Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming

In this paper, stochastic techniques have been developed to solve the 2-dimensional Bratu equations with the help of feed-forward artificial neural networks, optimized with particle swarm optimization (PSO) and sequential quadratic programming (SQP) algorithms. A hybrid of the above two algorithms, referred to as the PSO-SQP method is also studied. The original 2-dimensional equations are solved by first transforming them into equivalent one-dimensional boundary value problems (BVPs). These are then modeled using neural networks. The optimization problem for training the weights of the network has been addressed using particle swarm techniques for global search, integrated with an SQP method for rapid local convergence. The methodology is evaluated by applying on three different test cases of BVPs for the Bratu equations. Monte Carlo simulations and extensive analyses are carried out to validate the accuracy, convergence and effectiveness of the schemes. A comparative study of proposed results is made with available exact solution, as well as, reported numerical results.