rajkovic, P.M., Marinkovic, S.D. and Stankovic, M.S. (2007) Fractional Integrals and Derivatives in q-Calculus. Applicable Analysis and Discrete Mathematics, 1, 311-323.

http://dx.doi.org/10.2298/AADM0701311R

[2] Bostan, A., Salvy, B., Chowdhury, M.F.I., Schost, E., Lebreton, R. and Max, E. (2014) Power Series Solution of Singular q-Differential Equations. Journal of Combinatorial Theory, Series A, 121, 45-63.

http://dx.doi.org/10.1016/j.jcta.2013.09.005

[3] Kim, T. (2007) On the Analogs of Euler Number and Polynomials Associated with p-Adic q-Integral on Z

http://dx.doi.org/10.1016/j.jmaa.2006.09.027

[4] Lim, S.C., Eab, C.H., Mak, K.H., Li, M. and Chen, S.Y. (2012) Solving Linear Coupled Fractional Differential Equations by Direct Operational Method and Some Applications. Mathematical Problems in Engineering, 2012, Article ID: 653939. http://dx.doi.org/10.1155/2012/653939

[5] Foupouagnigni, M., Koepf, W. and Ronveaux, A. (2004) On Factorization and Solutions of q-Difference Equations Satisfied by Some Classes of Orthogonal Polynomials. Journal of Computational and Applied Mathematics, 162, 299- 326. http://dx.doi.org/10.1016/j.cam.2003.04.005

[6] Bowman, D. and Sohn, J. (1999) Partial q-Differences Equations for Basic Hypergeometric Function and Their q-Con- tinued Fractions. University of Illinois.

[7] De la Sen, M. (2014) On Nonnegative Solutions of Fractional q-Linear Time-Varying Dynamics. Hindawi Publisher Co. Abstract and Applied Analysis, 2014, Article ID: 247375.

http://dx.doi.org/10.1155/2014/247375

[8] Simsek, Y. (2006) q-Dedekind Type Sums Related to q-Zeta Function and Basic L-Series. Journal of Mathematical Analysis and Applications, 318, 333-351.

http://dx.doi.org/10.1016/j.jmaa.2005.06.007

[9] Abdeljavad, T., Benli, B. and Baleanu, D. (2012) A Generalized q-Mittag-Leffler Function by q-Captuo Fractional Linear Equations. Hindawi Publisher Co. Abstract and Applied Analysis, 2012, Article ID: 546062.

[10] Ismail, M.E.H. and Stanton, D. (2003) q-Taylor Theorems, Polynomial Expansions and Interpolation of Entire Functions. Journal of Approximation Theory, 123, 125-146.

http://dx.doi.org/10.1016/S0021-9045(03)00076-5

[11] Stankovic, M.S., Rajkovic, P.M. and Marinkovic, S.D. (2006) Inequalities Which Includes q-Integrals. Bulletin: Classe des sciences mathematiques et natturalles, 133, 137-146.

[12] Wu, G.-C. and Baleanu, D. (2013) New Application of the Variation Iteration Method from Differential Equation to q-Fractional Difference Equations. Advanced in Difference Equation, 21.

[13] Hashemiparast, S.M. (2011) Numerical Solution of the Integrals by Using Appropriate Nodes and Weights. Proceedings of the ICMS Conference, Istanbul.

[14] Ernst, T. (2003) A Method for q-Calculus. Journal of Nonlinear Mathematical Physics, 10, 487-525.

http://dx.doi.org/10.2991/jnmp.2003.10.4.5

[15] Jackson, F.H. (1910) On q-Definite Integrals. Quarterly Journal of Pure and Applied Mathematics, 41, 193-203.

[16] Koekoek, R., Alesky, P. and Swarrouw, R. (2010) Hyper Geometric Orthogonal Polynomials and Their q-Analogues. Cambridge University Press, Cambridge.

[17] Miao, Y. and Feng, Q. (2009) Several q-Integrals Inequalities. Journal of Mathematical Inequalities, 3, 115-121. http://dx.doi.org/10.7153/jmi-03-11

[18] Hashemiparast, S.M., Eslahchi, M.R. and Dehghan, M. (2007) Determination of Nodes in Numerical Integration Rules Using Difference Equations. Applied Mathematics and Computation, 176, 117-122.

[19] Rietsch, K. (2001) Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties. Journal of the American Mathematics Society, 16, 363-392.

[20] Andersen, J.E. and Berg, C. (2009) Quantum Hilbert Matrices and Orthogonal Polynomials. Journal of Computational and Applied Mathematics, 233, 723-729.

[21] Cooper, A.P. (2011) The Quantum Matrix. The Author and Copyrights ©2011.

[22] Gray, R.M. (2006) Toeplitz and Circulant Matrices: A Review. Department of Electrical Engineering, Stanford University, Stanford.

[23] Lv, X.-G. and Huang, T.-Z. (2007) A Note on Inversion of Toeplitz Matrices. Applied Mathematics Letters, 20, 1189- 1193. http://dx.doi.org/10.1016/j.aml.2006.10.008