For the boundary value problem (BVP) of a second-order partial differential equation on a plane triangle area, we propose a new algorithm based on the Adomian decomposition method (ADM) combined with a segmented technique. In addition, we present a new theorem that ensures the convergence of the algorithm. By this algorithm, the model for the effect of regional recharge on the plane triangle groundwater flow region is solved, from which we obtain the segmented exact solution of the problem, which satisfies the governing equation and all of the specified boundary conditions. Then, by the algorithm combined with Taylor's formula, the heterogeneous aquifer model on the plane triangle groundwater flow region is considered, from which we obtain the segmented high-precision approximate solution of the problem.
To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing the Lie group expressions of the solution, the trial solution of ODEs is split into two parts. The first part is a solution of other ODEs with initial values of original IVP. This is easily solved using the Lie group and known symbolic or numerical methods without any network parameters (weights and biases). The second part consists of an FNN with adjustable parameters. This is trained using the error back propagation method by minimizing an error (loss) function and updating the parameters. The method significantly reduces the number of the trainable parameters and can more quickly and accurately learn the real solution, compared to the existing similar methods. The numerical method is applied to several cases, including physical oscillation problems. The results have been graphically represented, and some conclusions have been made.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.