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.

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.

Modelling of linear dynamical systems is very important issue in science and engineering. The modelling process might be achieved by either the application of the governing laws describing the process or by using the input-output data sequence of the process. Most of the modelling algorithms reported in the literature focus on either determining the order or estimating the model parameters. In this paper, the authors present a new method for modelling. Given the input-output data sequence of the model in the absence of any information about the order, the correct order of the model as well as the correct parameters is determined simultaneously using genetic algorithm. The algorithm used in this paper has several advantages; first, it does not use complex mathematical procedures in detecting the order and the parameters; second, it can be used for low as well as high order systems; third, it can be applied to any linear dynamical system including the autoregressive, moving-average, and autoregressive moving-average models; fourth, it determines the order and the parameters in a simultaneous manner with a very high accuracy. Results presented in this paper show the potentiality, the generality, and the superiority of our method as compared with other well-known methods.

In this article, a new analytical method has been devised to solve higher-order initial value problems for ordinary differential equations. This method was implemented to construct a series solution for higher-order initial value problems in the form of a rapidly convergent series with easily computable components using symbolic computation software. The proposed method is based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial and reproduces the exact solution when the solution is polynomial. This technique is applied to a few test examples to illustrate the accuracy, efficiency, and applicability of the method. The results reveal that the method is very effective, straightforward, and simple.