AM  Vol.6 No.6 , June 2015
Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights
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.
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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.