Numerical Method for Evaluation of Double Integrals its Integrands have Singular Derivatives and Singular by Using Simpson's Rule when Number of Subintervals at the Two Dimensions Unequal

Abstract

The main aim of this research is to derive  numerical  rule to calculate values of double integrals its integrands have singular partial derivatives and  singular integrals at one end of limits region of integration by  using Simpson's rule with two dimensions (on the interior x and exterior y)   , and to find correction terms (formula of error) for it ØŒand using Romberg acceleration to improve the results of integrations by depending on correction terms that we found when the number of subintervals (m)on the dimension yequal to twice of  subintervals (n)on the dimension x,((n),(m) even integers number), preciselywhenh₁=2h₂ where h₁ is the distance between ordinates on x-axis and h₂ the distance between ordinates on y-axis .                            

we will use the symbol ( sim² ) to indicate this Simpson'srulewith two dimensionsandsymbolized the base with Romberg accelerationthe symbol (Rsim² ) ,and we can depend on this rule to calculate like these integrals because it gave high accuracy on the results with respect to the analytical values of integration with little subintervals.