Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms

•Reliable stochastic numerical treatment for the MHD Jeffery Hamel flows problems.•Formulation of an unsupervised NNs model for the Jeffery Hamel flow equations.•Design parameters of NNs model using memetic approach based on PSO and ASMs.•Proposed results are in good agreement with solution of numerical and analytical solvers.•The accuracy and convergence is validated through detailed statistical analysis.

In this paper a new stochastic technique for solving the nonlinear Jeffery–Hamel flow equations is presented, taking into consideration the magnetohydrodynamics (MHD) effects. A feed-forward artificial neural network (ANN) trained with particle swarm optimization (PSO) and the active-set method (ASM) is used for solving the problem. We first transform the original two-dimensional MHD Jeffery–Hamel problem into an equivalent third order boundary value problem (BVP). A mathematical model of the BVP is developed using a neural network formulation in an unsupervised manner. Optimal weights of the networks are learned with the PSO algorithm first, which is used as a tool for global search; the active-set method is employed for rapid local convergence in the second step. The designed scheme is evaluated on different cases of the problem by varying the angles of the walls, and the Reynolds and Hartmann numbers. Accuracy, convergence and effectiveness of the approach is established through extensive analyses based on a large number of independent runs. Comparative studies are carried out with numerical results of fully explicit Runge–Kutta method, as well as recently reported analytic solvers including variants of Adomian decomposition, Homotopy Perturbation, differential transform, Homotopy analysis and variational iteration methods to validate the correctness of the proposed scheme.