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A STUDY OF SIMPLIFIED SHALLOW WATER WAVES: ASSESSMENT OF ADOMIAN’S DECOMPOSITION METHOD FOR THE ANALYTICAL SOLUTION Mehdi Safari REFERENCES K. Abbaoui, Y. Cherruault, "The decomposition method applied to the Cauchy problem", Kybernetes , Vol.28, 1999, 103. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur," The inverse scattering transform-Fourier analysis for nonlinear problems ", Studies in Applied Mathematics, Vol.53, 1974, pp. 249-315. G. Adomian, "An analytical solution of the stochastic Navier-Stokes system ", Foundations of Physics, Vol.21, No.7, 1991, pp.831-843. G. Adomian, "Solution of physical problems by decomposition", Computers and Mathematics with Applications, Vol.27, No.9/10, 1994, pp.145-154. G. Adomian, "Solutions of nonlinear PDE", Applied Mathematics Letters, Vol.11, No.3, 1998, pp.121-123. G. Adomian, R. Rach, " Linear and nonlinear Schrödinger equations ", Foundations of Physics, Vol.21, 1991, pp.983-991. P.A. Clarkson and E.L. Mansfield, "on a shallow water wave equation", Nonlinearity, Vol.7, 1994, pp. 975-1000. D.D. Ganji, M. Safari, R. Ghayor, "Application of He's variational iteration method and Adomian's decomposition method to Sawada– Kotera–Ito seventh-order equation", Numerical Methods for Partial Differential Equations, Vol. 27, No,4,2011, pp: 887-897. D.D. Ganji, E. M. M. Sadeghi, M. Safari, "Application of He's variational iteration method and adomian's decomposition method method to Prochhammer Chree equation", International Journal of Modern Physics B, Vol. 23, No. 3, 2009, pp.435-446. R. Hirota and J.Satsuma," N-Soliton solutions of model equations for shallow water waves", Journal of the Physical Society of Japan, Vol.40, No.2, 1976, pp.611-612. D. Kaya, A. Yokus, "A numerical comparison of partial solutions in thedecomposition method for linear and nonlinear partial differential equations", Mathematics and Computers in Simulation, Vol.60, No.6, 2002, pp.507-512. M. Safari, D.D. Ganji, M. Moslemi , "Application of He's variational iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation", Computers and Mathematics with Applications, Vol.58, 2009, pp. 2091-2097. A.M.Wazwaz, "Partial Differential Equations: Methods and Applications", Balkema, Rottesda 2002. A.M.Wazwaz, "A First Course in Integral Equations", World Scientific, Singapore, 1997. A.M.Wazwaz, "A new algorithm for calculating Adomian polynomials for nonlinear operators ", Applied Mathematics and Computation, Vol.111, No.1, 2000, pp.33-51. A.M.Wazwaz, "A reliable modification of Adomian decomposition method", Applied Mathematics and Computation, Vol.102, No.1, 1999, pp.77-86. AcademyPublish.org – Journal of Engineering and Technology Vol.2, No.2 20
ON BICRITERIA LARGE SCALE TRANSSHIPMENT PROBLEMS Dr. Jasem M.S. Alrajhi, Dr. Hilal A. Abdelwali, Dr. Mohsen S. Al-Ardhi, Eng. Rafik El Shiaty On Bicriteria Large Scale Transshipment Problems Dr. Jasem M.S. Al-Rajhi* ajasem@gmail.com Dr. Hilal A. Abdelwali* haabdelwali@hotmail.com Dr. Mohsen S. Al-Ardhi* malardhi@hotmail.com Eng. Rafik El Shiaty** rmshiaty@engineer.com * Assistant Professor, Automotive and Marine Department, College of Technological Studies, PAAET, Kuwait. **Lecturer, Power and Refrigeration Technology Department, College of Technological Studies, PAAET, Kuwait. ABSTRACT In this paper, several bicriteria multistage transportation problems with transshipment (BMTSP) are formulated. An algorithm for solving a certain class of (BMTSP) is presented. The mathematical formulation of this class does not affect the special structure of the transshipment problem for each of the individual stages. The presented algorithm is mainly based on a fruitful application of the methods of solving bicriteria single stage transportation problems, available decomposition techniques for solving large scale linear programming problems, and the methods of treating the transshipment problems. An illustrative example is included. Keywords: Large Scale Transportation Problem, Transshipment Problem. Multiobjective Decision Making, Decomposition Technique of Linear Programming. INTRODUCTION The classical transportation problems allow only shipments that go directly from a supply point to a demand point, i.e. shipments do not take place between origins or between destinations, nor from destinations to origins. In many situations, shipments are allowed between supply points or between demand points. Sometimes there many also be points (called transshipment points) through which goods can be transshipped on their journey from a supply point to a demand point. Shipping problems with any or all of these characteristics are transshipment problems. A transshipment problem was first introduced by Orden (1965) [1]. He introduced an extension of the original transportation problem to include the possibility of transshipment. The problem of determining simultaneously the flows of primary products through processors to the market of final products has been formulated alternatively as a transshipment model by King and Logan [2] and as a reduced matrix model by Rhody (1963) [3]. An extension of this problem to a multi regional, multi product, and multi plant problem formulated in the form of general linear programming model has been proposed by Judge et al (1965) [4]. Afterwards, various alternative formulations of the transshipment problem within the framework of the transportation model that permits solution of problems of the type discussed by King and Logan without the need for subtraction of artificial variables were discussed by Hurt and Tramel (1965) [5]. Grag and Prakash (1985) [6] studied time m inimizing transshipment problem. Later dynamic transshipment problem was studied by Herer and Tzur (2001) [7]. Afterwards multi location transshipment problem with capacitated production and lost sales was studied by Ozdemir (2006) [8]. Osman M.S.A. et al (1984) [9] introduced an algorithm for solving bicriteria multistage transportation problems. Recently, Khurana et al (2011) [10] studied a transshipment problem with mixed constraints. Also. In (2012) Khurana et al [11] they introduced an algorithm for solving time minimizing capacitated transshipment problem. Yousria Abo-elnaga et al (2012) [12] introduced a trust region globalization strategy to solve multi-objective transportation, assignment, and transshipment problems. In this paper formulation of different structures of bicriteria large scale transshipment problems, and an algorithm for solving a class of them which can be solved using the decomposition technique of linear programming utilizing the special nature of transshipment problems are presented. The presented algorithm determines the points of the non-dominated set in the objective space. The method consists of solving the same multistage transshipment problem repeatedly but with different objectives and each iteration gives either a new non dominated extreme point or changes the direction of search in the objective space. An illustrative example is presented in this paper. Formulation of Bicriteria Multistage Transshipment Problems The formulation of different bicriteria multistage transportation problems with transshipment presented in this paper covers several real situations. Bicriteria Multistage Transportation Problem with Transshipment of the First kind (BMTSP 1) This case represents multistage transshipment problems without any restrictions on intermediate stages. In order to obtain the mathematical formulation of the problems representing this case let us assume that the availabilities are (a j ), j= 1, 2, 3, …., n; n is the number of (sources + destinations); the requirements are (b j ), j= 1, 2, 3, ….., n; the minimum transportation costs and deteriorations from i to j are (c ij ),(d ij) i= 1, 2, 3, …., n; j= 1, 2, 3, …., n; (x ij ) denotes the quatity shipped from i to j; and (x jj ) is the neat amount transshipped through point j, x ij ≥0. Then the problem takes the form: n n Min. Z 1 i 1 j 1 c ij x ij AcademyPublish.org – Journal of Engineering and Technology Vol.2, No.2 21
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Page 3 and 4: PARAMETRIC STUDY OF SANDWICH PANEL
Page 17 and 18: A STUDY OF SIMPLIFIED SHALLOW WATER
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Page 23 and 24: ON BICRITERIA LARGE SCALE TRANSSHIP
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Page 27 and 28: TRIBOLOGY OF HIGH SPEED MOVING MECH
Page 35 and 36: THE USE OF IRON IN PEAT WATER FOR F
Page 37 and 38: THE USE OF IRON IN PEAT WATER FOR F
Page 39 and 40: TRIBOLOGY OF HIGH-SPEED MOVING MECH
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