Applications of the neuro-evolutionary approach to the parabolic type partial differential equations - Waseem; Asad Ullah; Emad A.A. Ismail; Fuad A. Awwad (Ain Shams Engineering Journal)
This work aims to investigate the Cuckoo search-active set algorithm (CS-ASA), which is based on the artificial neural network (ANN) approach to the nonlinear partial differential equations (PDEs). CS-ASA technique modifies the Cuckoo search best possible solution to the next stage. This best solution is considered from all the best-chosen sets of Cuckoo search. This technique is applied to 1D and higher-order nonlinear PDEs that work as benchmark problems (heat, forced heat, Burger's, and Allen-Cahn). The results are obtained in the form of graphs by applying the CS-ASA algorithm. The state variables are plotted for a single hidden layer to reduce the computational cost. For the overall performance of CS-ASA, the fitness function is plotted against the number of iterations. It is clear that the fitness function minimizes with higher iterations, and hence achieve an optimal solution. The error in each case is presented at the grid points. Also, the fitness function for the description of the objective function is plotted. For best possible output, CS-ASA has chosen weight functions and its range in each case study accordingly. The obtained results (L2 error) are presented in the form of a table and compared with PSO, PSO-BP, and PSO-BP-CD. It is clear that CS-ASA perform better with a minimum L2 error for each problem under consideration.