An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear Troesch's problem

In this article, a simple and efficient approach for the approximate solution of a nonlinear differential equation known as Troesch's problem is proposed. In this article, a mathematical model of the Troesch's problem is described which arises in confinement of plasma column by radiation pressure. An artificial neural network (ANN) technique with gradient descent and particle swarm optimization is used to obtain the numerical solution of the Troesch's problem. This method overcomes the difficulty arising in the solution of Troesch's problem in the literature for eigenvalues of higher magnitude. The results obtained by the ANN method have been compared with the analytical solutions as well as with some other existing numerical techniques. It is observed that our results are more approximate and solution is provided on continuous finite time interval unlike the other numerical techniques. The main advantage of the proposed approach is that once the network is trained, it allows evaluating the solution at any required number of points for higher magnitude of eigenvalues with less computing time and memory.