By Bert B. Williams, John L. Gidley, Robert S. Schechter
SPE Monograph quantity 6. This monograph has been ready to function a uncomplicated reference for either the sector and learn engineer drawn to constructing a basic figuring out of the acidizing procedure and techniques for designing acid remedies to extend good productiveness.
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If very large values of n are employed, round-off error can cause a major deterioration in accuracy (see previous discussion of Romberg integration). 8 Numerical Solution of Differential Equations The two major categories of ordinary differential equations are 1. Initial value problems where conditions are specified at some starting value of the independent variable. 2. Boundary value problems where conditions are specified at two (or, rarely, more) values of the independent variable. ) Any nth -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition.
Butcher, J. , “On Runge-Kutta Processes of High Order,” Journal of the Australian Mathematical Society, Vol. 4, pp. 179–194, 1964. 7. Gear, C. , Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, 1971. 8. , Introduction to Numerical Analysis, Springer Verlag, New York, 1980. 9. , Numerical Solution of Ordinary and Partial Differential Equations, Addison Wesley, Reading, 1962. 10. de G. Allen, D. , Relaxation Methods in Engineering and Science, McGraw-Hill, New York, 1954.
In general Some other typical Gaussian quadrature formulas are (a, b) W(x) (−1, 1) √1 (0, ∞) (−∞, ∞) x c e−x 2 e−x 1−x 2 dy1 = f1 (y1 , y2 , . . , yn , t) dt GaussChebyshev dy2 = f2 (y1 , y2 , . . , yn , t) dt Laguerre (c = 0, 1, . . ) Hermite Weights and zeros for these formulas (and for other Gaussian formulas) may be found in references such as Stroud (Gaussian quadrature formulas, Prentice-Hall, 1966). Because the dominant error term in Gauss Quadrature involves very high-order derivatives, the best method for determining the accuracy of an integration is to compare dyn = fn (y1 , y2 , .
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