ANALYTICAL EVALUATION OF THE EINSTEIN INTEGRATE USING BINOMIAL EXPANSION THEOREM AND POWER SERIES

The objective of this paper is to provide an efficient and reliable analytical expression for the Einstein integrals using the binomial expansion theorem and power series. The obtained analytical expressions are valid for all values of their arguments. The algorithm can be used in the software and simulation programs. Furthermore, the comparison of the method with numerical calculations shows the applicability and accuracy of the method.

Anahtar Kelimeler:

Einstein integrals, Binomial expansion theorem, Bed load, Sediment transport

PDF

___

Reference10. Guseinov, I. I. and Mamedov, B. A. (2005). “Calculation of the generalized hubbell rectangular source integrals using binomial coefficients”. Applied Mathematics and Computation 161, 285-292
Reference11. Julien, P. Y. (1995). Erosion and Sedimentation, Cambridge University Press, CAMBRIDGE, U.K.
Reference12. Kiat, C.C., Ghani, A.AB. and Wen, L.H. (2007). 2nd International Conference on Managing Rivers in the 21st Century: Solutions Towards Sustainable River Basins, June 6-8, Riverside Kuching, Sarawak, Malaysia.
Reference13. Nakato, T. and Asce, M. (1984). “Numerical Integration of Einstein’s Integrals, I1 and I2 .” J. Hydraul. Eng., 110(12), 1863–1868.
Reference14. Yang, C.T. (1996). Sediment Transport Theory and Practice. The McGraw-Hill Companies, Inc., New York.
Reference1. Chang, H.H. (1988). Fluvial Processes in River Engineering, John Wiley and Sons, New York, NY.
Reference2. Chien, N. and Wan, Z. (1999). Mechanics of Sediment Transport, American Society of Civil Engineers, Reston, Va.
Reference3. Einstein, H.A. (1950). “The Bed Load Function For Sediment Transportation İn Open Channel Flows.” U.S. Department of Agriculture, Soil Conservation Service, Washington, D.C.
Reference4. Graf, W.H. (1971). Hydraulics of Sediment Transport, McGraw-Hill Book Co.
Reference5. Guo, J., and Hui, Y. J. (1991). “A Further Study on Einstein’s Sediment Transport Theory.” Adv. Water Sci., 2(2), 81–91 (In Chinese).
Reference6. Guo, J., and Wood, W.L. (1995). “Fine Suspended Sediment Transport Rates.” J. Hydraul. Eng., 121(12), 919–922
Reference7. Guo, J. (2002). “Approximations of Gamma Function and PSİ Function and Their Applications in Sediment Transport.” Advances in Hydraulics and Water Engineering, PROC. 13TH IAHR-APD Congress, World Scientific, Singapore, 1, 219–223.
Reference8. Guo, J., and Julien, P.Y. (2004). “Efficient Algorithm for Computing Einstein Integrals.” Journal of Hydraulic Engineering 130(12), 1198-1201.
Reference9. Gradshteyn, I.S. and Ryzhik, I.M. (1980). “Tables of Integrals, Series and Products, Academic Press, New York.