Numerical Integration Schemes for Unequal Data Spacing

In this Paper, we have derived the numerical integration formula for unequal data spacing. For doing so we use Newton’s Divided difference formula to evaluate general quadrature formula and then generate Trapezoidal and Simpson’s rules for the unequal data spacing with error terms. Finally we use numerical examples to compare the numerical results with analytical results and the well-known MonteCarlo integration results and find excellent agreement.


Introduction
Numerical integration is the process of computing the value of a definite integral from a set of numerical values of the integrand. It is necessary that the integrand has no singularities in the domain under consideration. The basic problem in numerical integration [3][4][5] is to compute an approximate value of a definite integral to a given accuracy. If is a smooth function integrand over a small number of dimensions and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision numerically. Most of the methods are using equally spaced data to obtain numerical integration. Method like Monte -Carlo [8] integration used random numbers over the interval to approximate the integral where the spaces between two points are not equal. Here we also use some random points over the interval to evaluate the integral numerically. We use Newton's Divided difference formula to derive the formula for unequal data spacing with the error term and use numerical examples to compare our solution with the exact solution and the well-known Monte -Carlo integration result.

General Quadrature Formula for Equal Space
The general problem of numerical integration may be stated as follows: Given a set of data points , , , , … … . , , of a function = , where is not known explicitly, it is required to compute the value of definite integral [10] = As in the case of numerical differentiation, one replaces by an interpolating polynomial and obtains an approximate value of the definite integral. Now different integration formulae can be obtained depending on the interpolation formula used. In this section we use Newton's forward difference formula. Let

General Quadrature Formula for Unequal Spacing
Consider the definite integral 2 , where the interval [ , ] be divided into unequal subintervals [4,9].
From Newton's Divided difference formula [4] we know that, Using (1) into the integral we get, This is the equation of general quadrature formula for unequal spacing. From the general formula, we can obtain different integration formulae by putting = 1, 2, 3, etc.

Trapezoidal Rule
Putting = 1 in (2) we get the formula of Trapezoidal rule [2,3]  Adding all terms we get This is the general formula of Trapezoidal rule for unequal spacing. ;

Error of the Trapezoidal Rule
But we have, So, using the value of in (6) we obtain, Now, for the error term subtracting (7) from (5) we get This is the error for the interval[ , ].
In the same way we get error for remaining intervals and the total error, This is the error of Trapezoidal rule for unequal spacing.

Simpson's 1/3-Rule
Putting = 2 in (2) Adding all terms we get This is the general formula of Simpson's 1/3 rule for unequal spacing.  (9)) i.e.  This is the error of Simpson's 1/3 rule for unequal spacing.

Example 1
Consider the integration W : We consider 100 random points over the intervals

Example 2
Consider the integration R + We consider 100 random points over the intervals

Example 3
Consider the Heat equation 4 . We want to calculate _ using our above methods. We consider 200 random points over the interval 0, 30 and get the following results Again, we consider 500 random points over the interval 0, 30 and get the following results In our first two examples we see that choosing 100 points we find better approximation using our methods comparing to the well-known Monte-Carlo results using the same &values. If we change our points randomly similar things happened. If we choose more points accuracy of the result is much higher which we see in our 3 rd example. We also observe that Simpson's 1/3 rule gives better approximation than the other methods we have discussed.

Conclusion
The main objectives of this paper are to develop and understand solving numerical integration for the unequal data space by different methods with their errors. Firstly we derive the methods with their errors and then we use these methods to solve the numerical examples using computer program (MATLAB software) and we get sufficient accuracy comparing to the exact solution and Monte Carlo integration of the problems.