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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.

Recently, much attention has been paid on q-calculus, especially on q-fractional calculus which finally most of them have changed to q-integral not easy and even possible to be solved analytically [

In this section we define the basic definitions and theorems related to the quantum integration

Jackson’s definition [

when

For the continuous function f in the interval [0, z] we have [

and

The generalized q-integral for

Similarly

For the integer number n, the quantum integer is defined as a [n]_{q} (bracket n) such that

In [

and the generalized q-derivative for

Similarly

Analytical calculation of q-integrals similar to the ordinary integrals leads to extending these integrals as a series expansion, [

For a given value of q the following approximation can be established

where

By calculating the limits in either side we get

Similar to the algorithm used in [

Theorem: The q-normal equations for

Proof: Without losing the generality, for

Hence, we have the following system of equations

This can be summarized as the following matrix form such that for

where

is a Toeplitz matrix [10,11], whose entries are quantum numbers so, we call it quantum Toeplitz matrix, an especial form of n-diameter quantum matrix and keeps the non-singularity or singularity properties of the original matrix, because all elements of matrix have been changed simultaneously, positive Toplitz matrices, quantum matrices and Inversion of Toeplitz matrices are considered in [

Now, the roots of above characteristics equations are the appropriate nodes, satisfying in the system of simultaneous equations, then having these nodes the weighs μ_{i} can be evaluated, and by applying these values in (12) the system of Equation (13) will be obtained to evaluate the approximate values of the quantum integral, obviously the unknown in the system of equations depend upon the quantum parameter

In Section 3 we illustrate the algorithm for the numerical approximation of q-integral and some examples to illuminate the exactness of the method.

We start the algorithm by the small values of n and similar method can be extended to any value of n, let n = 2, then for the evaluated values of x_{i}_{ }s and μ_{i}

characteristic equation is

The system of equations is:

Then

different approximation can be expected for the different values of q, which for some values of q, 0 < q < 1, q-integral may have minimum error, if

This gives

Then the characteristic equation is

Gives the following roots

By solving the linear system we obtain μ_{1}, μ_{2}

And the numerical q-integration formula for q = 0.1 can be evaluated from

Let n = 3, then for the evaluated values of x_{i}s and, x_{i}s

Characteristics equations is

For n = 3, the system of equations is

Similarly, for q = 0.1 numerical q-integration is

This gives

And a characteristics equation is:

With the roots as follows

Now, for calculating

And we will have

Finally the numerical q-integration for q = 0.1 takes the following form

Tables 1-3 give the q-integral approximation for n = 2, 3, 4 respectively and some values of q.

q = 0.1 | |

q = 0.2 | |

q = 0.3 | |

q = 0.4 | |

q = 0.5 | |

q = 0.6 | |

q = 0.7 | |

q = 0.8 | |

q = 0.9 |

q = 0.1 | |

q = 0.2 | |

q = 0.3 | |

q = 0.4 | |

q = 0.5 | |

q = 0.6 | |

q = 0.7 | |

q = 0.8 | |

q = 0.9 |

q | |
---|---|

q = 0.1 | |

q = 0.2 | |

q = 0.3 | |

q = 0.4 | |

q = 0.5 | |

q = 0.6 | |

q = 0.7 | |

q = 0.8 | |

q = 0.9 |

q | |
---|---|

0.99999 |

Integral | Value of integral for q = 1 (ordinary integral) | q-integral approximation | Error |
---|---|---|---|

0.946083 | 0.918540689 | 0.02 | |

1.1491512305 | 1.2055660182 | 0.05 | |

0.69314718 | 0.486661583 | 0.20 |

The numerical values show for all values of n the error of q-integrations fluctuate for different values of q, it seems q = 0.70 gives the worse error almost for all values of n, the errors decreases as q approaches to the extreme values 0 and 1. Using this result and (3) the q-integral can be calculated for very large value of q approaching to 1 which will approximate the ordinary integrals whose q-integrals is easier than ordinary integrals by using q-integral approximation for n = 2 and different values of q, as illustrated in

In this paper, a new algorithm for the numerical approximation of q-integration based on q-calculation of appropriate nodes and weights is introduced. The evaluation of nodes and weight is based on q-integral error minimization, as expected in the numerical examples which give a good approximation in comparison with exact solutions for the given values of q and fixed n. As the q-fractional integration can be transferred to q-integrals, the procedure is also applicable for q-fractional integration, and also improper integrals for the large values of q.