Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm

A new kind of optimization technique, namely, continuous genetic algorithm, is presented in this paper for numerically approximating the solutions of Troesch’s and Bratu’s problems. The underlying idea of the method is to convert the two differential problems into discrete versions by replacing each of the second derivatives by an appropriate difference quotient approximation. The new method has the following characteristics. First, it should not resort to more advanced mathematical tools; that is, the algorithm should be simple to understand and implement and should be thus easily accepted in the mathematical and physical application fields. Second, the algorithm is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical and physical problems. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. The algorithm is tested on different versions of Troesch’s and Bratu’s problems. Experimental results show that the proposed algorithm is effective, straightforward, and simple.

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Optimization Solution of Troesch’s and Bratu’s Problems of Ordinary Type Using Novel Continuous Genetic Algorithm

In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. This novel approach possesses main advantages; it can be applied without any limitation on the nature of the problem, the type of singularity, and the number of mesh points. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the presented technique. The results reveal that the algorithm is very effective, straightforward, and simple.

/pdf/solving-singular-two-point-boundary-value-problems-using-122bbe2o3b.pdf

Solving Singular Two-Point Boundary Value Problems Using Continuous Genetic Algorithm

In this paper, the solution of inverse kinematics problem of robot manipulators using genetic algorithms (GA) is presen ted. Two versions of genetic algorithms are used which include th e conventional GA and the continuous GA. The inverse kinemat ics problem is formulated as an optimization problem based on the concep t of minimizing the accumulative path deviation in the absen ce of any obstacles in the workspace. Simulation results show that th e continuous GA outperforms the conventional GA from all asp ects. The superiority of the continuous GA is seen in that it will alway s provide smooth and faster solutions as compared with the co nventional GA.

Solution of Inverse Kinematics Problem using Genetic Algorithms

In this paper, an optimization algorithm is presented for solving systems of s ingular boundary value problems. In this technique, the system is formulated as an optimization problem by the direct minimization of the overall individual residual error subject to the given constraints boundary conditions, and is then solved u sing continuous genetic algorithm in the sense that each of the derivatives is replaced by an appropriate difference quotient appro ximation. Two numerical experiments are carried out to verify the mathematical results, and the theoretical statements for the solutions are su pported by the results of numerical experiments. Meanwhile, the statistical analysis is provided in order to capture the behavior of the solu tions and to discover the effect of system parameters on the convergence speed of the algorithm. The numerical results demonstrate that the algorithm is quite accurate and efficient for solving systems of singular boundary value problems.

/pdf/an-optimization-algorithm-for-solving-systems-of-singular-185wv6ex59.pdf

An Optimization Algorithm for Solving Systems of Singular Boundary Value Problems

SUMMARY Second-order, two-point boundary-value problems are encountered in many engineering applications including the study of beam deflections, heat flow, and various dynamic systems. Two classical numerical techniques are widely used in the engineering community for the solution of such problems; the shooting method and finite difference method. These methods are suited for linear problems. However, when solving the non-linear problems, these methods require some major modifications that include the use of some root-finding technique. Furthermore, they require the use of other basic numerical techniques in order to obtain the solution. In this paper, the author introduces a novel method based on continuous genetic algorithms for numerically approximating a solution to this problem. The new method has the following characteristics; first, it does not require any modification while switching from the linear to the non-linear case; as a result, it is of versatile nature. Second, this approach does not resort to more advanced mathematical tools and is thus easily accepted in the engineering application field. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. However, being a variant of the finite difference scheme with truncation error of the order O(h 2 ), the method provides solutions with moderate accuracy. Numerical examples presented in the paper illustrate the applicability and generality of the proposed method. Copyright 2004 John Wiley & Sons, Ltd.

Numerical solution of second-order, two-point boundary value problems using continuous genetic algorithms

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Numerical solution of second-order, two-point boundary value problems using continuous genetic algorithms