Numerical Solution of Non-Linear Algebraic Equations by Modified Genetic Algorithm

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Numerous unit operations in chemical and process engineering can be represented as a system of non-linear algebraic equations, when modeled for steady state operation ,e.g. isothermal and non-isothermal operations of a series of CSTRs, batteries of evaporators, networks of various separation operations (flash drum, mixers), distillation, extraction and absorption columns, and pumps and piping networks etc. These governing equations are sometimes very difficult to solve due to the nonlinear and uneven nature associated with them. The difficulty level increases when the resulting set of equations become flat near their zeros and thus derivatives based schemes mostly diverge or give poor results. Many a times, even a good initial guess in conventional numerical techniques do not guarantee to have a true solution and problem specific methods have to be designed. This whole scenario can also be viewed as an optimization problem having equality constraints only and casting the equations in the form of norm of function vector, which formulates an objective function to be minimized. The true minimum thus found gives us the correct solution vector. Recently, Genetic Algorithms have been quite effectively used to solve many complex engineering optimization problems. In continuation of our earlier research work where an elitist genetic algorithm was developed for the solutions of various difficult MINLP problems (Danish et al. 2006a and b), this research work extends its application for the solution of difficult non-linear algebraic equations. A novel scheme of dynamic mutation parameter as a function of fitness along with dynamic penalty has been proposed. The small value of mutation parameter in initial stages enables the algorithm to search globally, and the solution thus found is refined by keeping its value higher in later generations. This new scheme is found to be very effective in the sense that the algorithm requires very small population size and comparatively lesser number of generations to give reasonably good solutions. To test the efficacy of algorithm we have solved five sets of difficult non-linear algebraic equations (Dennis and Schnabel, 1983). It is worthwhile to mention that one of these equations was having both its Jacobian and Hessian as zero, at its true solution. Applicability of the developed GA was also demonstrated by simulating an industrial case study of triple effect evaporator used for concentrating the caustic soda solution (Zain and Kumar, 1996), which also poses difficulty during numerical simulation by Newton-Raphson method.